Extrinsic Geometric Flows

Andrews, Ben; Chow, Bennett; Guenther, Christine; Langford, Mat

doi:10.1090/gsm/206

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Extrinsic Geometric Flows

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Ben Andrews, The Australian National University, Canberra, Australia, Bennett Chow, University of California, San Diego, La Jolla, CA, Christine Guenther, Pacific University, Forest Grove, OR and Mat Langford, University of Tennessee, Knoxville, TN

Publication: Graduate Studies in Mathematics
Publication Year: 2020; Volume 206
ISBNs: 978-1-4704-5596-5 (print); 978-1-4704-5686-3 (online)
DOI: https://doi.org/10.1090/gsm/206

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Mark A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations 25 (2006), no. 2, 205–246. MR 2188747, DOI 10.1007/s00526-005-0351-8
U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704
Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
Roberta Alessandroni and Carlo Sinestrari, Convexity estimates for a nonhomogeneous mean curvature flow, Math. Z. 266 (2010), no. 1, 65–82. MR 2670672, DOI 10.1007/s00209-009-0554-3
Roberta Alessandroni and Carlo Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 541–571. MR 2722655
Roberta Alessandroni and Carlo Sinestrari, Evolution of convex entire graphs by curvature flows, Geom. Flows 1 (2015), no. 1, 111–125. MR 3420371, DOI 10.1515/geofl-2015-0006
J. W. Alexander, A proof and extension of the Jordan-Brouwer separation theorem, Trans. Amer. Math. Soc. 23 (1922), no. 4, 333–349. MR 1501206, DOI 10.1090/S0002-9947-1922-1501206-6
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3–35 (Russian). MR 0003051
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Vestnik Leningrad. Univ. 11 (1956), no. 19, 5–17 (Russian). MR 0086338
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. II, Vestnik Leningrad. Univ. 12 (1957), no. 7, 15–44 (Russian, with English summary). MR 0102111
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. III, Vestnik Leningrad. Univ. 13 (1958), no. 7, 14–26 (Russian, with English summary). MR 0102112
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad. Univ. 13 (1958), no. 19, 5–8 (Russian, with English summary). MR 0102114
A. D. Aleksandrov and Ju. A. Volkov, Uniqueness theorems for surfaces in the large. IV, Vestnik Leningrad. Univ. 13 (1958), no. 13, 27–34 (Russian, with English summary). MR 0102113
Nicholas D. Alikakos and Alexandre Freire, The normalized mean curvature flow for a small bubble in a Riemannian manifold, J. Differential Geom. 64 (2003), no. 2, 247–303. MR 2029906
William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
William K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446. MR 397520, DOI 10.2307/1970934
Brian Allen, IMCF and the stability of the PMT and RPI under $L^2$ convergence, Ann. Henri Poincaré 19 (2018), no. 4, 1283–1306. MR 3775157, DOI 10.1007/s00023-017-0641-7
Dylan J. Altschuler, Steven J. Altschuler, Sigurd B. Angenent, and Lani F. Wu, The zoo of solitons for curve shortening in $\Bbb R^n$, Nonlinearity 26 (2013), no. 5, 1189–1226. MR 3043378, DOI 10.1088/0951-7715/26/5/1189
Steven J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991), no. 2, 491–514. MR 1131441
Steven Altschuler, Sigurd B. Angenent, and Yoshikazu Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal. 5 (1995), no. 3, 293–358. MR 1360824, DOI 10.1007/BF02921800
Steven J. Altschuler and Matthew A. Grayson, Shortening space curves and flow through singularities, J. Differential Geom. 35 (1992), no. 2, 283–298. MR 1158337
Steven J. Altschuler and Lang F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations 2 (1994), no. 1, 101–111. MR 1384396, DOI 10.1007/BF01234317
Koichi Anada, Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions, Calc. Var. Partial Differential Equations 12 (2001), no. 2, 109–116. MR 1821233, DOI 10.1007/PL00009908
Koichi Anada and Tetsuya Ishiwata, Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation, J. Differential Equations 262 (2017), no. 1, 181–271. MR 3567485, DOI 10.1016/j.jde.2016.09.023
Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, DOI 10.1007/BF01233434
Ben Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. MR 1385524, DOI 10.1007/BF01191340
Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. MR 1267897
Ben Andrews, Entropy estimates for evolving hypersurfaces, Comm. Anal. Geom. 2 (1994), no. 1, 53–64. MR 1312677, DOI 10.4310/CAG.1994.v2.n1.a3
Ben Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), no. 2, 179–197. MR 1296393, DOI 10.1007/BF02571941
Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207–230. MR 1424425
Ben Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 20 (1997), 1001–1031. MR 1486693, DOI 10.1155/S1073792897000640
Ben Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315–371. MR 1660843, DOI 10.1007/s005260050111
Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. MR 1714339, DOI 10.1007/s002220050344
Ben Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1–34. MR 1781612, DOI 10.2140/pjm.2000.195.1
B. Andrews, Positively curved surfaces in the three-sphere, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 221–230. MR 1957035
Ben Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), no. 2, 443–459. MR 1949167, DOI 10.1090/S0894-0347-02-00415-0
B. Andrews, Fully nonlinear parabolic equations in two space variables, arXiv:0402235v1, 2004.
Ben Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. MR 2339467, DOI 10.1515/CRELLE.2007.051
Ben Andrews, Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations 39 (2010), no. 3-4, 649–657. MR 2729317, DOI 10.1007/s00526-010-0329-z
Ben Andrews, Gradient and oscillation estimates and their applications in geometric PDE, Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence, RI, 2012, pp. 3–19. MR 2908056
Ben Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012), no. 3, 1413–1418. MR 2967056, DOI 10.2140/gt.2012.16.1413
Ben Andrews, Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, Int. Press, Somerville, MA, 2015, pp. 1–47. MR 3381494, DOI 10.4310/SDG.2014.v19.n1.a1
Ben Andrews and Charles Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010), no. 3, 357–395. MR 2739807
Ben Andrews and Paul Bryan, A comparison theorem for the isoperimetric profile under curve-shortening flow, Comm. Anal. Geom. 19 (2011), no. 3, 503–539. MR 2843240, DOI 10.4310/CAG.2011.v19.n3.a3
Ben Andrews and Paul Bryan, Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson’s theorem, J. Reine Angew. Math. 653 (2011), 179–187. MR 2794630, DOI 10.1515/CRELLE.2011.026
Ben Andrews and Xuzhong Chen, Surfaces moving by powers of Gauss curvature, Pure Appl. Math. Q. 8 (2012), no. 4, 825–834. MR 2959911, DOI 10.4310/PAMQ.2012.v8.n4.a1
Ben Andrews and Julie Clutterbuck, Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable, J. Differential Equations 246 (2009), no. 11, 4268–4283. MR 2517770, DOI 10.1016/j.jde.2009.01.024
Ben Andrews and Julie Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J. 58 (2009), no. 1, 351–380. MR 2504416, DOI 10.1512/iumj.2009.58.3756
Ben Andrews and Julie Clutterbuck, Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (2011), no. 3, 899–916. MR 2784332, DOI 10.1090/S0894-0347-2011-00699-1
Ben Andrews and Julie Clutterbuck, Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue, Anal. PDE 6 (2013), no. 5, 1013–1024. MR 3125548, DOI 10.2140/apde.2013.6.1013
Ben Andrews, Pengfei Guan, and Lei Ni, Flow by powers of the Gauss curvature, Adv. Math. 299 (2016), 174–201. MR 3519467, DOI 10.1016/j.aim.2016.05.008
Ben Andrews and Christopher Hopper, The Ricci flow in Riemannian geometry, Lecture Notes in Mathematics, vol. 2011, Springer, Heidelberg, 2011. A complete proof of the differentiable 1/4-pinching sphere theorem. MR 2760593, DOI 10.1007/978-3-642-16286-2
Ben Andrews and Mat Langford, Cylindrical estimates for hypersurfaces moving by convex curvature functions, Anal. PDE 7 (2014), no. 5, 1091–1107. MR 3265960, DOI 10.2140/apde.2014.7.1091
Ben Andrews and Mat Langford, Two-sided non-collapsing curvature flows, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 543–560. MR 3495438
Ben Andrews, Mat Langford, and James McCoy, Non-collapsing in fully non-linear curvature flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 1, 23–32. MR 3011290, DOI 10.1016/j.anihpc.2012.05.003
Ben Andrews, Mat Langford, and James McCoy, Convexity estimates for hypersurfaces moving by convex curvature functions, Anal. PDE 7 (2014), no. 2, 407–433. MR 3218814, DOI 10.2140/apde.2014.7.407
Ben Andrews, Mat Langford, and James McCoy, Convexity estimates for surfaces moving by curvature functions, J. Differential Geom. 99 (2015), no. 1, 47–75. MR 3299822
Ben Andrews and Haizhong Li, Embedded constant mean curvature tori in the three-sphere, J. Differential Geom. 99 (2015), no. 2, 169–189. MR 3302037
Ben Andrews, Haizhong Li, and Yong Wei, $\scr F$-stability for self-shrinking solutions to mean curvature flow, Asian J. Math. 18 (2014), no. 5, 757–777. MR 3287002, DOI 10.4310/AJM.2014.v18.n5.a1
Ben Andrews and James McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3427–3447. MR 2901219, DOI 10.1090/S0002-9947-2012-05375-X
Ben Andrews, James McCoy, and Yu Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 611–665. MR 3070558, DOI 10.1007/s00526-012-0530-3
Sigurd Angenent, Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow, Netw. Heterog. Media 8 (2013), no. 1, 1–8. MR 3043925, DOI 10.3934/nhm.2013.8.1
Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum, Unique asymptotics of ancient convex mean curvature flow solutions, J. Differential Geom. 111 (2019), no. 3, 381–455. MR 3934596, DOI 10.4310/jdg/1552442605
Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
Sigurd Angenent, Parabolic equations for curves on surfaces. I. Curves with $p$-integrable curvature, Ann. of Math. (2) 132 (1990), no. 3, 451–483. MR 1078266, DOI 10.2307/1971426
Sigurd Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), no. 3, 601–633. MR 1100205
Sigurd Angenent, Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math. (2) 133 (1991), no. 1, 171–215. MR 1087347, DOI 10.2307/2944327
Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38. MR 1167827
Sigurd B. Angenent, Curve shortening and the topology of closed geodesics on surfaces, Ann. of Math. (2) 162 (2005), no. 3, 1187–1241. MR 2179729, DOI 10.4007/annals.2005.162.1187
Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum, Unique asymptotics of ancient convex mean curvature flow solutions, J. Differential Geom. 111 (2019), no. 3, 381–455. MR 3934596, DOI 10.4310/jdg/1552442605
S. Angenent, T. Ilmanen, and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in $\mathbf R^3$, Comm. Partial Differential Equations 20 (1995), no. 11-12, 1937–1958. MR 1361726, DOI 10.1080/03605309508821158
S. B. Angenent, T. Ilmanen, and J. J. L. Velázquez, Fattening from smooth initial data in mean curvature flow, preprint.
Sigurd Angenent, Guillermo Sapiro, and Allen Tannenbaum, On the affine heat equation for non-convex curves, J. Amer. Math. Soc. 11 (1998), no. 3, 601–634. MR 1491538, DOI 10.1090/S0894-0347-98-00262-8
S. B. Angenent and J. J. L. Velázquez, Asymptotic shape of cusp singularities in curve shortening, Duke Math. J. 77 (1995), no. 1, 71–110. MR 1317628, DOI 10.1215/S0012-7094-95-07704-7
S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math. 482 (1997), 15–66. MR 1427656, DOI 10.1515/crll.1997.482.15
S. B. Angenent and Q. You, Ancient solutions to curve shortening with finite total curvature, arXiv:1803.01399v1, 2018.
Maria Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces, Comment. Math. Helv. 72 (1997), no. 1, 52–66. MR 1456315, DOI 10.1007/PL00000366
Maria Athanassenas, Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow, Calc. Var. Partial Differential Equations 17 (2003), no. 1, 1–16. MR 1979113, DOI 10.1007/s00526-002-0098-4
Maria Athanassenas and Sevvandi Kandanaarachchi, Convergence of axially symmetric volume-preserving mean curvature flow, Pacific J. Math. 259 (2012), no. 1, 41–54. MR 2988482, DOI 10.2140/pjm.2012.259.41
Thomas Kwok-Keung Au, On the saddle point property of Abresch-Langer curves under the curve shortening flow, Comm. Anal. Geom. 18 (2010), no. 1, 1–21. MR 2660456, DOI 10.4310/CAG.2010.v18.n1.a1
C. Baker, The mean curvature flow of submanifolds of high codimension, Australian National University, 2011. Thesis (Ph.D.)–Australian National University.
Charles Baker and Huy The Nguyen, Codimension two surfaces pinched by normal curvature evolving by mean curvature flow, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 6, 1599–1610. MR 3712012, DOI 10.1016/j.anihpc.2016.10.010
Pietro Baldi, Emanuele Haus, and Carlo Mantegazza, Networks self-similarly moving by curvature with two triple junctions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 2, 323–338. MR 3649351, DOI 10.4171/RLM/765
Pietro Baldi, Emanuele Haus, and Carlo Mantegazza, On the classification of networks self-similarly moving by curvature, Geom. Flows 2 (2017), no. 1, 125–137. MR 3745455, DOI 10.1515/geofl-2017-0006
Pietro Baldi, Emanuele Haus, and Carlo Mantegazza, Non-existence of $theta$-shaped self-similarly shrinking networks moving by curvature, Comm. Partial Differential Equations 43 (2018), no. 3, 403–427. MR 3804202, DOI 10.1080/03605302.2018.1446162
J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), no. 3, 699–728. MR 757959, DOI 10.1215/S0012-7094-84-05134-2
Jacob Bernstein and Thomas Mettler, Characterizing classical minimal surfaces via the entropy differential, J. Geom. Anal. 27 (2017), no. 3, 2235–2268. MR 3667429, DOI 10.1007/s12220-017-9759-6
Jacob Bernstein and Lu Wang, A topological property of asymptotically conical self-shrinkers of small entropy, Duke Math. J. 166 (2017), no. 3, 403–435. MR 3606722, DOI 10.1215/00127094-3715082
J. Bernstein and L. Wang, An integer degree for asymptotically conical self-expanders, arXiv:1807.06494v1, 2018.
Maria Chiara Bertini and Carlo Sinestrari, Volume preserving flow by powers of symmetric polynomials in the principal curvatures, Math. Z. 289 (2018), no. 3-4, 1219–1236. MR 3830246, DOI 10.1007/s00209-017-1995-8
Maria Chiara Bertini and Carlo Sinestrari, Volume-preserving nonhomogeneous mean curvature flow of convex hypersurfaces, Ann. Mat. Pura Appl. (4) 197 (2018), no. 4, 1295–1309. MR 3829571, DOI 10.1007/s10231-018-0725-0
W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie II, 1 ed., vol. 7 of Grundlehren der mathematischen Wissenschaften. Julius Springer Verlag, 1923.
Christoph Böhm and Burkhard Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2) 167 (2008), no. 3, 1079–1097. MR 2415394, DOI 10.4007/annals.2008.167.1079
William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, 2nd ed., Pure and Applied Mathematics, vol. 120, Academic Press, Inc., Orlando, FL, 1986. MR 861409
Theodora Bourni, Allard-type boundary regularity for $C^{1,\alpha }$ boundaries, Adv. Calc. Var. 9 (2016), no. 2, 143–161. MR 3483600, DOI 10.1515/acv-2014-0032
Theodora Bourni and Mat Langford, Type-II singularities of two-convex immersed mean curvature flow, Geom. Flows 2 (2017), no. 1, 1–17. Previously published in 2 (2016). MR 3562950, DOI 10.1515/geofl-2016-0001
T. Bourni, M. Langford, and G. Tinaglia, A collapsing ancient solution of mean curvature flow in $\mathbb {R}^3$, arXiv:1705.06981v2, 2017.
T. Bourni, M. Langford, and G. Tinaglia, On the existence of translating solutions of mean curvature flow in slab regions, to appear in Anal. PDE, arXiv:1805.05173v3, 2018.
T. Bourni, M. Langford, and G. Tinaglia Convex ancient solutions to curve shortening flow, arXiv:1903.02022v1, 2019.
T. Bourni, M. Langford, and G. Tinaglia, Convex ancient solutions to mean curvature flow, arXiv:1907.03932v1, 2019.
Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 0485012
Hubert L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no. 2, 177–267. MR 1908823
Hubert L. Bray, On the positive mass, Penrose, and ZAS inequalities in general dimension, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 1–27. MR 2906919
Simon Brendle, Embedded minimal tori in $S^3$ and the Lawson conjecture, Acta Math. 211 (2013), no. 2, 177–190. MR 3143888, DOI 10.1007/s11511-013-0101-2
Simon Brendle, Embedded Weingarten tori in $S^3$, Adv. Math. 257 (2014), 462–475. MR 3187655, DOI 10.1016/j.aim.2014.02.025
Simon Brendle, Two-point functions and their applications in geometry, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 581–596. MR 3237760, DOI 10.1090/S0273-0979-2014-01461-2
Simon Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math. 202 (2015), no. 1, 217–237. MR 3402798, DOI 10.1007/s00222-014-0570-8
Simon Brendle, Embedded self-similar shrinkers of genus 0, Ann. of Math. (2) 183 (2016), no. 2, 715–728. MR 3450486, DOI 10.4007/annals.2016.183.2.6
Simon Brendle, An inscribed radius estimate for mean curvature flow in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1447–1472. MR 3616339
S. Brendle, Ancient solutions to the Ricci flow in dimension $3$, arXiv:1811.02559v2, 2018.
S. Brendle and K. Choi, Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions, arXiv:1804.00018v2, 2018.
Simon Brendle and Kyeongsu Choi, Uniqueness of convex ancient solutions to mean curvature flow in $\Bbb R^3$, Invent. Math. 217 (2019), no. 1, 35–76. MR 3958790, DOI 10.1007/s00222-019-00859-4
Simon Brendle, Kyeongsu Choi, and Panagiota Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math. 219 (2017), no. 1, 1–16. MR 3765656, DOI 10.4310/ACTA.2017.v219.n1.a1
Simon Brendle and Gerhard Huisken, Mean curvature flow with surgery of mean convex surfaces in $\Bbb R^3$, Invent. Math. 203 (2016), no. 2, 615–654. MR 3455158, DOI 10.1007/s00222-015-0599-3
Simon Brendle and Gerhard Huisken, A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds, Invent. Math. 210 (2017), no. 2, 559–613. MR 3714512, DOI 10.1007/s00222-017-0736-2
Simon Brendle and Pei-Ken Hung, A sharp inscribed radius estimate for fully nonlinear flows, Amer. J. Math. 141 (2019), no. 1, 41–53. MR 3904766, DOI 10.1353/ajm.2019.0001
Simon Brendle and Richard Schoen, Manifolds with $1/4$-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307. MR 2449060, DOI 10.1090/S0894-0347-08-00613-9
Patrick Breuning, Immersions with bounded second fundamental form, J. Geom. Anal. 25 (2015), no. 2, 1344–1386. MR 3319975, DOI 10.1007/s12220-014-9472-7
Philip Broadbridge and Peter Vassiliou, The role of symmetry and separation in surface evolution and curve shortening, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 052, 19. MR 2804584, DOI 10.3842/SIGMA.2011.052
Lia Bronsard and Fernando Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal. 124 (1993), no. 4, 355–379. MR 1240580, DOI 10.1007/BF00375607
Morton Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74–76. MR 117695, DOI 10.1090/S0002-9904-1960-10400-4
P. Bryan and M. Ivaki, Harnack estimate for mean curvature flow on the sphere, arXiv:1508.02821v4, 2015.
P. Bryan, M. Ivaki, and J. Scheuer, On the classification of ancient solutions to curvature flows on the sphere, arXiv:1604.01694v2, 2016.
P. Bryan, M. Ivaki, and J. Scheuer, Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds, arXiv:1703.07493v1, 2017.
Paul Bryan, Mohammad N. Ivaki, and Julian Scheuer, Harnack inequalities for evolving hypersurfaces on the sphere, Comm. Anal. Geom. 26 (2018), no. 5, 1047–1077. MR 3900479, DOI 10.4310/CAG.2018.v26.n5.a2
Paul Bryan and Janelle Louie, Classification of convex ancient solutions to curve shortening flow on the sphere, J. Geom. Anal. 26 (2016), no. 2, 858–872. MR 3472819, DOI 10.1007/s12220-015-9574-x
John A. Buckland, Mean curvature flow with free boundary on smooth hypersurfaces, J. Reine Angew. Math. 586 (2005), 71–90. MR 2180601, DOI 10.1515/crll.2005.2005.586.71
John A. Buckland, Short-time existence of solutions to the cross curvature flow on 3-manifolds, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1803–1807. MR 2207496, DOI 10.1090/S0002-9939-05-08204-3
Esther Cabezas-Rivas and Vicente Miquel, Volume preserving mean curvature flow in the hyperbolic space, Indiana Univ. Math. J. 56 (2007), no. 5, 2061–2086. MR 2359723, DOI 10.1512/iumj.2007.56.3060
Esther Cabezas-Rivas and Vicente Miquel, Volume-preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space, Math. Z. 261 (2009), no. 3, 489–510. MR 2471083, DOI 10.1007/s00209-008-0333-6
Esther Cabezas-Rivas and Vicente Miquel, Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants, Calc. Var. Partial Differential Equations 43 (2012), no. 1-2, 185–210. MR 2886115, DOI 10.1007/s00526-011-0408-9
Esther Cabezas-Rivas and Vicente Miquel, Non-preserved curvature conditions under constrained mean curvature flows, Differential Geom. Appl. 49 (2016), 287–300. MR 3573835, DOI 10.1016/j.difgeo.2016.08.006
Esther Cabezas-Rivas and Carlo Sinestrari, Volume-preserving flow by powers of the $m$th mean curvature, Calc. Var. Partial Differential Equations 38 (2010), no. 3-4, 441–469. MR 2647128, DOI 10.1007/s00526-009-0294-6
E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56. MR 92069
John Rozier Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. With a foreword by Felix E. Browder. MR 747979, DOI 10.1017/CBO9781139086967
Marcos P. Cavalcante and José M. Espinar, Halfspace type theorems for self-shrinkers, Bull. Lond. Math. Soc. 48 (2016), no. 2, 242–250. MR 3483061, DOI 10.1112/blms/bdv099
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, 2008. Revised reprint of the 1975 original. MR 2394158, DOI 10.1090/chel/365
Bing-Long Chen and Xi-Ping Zhu, Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math. 140 (2000), no. 2, 423–452. MR 1757002, DOI 10.1007/s002220000061
Jingyi Chen and Jiayu Li, Singularity of mean curvature flow of Lagrangian submanifolds, Invent. Math. 156 (2004), no. 1, 25–51. MR 2047657, DOI 10.1007/s00222-003-0332-5
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
F. Chini and N. M. Møller, Ancient mean curvature flows and their spacetime tracks, arXiv:1901.05481v2, 2019.
F Chini and N. M. Møller, Bi-halfspace and convex hull theorems for translating solitons, arXiv:1809.01069v2, 2019.
O. Chodosh and F. Schulze, Uniqueness of asymptotically conical tangent flows, arXiv:1901.06369v1, 2019.
K. Choi and P. Daskalopoulos, Uniqueness of closed self-similar solutions to the Gauss curvature flow, arXiv:1609.05487v1, 2016.
Kyeongsu Choi, Panagiota Daskalopoulos, Lami Kim, and Ki-Ahm Lee, The evolution of complete non-compact graphs by powers of Gauss curvature, J. Reine Angew. Math. 757 (2019), 131–158. MR 4036572, DOI 10.1515/crelle-2017-0032
K. Choi, P. Daskalopoulos, and K.-A. Lee, Translating solutions to the Gauss curvature flow with flat sides, arXiv:1610.07206v2, 2016.
K. Choi, R. Haslhofer, and O. Hershkovits, Ancient low entropy flows, mean convex neighborhoods, and uniqueness, arXiv:1810.08467v1, 2018.
K. Choi and C. Mantoulidis, Ancient gradient flows of elliptic functionals and Morse index, arXiv:1902.07697v2, 2019.
David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1–15. MR 1302814
Kai-Seng Chou and Xu-Jia Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 6, 733–751. MR 1804653, DOI 10.1016/S0294-1449(00)00053-6
Kai-Seng Chou and Xi-Ping Zhu, A convexity theorem for a class of anisotropic flows of plane curves, Indiana Univ. Math. J. 48 (1999), no. 1, 139–154. MR 1722196, DOI 10.1512/iumj.1999.48.1273
Kai-Seng Chou and Xi-Ping Zhu, The curve shortening problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888641, DOI 10.1201/9781420035704
Bennett Chow, Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. MR 826427
Bennett Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63–82. MR 862712, DOI 10.1007/BF01389153
Bennett Chow, On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469–483. MR 1100812, DOI 10.1002/cpa.3160440405
Bennett Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom. 5 (1997), no. 2, 389–409. MR 1483984, DOI 10.4310/CAG.1997.v5.n2.a5
Bennett Chow and Sun-Chin Chu, A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Lett. 2 (1995), no. 6, 701–718. MR 1362964, DOI 10.4310/MRL.1995.v2.n6.a4
Bennett Chow and Sun-Chin Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. Anal. 11 (2001), no. 2, 219–231. MR 1856176, DOI 10.1007/BF02921963
B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications. Parts I–IV, Mathematical Surveys and Monographs, vols. 135, 144, 163, 206, American Mathematical Society, Providence, RI, 2007, 2008, 2010, 2015.
Bennett Chow and Robert Gulliver, Aleksandrov reflection and nonlinear evolution equations. I. The $n$-sphere and $n$-ball, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 249–264. MR 1386736, DOI 10.1007/BF01254346
Bennett Chow and Robert Gulliver, Aleksandrov reflection and geometric evolution of hypersurfaces, Comm. Anal. Geom. 9 (2001), no. 2, 261–280. MR 1846204, DOI 10.4310/CAG.2001.v9.n2.a2
Bennett Chow and Richard S. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turkish J. Math. 28 (2004), no. 1, 1–10. MR 2055396
Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. MR 2061425, DOI 10.1090/surv/110
Bennett Chow, Lii-Perng Liou, and Dong-Ho Tsai, Expansion of embedded curves with turning angle greater than $-\pi$, Invent. Math. 123 (1996), no. 3, 415–429. MR 1383955, DOI 10.1007/s002220050034
Bennett Chow, Lii-Perng Liou, and Dong-Ho Tsai, On the nonlinear parabolic equation $\partial _tu=F(\Delta u+nu)$ on $S^n$, Comm. Anal. Geom. 4 (1996), no. 3, 415–434. MR 1415750, DOI 10.4310/CAG.1996.v4.n3.a3
Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
Bennett Chow and Dong-Ho Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), no. 2, 312–330. MR 1425578
Bennett Chow and Dong-Ho Tsai, Expansion of convex hypersurfaces by nonhomogeneous functions of curvature, Asian J. Math. 1 (1997), no. 4, 769–784. MR 1621575, DOI 10.4310/AJM.1997.v1.n4.a7
Bennett Chow and Dong-Ho Tsai, Nonhomogeneous Gauss curvature flows, Indiana Univ. Math. J. 47 (1998), no. 3, 965–994. MR 1665729, DOI 10.1512/iumj.1998.47.1546
J. Clutterbuck, Parabolic equations with continuous initial data, PhD thesis, The Australian National University, 2005.
J. Clutterbuck and O. C. Schnürer, Stability of mean convex cones under mean curvature flow, Math. Z. 267 (2011), no. 3-4, 535–547. MR 2776047, DOI 10.1007/s00209-009-0634-4
Julie Clutterbuck, Oliver C. Schnürer, and Felix Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations 29 (2007), no. 3, 281–293. MR 2321890, DOI 10.1007/s00526-006-0033-1
Tobias Holck Colding, Tom Ilmanen, and William P. Minicozzi II, Rigidity of generic singularities of mean curvature flow, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 363–382. MR 3349836, DOI 10.1007/s10240-015-0071-3
Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi II, and Brian White, The round sphere minimizes entropy among closed self-shrinkers, J. Differential Geom. 95 (2013), no. 1, 53–69. MR 3128979
Tobias H. Colding and William P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics, vol. 4, New York University, Courant Institute of Mathematical Sciences, New York, 1999. MR 1683966
Tobias H. Colding and William P. Minicozzi II, Sharp estimates for mean curvature flow of graphs, J. Reine Angew. Math. 574 (2004), 187–195. MR 2099114, DOI 10.1515/crll.2004.069
Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
Tobias H. Colding and William P. Minicozzi II, Smooth compactness of self-shrinkers, Comment. Math. Helv. 87 (2012), no. 2, 463–475. MR 2914856, DOI 10.4171/CMH/260
Tobias Holck Colding and William P. Minicozzi II, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. (2) 182 (2015), no. 1, 221–285. MR 3374960, DOI 10.4007/annals.2015.182.1.5
Tobias Holck Colding and William P. Minicozzi II, Differentiability of the arrival time, Comm. Pure Appl. Math. 69 (2016), no. 12, 2349–2363. MR 3570481, DOI 10.1002/cpa.21635
Tobias Holck Colding and William P. Minicozzi II, Regularity of the level set flow, Comm. Pure Appl. Math. 71 (2018), no. 4, 814–824. MR 3772402, DOI 10.1002/cpa.21703
T. H. Colding and W. P. Minicozzi, II, Entropy and codimension bounds for generic singularities, arXiv:1906.07609v2, 2019.
T. H. Colding and W. P. Minicozzi, II, Optimal bounds for ancient caloric functions, arXiv:1902.01736v1, 2019.
Tobias Holck Colding, William P. Minicozzi II, and Erik Kjær Pedersen, Mean curvature flow, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 2, 297–333. MR 3312634, DOI 10.1090/S0273-0979-2015-01468-0
Andrew A. Cooper, A characterization of the singular time of the mean curvature flow, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2933–2942. MR 2801634, DOI 10.1090/S0002-9939-2010-10714-1
Andrew Allen Cooper, Mean curvature flow in higher codimension, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Michigan State University. MR 2941847
P. Daskalopoulos and R. Hamilton, The free boundary in the Gauss curvature flow with flat sides, J. Reine Angew. Math. 510 (1999), 187–227. MR 1696096, DOI 10.1515/crll.1999.046
Panagiota Daskalopoulos, Richard Hamilton, and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom. 84 (2010), no. 3, 455–464. MR 2669361
P. Daskalopoulos and G. Huisken, Inverse mean curvature evolution of entire graphs, arXiv:1709.06665v1, 2017.
P. Daskalopoulos and Ki-Ahm Lee, Worn stones with flat sides all time regularity of the interface, Invent. Math. 156 (2004), no. 3, 445–493. MR 2061326, DOI 10.1007/s00222-003-0328-1
Juan Dávila, Manuel del Pino, and Xuan Hien Nguyen, Finite topology self-translating surfaces for the mean curvature flow in $\Bbb {R}^3$, Adv. Math. 320 (2017), 674–729. MR 3709119, DOI 10.1016/j.aim.2017.09.014
Q. Ding, Minimal cones and self-expanding solutions for mean curvature flows, arXiv:1503.02612v3, 2015.
F. Dittberner, Curve flows with a global forcing term, arXiv:1809.08643v1, 2018.
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
Gregory Drugan, Self-shrinking Solutions to Mean Curvature Flow, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University of Washington. MR 3271860
Gregory Drugan, An immersed $S^2$ self-shrinker, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3139–3159. MR 3314804, DOI 10.1090/S0002-9947-2014-06051-0
Gregory Drugan and Stephen J. Kleene, Immersed self-shrinkers, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7213–7250. MR 3683108, DOI 10.1090/tran/6907
Gregory Drugan, Hojoo Lee, and Xuan Hien Nguyen, A survey of closed self-shrinkers with symmetry, Results Math. 73 (2018), no. 1, Paper No. 32, 32. MR 3763107, DOI 10.1007/s00025-018-0763-3
Gregory Drugan and Xuan Hien Nguyen, Mean curvature flow of entire graphs evolving away from the heat flow, Proc. Amer. Math. Soc. 145 (2017), no. 2, 861–869. MR 3577885, DOI 10.1090/proc/13238
Klaus Ecker, A local monotonicity formula for mean curvature flow, Ann. of Math. (2) 154 (2001), no. 2, 503–525. MR 1865979, DOI 10.2307/3062105
Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995, DOI 10.1007/978-0-8176-8210-1
Klaus Ecker, Local monotonicity formulas for some nonlinear diffusion equations, Calc. Var. Partial Differential Equations 23 (2005), no. 1, 67–81. MR 2133662, DOI 10.1007/s00526-004-0290-9
Klaus Ecker and Gerhard Huisken, Immersed hypersurfaces with constant Weingarten curvature, Math. Ann. 283 (1989), no. 2, 329–332. MR 980601, DOI 10.1007/BF01446438
Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
Klaus Ecker and Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. MR 1117150, DOI 10.1007/BF01232278
Klaus Ecker, Dan Knopf, Lei Ni, and Peter Topping, Local monotonicity and mean value formulas for evolving Riemannian manifolds, J. Reine Angew. Math. 616 (2008), 89–130. MR 2369488, DOI 10.1515/CRELLE.2008.019
Nick Edelen, Convexity estimates for mean curvature flow with free boundary, Adv. Math. 294 (2016), 1–36. MR 3479560, DOI 10.1016/j.aim.2016.02.026
C. L. Epstein and M. I. Weinstein, A stable manifold theorem for the curve shortening equation, Comm. Pure Appl. Math. 40 (1987), no. 1, 119–139. MR 865360, DOI 10.1002/cpa.3160400106
J.-H. Eschenburg, Local convexity and nonnegative curvature—Gromov’s proof of the sphere theorem, Invent. Math. 84 (1986), no. 3, 507–522. MR 837525, DOI 10.1007/BF01388744
Joachim Escher and Gieri Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2789–2796. MR 1485470, DOI 10.1090/S0002-9939-98-04727-3
Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Lawrence Christopher Evans, A strong maximum principle for parabolic systems in a convex set with arbitrary boundary, Proc. Amer. Math. Soc. 138 (2010), no. 9, 3179–3185. MR 2653943, DOI 10.1090/S0002-9939-2010-10495-1
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. MR 3409135
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal. 2 (1992), no. 2, 121–150. MR 1151756, DOI 10.1007/BF02921385
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI 10.1112/S0025579300005714
Frederick Tsz-Ho Fong and Peter McGrath, Rotational symmetry of asymptotically conical mean curvature flow self-expanders, Comm. Anal. Geom. 27 (2019), no. 3, 599–618. MR 4003004, DOI 10.4310/CAG.2019.v27.n3.a3
Alexandre Freire, Mean curvature motion of graphs with constant contact angle at a free boundary, Anal. PDE 3 (2010), no. 4, 359–407. MR 2718258, DOI 10.2140/apde.2010.3.359
Alexandre Freire, Mean curvature motion of triple junctions of graphs in two dimensions, Comm. Partial Differential Equations 35 (2010), no. 2, 302–327. MR 2748626, DOI 10.1080/03605300903419775
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Charles Frohman and William H. Meeks III, The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in $\mathbf R^3$, J. Amer. Math. Soc. 10 (1997), no. 3, 495–512. MR 1443545, DOI 10.1090/S0894-0347-97-00215-4
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Michael E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225–1229. MR 726325, DOI 10.1215/S0012-7094-83-05052-4
M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), no. 2, 357–364. MR 742856, DOI 10.1007/BF01388602
Michael E. Gage, Curve shortening on surfaces, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 2, 229–256. MR 1046497
Michael E. Gage, Deforming curves on convex surfaces to simple closed geodesics, Indiana Univ. Math. J. 39 (1990), no. 4, 1037–1059. MR 1087184, DOI 10.1512/iumj.1990.39.39049
E. Gama and F. Martín Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb {R}^{n+1}$, arXiv:1802.08468v3, 2018.
Shanze Gao, Haizhong Li, and Hui Ma, Uniqueness of closed self-similar solutions to $\sigma _k^{\alpha }$-curvature flow, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 5, Paper No. 45, 26. MR 3845754, DOI 10.1007/s00030-018-0535-5
Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886, DOI 10.1017/CBO9781107341029
Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314. MR 1064876
Claus Gerhardt, Curvature problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR 2284727
Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. MR 2879249
Claus Gerhardt, Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 471–489. MR 3148124, DOI 10.1007/s00526-012-0589-x
Claus Gerhardt, Curvature flows in the sphere, J. Differential Geom. 100 (2015), no. 2, 301–347. MR 3343834
R. Geroch, Energy extraction, Ann. New York Acad. Sci. 224 (1973), 108–117.
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, DOI 10.1002/cpa.3160380304
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Georges Glaeser, Fonctions composées différentiables, Ann. of Math. (2) 77 (1963), 193–209 (French). MR 143058, DOI 10.2307/1970204
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. MR 979601, DOI 10.2307/1971486
R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), no. 1, 119–141. MR 917868
K. Groh, M. Schwarz, K. Smoczyk, and K. Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007), no. 2, 295–327. MR 2324804, DOI 10.1007/s00209-007-0126-3
Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
Bo Guan, Mean curvature motion of nonparametric hypersurfaces with contact angle condition, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994) A K Peters, Wellesley, MA, 1996, pp. 47–56. MR 1417947
Pengfei Guan and Junfang Li, The quermassintegral inequalities for $k$-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. MR 2522433, DOI 10.1016/j.aim.2009.03.005
Pengfei Guan and Lei Ni, Entropy and a convergence theorem for Gauss curvature flow in high dimension, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3735–3761. MR 3730513, DOI 10.4171/JEMS/752
Victor Guillemin and Alan Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original. MR 2680546, DOI 10.1090/chel/370
Hongxin Guo, Robert Philipowski, and Anton Thalmaier, A note on Chow’s entropy functional for the Gauss curvature flow, C. R. Math. Acad. Sci. Paris 351 (2013), no. 21-22, 833–835 (English, with English and French summaries). MR 3128971, DOI 10.1016/j.crma.2013.10.003
J. Hadamard, Sur certaines propriétés des trajectoires en dynamique, J. Math. Pures Appl. 3 (1897), 331–388.
Hoeskuldur P. Halldorsson, Self-similar solutions to the curve shortening flow, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5285–5309. MR 2931330, DOI 10.1090/S0002-9947-2012-05632-7
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
Richard S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993), no. 1, 1–11. MR 1231700
Richard S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225–243. MR 1198607
Richard S. Hamilton, Monotonicity formulas for parabolic flows on manifolds, Comm. Anal. Geom. 1 (1993), no. 1, 127–137. MR 1230277, DOI 10.4310/CAG.1993.v1.n1.a7
Richard S. Hamilton, Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2 (1994), no. 1, 167–172. MR 1312684, DOI 10.4310/CAG.1994.v2.n1.a10
Richard S. Hamilton, Remarks on the entropy and Harnack estimates for the Gauss curvature flow, Comm. Anal. Geom. 2 (1994), no. 1, 155–165. MR 1312683, DOI 10.4310/CAG.1994.v2.n1.a9
Richard S. Hamilton, Worn stones with flat sides, A tribute to Ilya Bakelman (College Station, TX, 1993) Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 69–78. MR 1423370
Richard S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), no. 3, 545–572. MR 1333936, DOI 10.2307/2375080
Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
Richard S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215–226. MR 1316556
Richard S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Modern methods in complex analysis (Princeton, NJ, 1992) Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 201–222. MR 1369140, DOI 10.1016/1053-8127(94)00130-3
Richard S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), no. 1, 1–92. MR 1456308, DOI 10.4310/CAG.1997.v5.n1.a1
Qing Han, Deforming convex hypersurfaces by curvature functions, Analysis 17 (1997), no. 2-3, 113–127. MR 1486359, DOI 10.1524/anly.1997.17.23.113
Robert Haslhofer, Uniqueness of the bowl soliton, Geom. Topol. 19 (2015), no. 4, 2393–2406. MR 3375531, DOI 10.2140/gt.2015.19.2393
Robert Haslhofer and Or Hershkovits, Ancient solutions of the mean curvature flow, Comm. Anal. Geom. 24 (2016), no. 3, 593–604. MR 3521319, DOI 10.4310/CAG.2016.v24.n3.a6
Robert Haslhofer and Bruce Kleiner, On Brendle’s estimate for the inscribed radius under mean curvature flow, Int. Math. Res. Not. IMRN 15 (2015), 6558–6561. MR 3384488, DOI 10.1093/imrn/rnu139
Robert Haslhofer and Bruce Kleiner, Mean curvature flow of mean convex hypersurfaces, Comm. Pure Appl. Math. 70 (2017), no. 3, 511–546. MR 3602529, DOI 10.1002/cpa.21650
Robert Haslhofer and Bruce Kleiner, Mean curvature flow with surgery, Duke Math. J. 166 (2017), no. 9, 1591–1626. MR 3662439, DOI 10.1215/00127094-0000008X
A. Hatcher, Notes on basic 3-manifold topology. http://pi.math.cornell.edu/$\sim$hatcher/3M/ 3M.pdf, 2000.
Laurent Hauswirth and Frank Pacard, Higher genus Riemann minimal surfaces, Invent. Math. 169 (2007), no. 3, 569–620. MR 2336041, DOI 10.1007/s00222-007-0056-z
John Head, On the mean curvature evolution of two-convex hypersurfaces, J. Differential Geom. 94 (2013), no. 2, 241–266. MR 3080482
M. Heidusch, Zur Regularität des Inversen Mittleren Krümmungsflusses, PhD thesis, Eberhard-Karls-Universität Tübingen, 2001.
Sebastian Helmensdorfer, A model for the behavior of fluid droplets based on mean curvature flow, SIAM J. Math. Anal. 44 (2012), no. 3, 1359–1371. MR 2982716, DOI 10.1137/110824905
S. Helmensdorfer and P. Topping, The geometry of differential Harnack estimates, Actes du séminaire de Théorie spectrale et géométrie 30 (2011-2012), 77–89.
O. Hershkovits, Translators asymptotic to cylinders, arXiv:1805.10553v1, 2018.
Noel J. Hicks, Notes on differential geometry, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
D. Hoffman, T. Ilmanen, F. Martín, and B. White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 117, 29. MR 3962912, DOI 10.1007/s00526-019-1560-x
D. Hoffman, F. Martín, and B. White, Scherk-like translators for mean curvature flow, arXiv:1903.04617v3, 2018.
D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. MR 1062966, DOI 10.1007/BF01231506
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388. MR 3573332, DOI 10.1007/s11511-016-0140-6
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
Gerhard Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. MR 837523, DOI 10.1007/BF01388742
Gerhard Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math. 382 (1987), 35–48. MR 921165, DOI 10.1515/crll.1987.382.35
Gerhard Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations 77 (1989), no. 2, 369–378. MR 983300, DOI 10.1016/0022-0396(89)90149-6
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR 1216584, DOI 10.1090/pspum/054.1/1216584
Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127–133. MR 1656553, DOI 10.4310/AJM.1998.v2.n1.a2
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Gerhard Huisken and Alexander Polden, Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems (Cetraro, 1996) Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, pp. 45–84. MR 1731639, DOI 10.1007/BFb0092669
Gerhard Huisken and Carlo Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45–70. MR 1719551, DOI 10.1007/BF02392946
Gerhard Huisken and Carlo Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14. MR 1666878, DOI 10.1007/s005260050113
Gerhard Huisken and Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009), no. 1, 137–221. MR 2461428, DOI 10.1007/s00222-008-0148-4
Gerhard Huisken and Carlo Sinestrari, Convex ancient solutions of the mean curvature flow, J. Differential Geom. 101 (2015), no. 2, 267–287. MR 3399098
John Hutchinson, Poincaré-Sobolev and related inequalities for submanifolds of $\textbf {R}^N$, Pacific J. Math. 145 (1990), no. 1, 59–69. MR 1066398
Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. MR 1237490
Tom Ilmanen, The level-set flow on a manifold, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 193–204. MR 1216585, DOI 10.1090/pspum/054.1/1216585
Tom Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520, x+90. MR 1196160, DOI 10.1090/memo/0520
T. Ilmanen, Singularities of mean curvature flow of surfaces, https://people.math.ethz.ch/ $\sim$ilmanen/papers/sing.ps, 1995.
T. Ilmanen, Lectures on mean curvature flow and related equations, https:// people.math.ethz.ch/$\sim$ilmanen/papers/notes.pdf, 1998.
T. Ilmanen, Problems in mean curvature flow, https://people.math.ethz.ch/$\sim$ilmanen/ classes/eil03/problems03.pdf, 2003.
Tom Ilmanen, André Neves, and Felix Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 (2019), no. 1, 39–89. MR 3909904, DOI 10.4310/jdg/1547607687
Naoyuki Ishimura, Curvature evolution of plane curves with prescribed opening angle, Bull. Austral. Math. Soc. 52 (1995), no. 2, 287–296. MR 1348488, DOI 10.1017/S0004972700014714
Naoyuki Ishimura, Self-similar solutions for the Gauss curvature evolution of rotationally symmetric surfaces, Nonlinear Anal. 33 (1998), no. 1, 97–104. MR 1623054, DOI 10.1016/S0362-546X(97)00539-7
Mohammad N. Ivaki, An application of dual convex bodies to the inverse Gauss curvature flow, Proc. Amer. Math. Soc. 143 (2015), no. 3, 1257–1271. MR 3293740, DOI 10.1090/S0002-9939-2014-12314-8
Mohammad N. Ivaki, Deforming a hypersurface by Gauss curvature and support function, J. Funct. Anal. 271 (2016), no. 8, 2133–2165. MR 3539348, DOI 10.1016/j.jfa.2016.07.003
Mohammad N. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 1, 18. MR 3880311, DOI 10.1007/s00526-018-1462-3
N. M. Ivochkina, Th. Nehring, and F. Tomi, Evolution of starshaped hypersurfaces by nonhomogeneous curvature functions, Algebra i Analiz 12 (2000), no. 1, 185–203; English transl., St. Petersburg Math. J. 12 (2001), no. 1, 145–160. MR 1758567
Pong Soo Jang and Robert M. Wald, The positive energy conjecture and the cosmic censor hypothesis, J. Mathematical Phys. 18 (1977), no. 1, 41–44. MR 523907, DOI 10.1063/1.523134
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 187–204. MR 0030135
Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1991. MR 1185075
Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62. MR 3354954, DOI 10.4171/EMSS/8
Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. MR 1043269, DOI 10.2307/1971494
Nicolaos Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), no. 3, 683–715. MR 1100207
Nikolaos Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95–169. MR 1601434
Nikolaos Kapouleas, Stephen James Kleene, and Niels Martin Møller, Mean curvature self-shrinkers of high genus: non-compact examples, J. Reine Angew. Math. 739 (2018), 1–39. MR 3808256, DOI 10.1515/crelle-2015-0050
Kota Kasai and Yoshihiro Tonegawa, A general regularity theory for weak mean curvature flow, Calc. Var. Partial Differential Equations 50 (2014), no. 1-2, 1–68. MR 3194675, DOI 10.1007/s00526-013-0626-4
Atsushi Katsuda, Gromov’s convergence theorem and its application, Nagoya Math. J. 100 (1985), 11–48. MR 818156, DOI 10.1017/S0027763000000209
D. Ketover, Self-shrinking platonic solids, arXiv:1602.07271v1, 2016.
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original. MR 1786735, DOI 10.1137/1.9780898719451
J. R. King, Emerging areas of mathematical modelling, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 358 (2000), no. 1765, 3–19. Science into the next millennium: young scientists give their visions of the future, Part II. MR 1759647, DOI 10.1098/rsta.2000.0516
Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
Stephen Kleene and Niels Martin Møller, Self-shrinkers with a rotational symmetry, Trans. Amer. Math. Soc. 366 (2014), no. 8, 3943–3963. MR 3206448, DOI 10.1090/S0002-9947-2014-05721-8
Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 59 (2006), no. 3, 344–407. MR 2200259, DOI 10.1002/cpa.20101
Nicholas J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), no. 4, 603–614. MR 703287, DOI 10.1512/iumj.1983.32.32042
Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
B. Kotschwar, Harnack inequalities for evolving convex surfaces from the space-time perspective, math.la.asu.edu/ kotschwar/pub, 2009.
S. N. Kružkov, Nonlinear parabolic equations with two independent variables, Trudy Moskov. Mat. Obšč. 16 (1967), 329–346 (Russian). MR 0226208
N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759, DOI 10.1007/978-94-010-9557-0
N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR 1406091, DOI 10.1090/gsm/012
N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. MR 2435520, DOI 10.1090/gsm/096
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
A. Lahiri, A new version of Brakke’s local regularity theorem, arXiv:1601.06710v2, 2016.
A. Lahiri, Equality of the usual definitions of Brakke flow, arXiv:1705.08789v1, 2017.
Ben Lambert and Julian Scheuer, The inverse mean curvature flow perpendicular to the sphere, Math. Ann. 364 (2016), no. 3-4, 1069–1093. MR 3466860, DOI 10.1007/s00208-015-1248-2
Ben Lambert and Julian Scheuer, A geometric inequality for convex free boundary hypersurfaces in the unit ball, Proc. Amer. Math. Soc. 145 (2017), no. 9, 4009–4020. MR 3665052, DOI 10.1090/proc/13516
Ben Lambert, Jason Lotay, and Felix Schulze, Acient solutions in Lagrangian mean curvature flow, preprint, arXiv.org:1901.05383.
Joel Langer, A compactness theorem for surfaces with $L_p$-bounded second fundamental form, Math. Ann. 270 (1985), no. 2, 223–234. MR 771980, DOI 10.1007/BF01456183
Mat Langford, Motion of hypersurfaces by curvature, Bull. Aust. Math. Soc. 92 (2015), no. 3, 516–517. MR 3415631, DOI 10.1017/S000497271500091X
Mat Langford, The optimal interior ball estimate for a $k$-convex mean curvature flow, Proc. Amer. Math. Soc. 143 (2015), no. 12, 5395–5398. MR 3411154, DOI 10.1090/proc/12624
Mat Langford, A general pinching principle for mean curvature flow and applications, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Paper No. 107, 31. MR 3669776, DOI 10.1007/s00526-017-1193-x
M. Langford and S. Lynch, Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions, J. Reine. Angew. Math. (to appear).
Joseph Lauer, Convergence of mean curvature flows with surgery, Comm. Anal. Geom. 21 (2013), no. 2, 355–363. MR 3043750, DOI 10.4310/CAG.2013.v21.n2.a4
Joseph Lauer, A new length estimate for curve shortening flow and low regularity initial data, Geom. Funct. Anal. 23 (2013), no. 6, 1934–1961. MR 3132906, DOI 10.1007/s00039-013-0248-1
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
H. Blaine Lawson Jr., The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183–187. MR 287447, DOI 10.1007/BF01404649
Nam Q. Le and Natasa Sesum, The mean curvature at the first singular time of the mean curvature flow, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 6, 1441–1459. MR 2738327, DOI 10.1016/j.anihpc.2010.09.002
Nam Q. Le and Natasa Sesum, Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers, Comm. Anal. Geom. 19 (2011), no. 4, 633–659. MR 2880211, DOI 10.4310/CAG.2011.v19.n4.a1
Nam Q. Le and Natasa Sesum, On the extension of the mean curvature flow, Math. Z. 267 (2011), no. 3-4, 583–604. MR 2776050, DOI 10.1007/s00209-009-0637-1
John M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR 1468735, DOI 10.1007/b98852
G. Li and Y. Lv, Contracting convex hypersurfaces in space form by non-homogeneous curvature function, The Journal of Geometric Analysis (Jan 2019).
Haizhong Li, Xianfeng Wang, and Yong Wei, Surfaces expanding by non-concave curvature functions, Ann. Global Anal. Geom. 55 (2019), no. 2, 243–279. MR 3923539, DOI 10.1007/s10455-018-9625-1
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Qi-Rui Li, Surfaces expanding by the power of the Gauss curvature flow, Proc. Amer. Math. Soc. 138 (2010), no. 11, 4089–4102. MR 2679630, DOI 10.1090/S0002-9939-2010-10431-8
Q.-R. Li, W. Sheng, and X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, arXiv:1712.07774v1, 2017.
Xiaolong Li and Kui Wang, Nonparametric hypersurfaces moving by powers of Gauss curvature, Michigan Math. J. 66 (2017), no. 4, 675–682. MR 3720319, DOI 10.1307/mmj/1508810813
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
L. Lin, Mean curvature flow of star-shaped hypersurfaces, arXiv:1508.01225, 2015.
Longzhi Lin and Natasa Sesum, Blow-up of the mean curvature at the first singular time of the mean curvature flow, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 65, 16. MR 3509039, DOI 10.1007/s00526-016-1003-x
Yu-Chu Lin and Dong-Ho Tsai, Using Aleksandrov reflection to estimate the location of the center of expansion, Proc. Amer. Math. Soc. 138 (2010), no. 2, 557–565. MR 2557173, DOI 10.1090/S0002-9939-09-10155-7
P. Lu and J. Zhou, Ancient solutions for Andrews’ hypersurface flow, arXiv:1812.04926v1, 2018.
S. L. Lukyanov, E. S. Vitchev, and A. B. Zamolodchikov, Integrable model of boundary interaction: the paperclip, Nuclear Phys. B 683 (2004), no. 3, 423–454. MR 2057110, DOI 10.1016/j.nuclphysb.2004.02.010
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547]. MR 3012216
L. Lusternik and L. Schnirelmann, Sur le problème de trois géodésiques fermées sur les surfaces de genre 0, C. R. Acad. Sci. Paris 189 (1929), 269–271.
S. Lynch and H. Nguyen, Pinched ancient solutions to the high codimension mean curvature flow, arxiv:1709.09697v1, 2017.
Elena Mäder-Baumdicker, The area preserving curve shortening flow with Neumann free boundary conditions, Geom. Flows 1 (2015), no. 1, 34–79. MR 3351503, DOI 10.1515/geofl-2015-0004
Elena Mäder-Baumdicker, Singularities of the area preserving curve shortening flow with a free boundary condition, Math. Ann. 371 (2018), no. 3-4, 1429–1448. MR 3831277, DOI 10.1007/s00208-017-1637-9
Annibale Magni, Carlo Mantegazza, and Matteo Novaga, Motion by curvature of planar networks, II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 117–144. MR 3495423
Matthias Makowski and Julian Scheuer, Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere, Asian J. Math. 20 (2016), no. 5, 869–892. MR 3622318, DOI 10.4310/AJM.2016.v20.n5.a2
Carlo Mantegazza, Evolution by curvature of networks of curves in the plane, Variational problems in Riemannian geometry, Progr. Nonlinear Differential Equations Appl., vol. 59, Birkhäuser, Basel, 2004, pp. 95–109. Joint project with Matteo Novaga and Vincenzo Maria Tortorelli. MR 2076269
Carlo Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949, DOI 10.1007/978-3-0348-0145-4
Carlo Mantegazza and Luca Martinazzi, A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 857–874. MR 3060703
Carlo Mantegazza, Matteo Novaga, and Alessandra Pluda, Motion by curvature of networks with two triple junctions, Geom. Flows 2 (2017), no. 1, 18–48. Previously published in 2 (2016). MR 3565976, DOI 10.1515/geofl-2016-0002
Carlo Mantegazza, Matteo Novaga, and Vincenzo Maria Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 2, 235–324. MR 2075985
Christophe Margerin, Pointwise pinched manifolds are space forms, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 307–328. MR 840282, DOI 10.1090/pspum/044/840282
Thomas Marquardt, Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone, J. Geom. Anal. 23 (2013), no. 3, 1303–1313. MR 3078355, DOI 10.1007/s12220-011-9288-7
Thomas Marquardt, Weak solutions of inverse mean curvature flow for hypersurfaces with boundary, J. Reine Angew. Math. 728 (2017), 237–261. MR 3668996, DOI 10.1515/crelle-2014-0116
Francisco Martín, Jesús Pérez-García, Andreas Savas-Halilaj, and Knut Smoczyk, A characterization of the grim reaper cylinder, J. Reine Angew. Math. 746 (2019), 209–234. MR 3895630, DOI 10.1515/crelle-2016-0011
Francisco Martín, Andreas Savas-Halilaj, and Knut Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2853–2882. MR 3412395, DOI 10.1007/s00526-015-0886-2
Barry Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59–65. MR 117693, DOI 10.1090/S0002-9904-1959-10274-3
B. C. Mazur, On embeddings of spheres, Acta Math. 105 (1961), 1–17. MR 125570, DOI 10.1007/BF02559532
Rafe Mazzeo and Mariel Saez, Self-similar expanding solutions for the planar network flow, Analytic aspects of problems in Riemannian geometry: elliptic PDEs, solitons and computer imaging, Sémin. Congr., vol. 22, Soc. Math. France, Paris, 2011, pp. 159–173 (English, with English and French summaries). MR 3060453
James McCoy, The surface area preserving mean curvature flow, Asian J. Math. 7 (2003), no. 1, 7–30. MR 2015239, DOI 10.4310/AJM.2003.v7.n1.a2
James A. McCoy, The mixed volume preserving mean curvature flow, Math. Z. 246 (2004), no. 1-2, 155–166. MR 2031450, DOI 10.1007/s00209-003-0592-1
James Alexander McCoy, Self-similar solutions of fully nonlinear curvature flows, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 2, 317–333. MR 2856150
P. McGrath, Closed mean curvature self-shrinking surfaces of generalized rotational type, arXiv:1507.00681v1, 2015.
William W. Meeks III and Shing Tung Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151–168. MR 645492, DOI 10.1007/BF01214308
Frank Merle and Hatem Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), no. 2, 139–196. MR 1488298, DOI 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C
J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
N. M. Møller, Closed self-shrinking surfaces in $\mathbb {R}^3$ via the torus, arXiv:1111.7318v2, 2011.
Frank Morgan, Geometric measure theory, 5th ed., Elsevier/Academic Press, Amsterdam, 2016. A beginner’s guide; Illustrated by James F. Bredt. MR 3497381
John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
Alexander Mramor, A finiteness theorem via the mean curvature flow with surgery, J. Geom. Anal. 28 (2018), no. 4, 3348–3372. MR 3881975, DOI 10.1007/s12220-017-9962-5
A. Mramor and A. Payne, Ancient and eternal solutions to mean curvature flow from minimal surfaces, arXiv:1904.08439v2, 2019.
A. Mramor and S. Wang, On the topological rigidity of compact self-shrinkers in $\mathbb {R}^{3}$, International Mathematics Research Notices (2018), rny050.
T. Mullins, On the inverse mean curvature flow in warped product manifolds, arXiv:1610.05234v1, 2016.
W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900–904. MR 78836
Nikolai Nadirashvili and Serge Vlăduţ, Singular solutions of Hessian elliptic equations in five dimensions, J. Math. Pures Appl. (9) 100 (2013), no. 6, 769–784 (English, with English and French summaries). MR 3125267, DOI 10.1016/j.matpur.2013.03.001
Kazuaki Nakayama, Takeshi Iizuka, and Miki Wadati, Curve lengthening equation and its solutions, J. Phys. Soc. Japan 63 (1994), no. 4, 1311–1321. MR 1280385, DOI 10.1143/JPSJ.63.1311
Mitsunori Nara and Masaharu Taniguchi, The condition on the stability of stationary lines in a curvature flow in the whole plane, J. Differential Equations 237 (2007), no. 1, 61–76. MR 2327727, DOI 10.1016/j.jde.2007.02.012
André Neves, Singularities of Lagrangian mean curvature flow: monotone case, Math. Res. Lett. 17 (2010), no. 1, 109–126. MR 2592731, DOI 10.4310/MRL.2010.v17.n1.a9
André Neves, Recent progress on singularities of Lagrangian mean curvature flow, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 413–438. MR 2906935
André Neves, Finite time singularities for Lagrangian mean curvature flow, Ann. of Math. (2) 177 (2013), no. 3, 1029–1076. MR 3034293, DOI 10.4007/annals.2013.177.3.5
Huy The Nguyen, Convexity and cylindrical estimates for mean curvature flow in the sphere, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4517–4536. MR 3335392, DOI 10.1090/S0002-9947-2015-05927-3
Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. I, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1683–1701. MR 2465812, DOI 10.1090/S0002-9947-08-04748-X
Xuan Hien Nguyen, Translating tridents, Comm. Partial Differential Equations 34 (2009), no. 1-3, 257–280. MR 2512861, DOI 10.1080/03605300902768685
Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. II, Adv. Differential Equations 15 (2010), no. 5-6, 503–530. MR 2643233
Xuan Hien Nguyen, Complete embedded self-translating surfaces under mean curvature flow, J. Geom. Anal. 23 (2013), no. 3, 1379–1426. MR 3078359, DOI 10.1007/s12220-011-9292-y
Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow, Part III, Duke Math. J. 163 (2014), no. 11, 2023–2056. MR 3263027, DOI 10.1215/00127094-2795108
Xuan Hien Nguyen, Doubly periodic self-translating surfaces for the mean curvature flow, Geom. Dedicata 174 (2015), 177–185. MR 3303047, DOI 10.1007/s10711-014-0011-2
Yilong Ni and Meijun Zhu, One-dimensional conformal metric flow, Adv. Math. 218 (2008), no. 4, 983–1011. MR 2419376, DOI 10.1016/j.aim.2008.02.006
Chia-Hsing Nien and Dong-Ho Tsai, Convex curves moving translationally in the plane, J. Differential Equations 225 (2006), no. 2, 605–623. MR 2225802, DOI 10.1016/j.jde.2006.03.005
Seiki Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 343–352. MR 840284, DOI 10.1090/pspum/044/840284
Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195–270. MR 173993, DOI 10.1090/S0002-9904-1965-11276-9
Jeffrey A. Oaks, Singularities and self-intersections of curves evolving on surfaces, Indiana Univ. Math. J. 43 (1994), no. 3, 959–981. MR 1305955, DOI 10.1512/iumj.1994.43.43042
Vladimir Oliker, Evolution of nonparametric surfaces with speed depending on curvature. I. The Gauss curvature case, Indiana Univ. Math. J. 40 (1991), no. 1, 237–258. MR 1101228, DOI 10.1512/iumj.1991.40.40010
Vladimir Oliker, Self-similar solutions and asymptotic behavior of flows of nonparametric surfaces driven by Gauss or mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 389–402. MR 1216597, DOI 10.1090/pspum/054.1/1216597
Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. MR 1240056, DOI 10.1007/978-1-4612-4350-2
Kevin Daniel Olwell, A family of solitons for the Gauss curvature flow, ProQuest LLC, Ann Arbor, MI, 1993. Thesis (Ph.D.)–University of California, San Diego. MR 2690491
Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. MR 965860, DOI 10.1016/0021-9991(88)90002-2
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159v1, 2002.
G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109v1, 2003.
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245v1, 2003.
Stefan Peters, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), no. 1, 3–16. MR 892147
Peter Petersen, Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016. MR 3469435, DOI 10.1007/978-3-319-26654-1
Yo’av Rieck, A proof of Waldhausen’s uniqueness of splittings of $S^3$ (after Rubinstein and Scharlemann), Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 277–284. MR 2408250, DOI 10.2140/gtm.2007.12.277
S. Risa, Ancient solutions of curvature flows, PhD thesis, Università di Roma, Tor Vergata, 2016.
Susanna Risa and Carlo Sinestrari, Ancient solutions of geometric flows with curvature pinching, J. Geom. Anal. 29 (2019), no. 2, 1206–1232. MR 3935256, DOI 10.1007/s12220-018-0036-0
S. Risa and C. Sinestrari, Strong spherical rigidity of ancient solutions of expansive curvature flows, arXiv:1907.12319v1, 2019.
R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
Hyam Rubinstein, Some of Hyam’s favourite problems, Geometry and topology down under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, pp. 165–175. MR 3186672, DOI 10.1090/conm/597/11778
Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
Richard Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609–630. MR 116292, DOI 10.2307/2372973
Mariel Sáez and Oliver C. Schnürer, Mean curvature flow without singularities, J. Differential Geom. 97 (2014), no. 3, 545–570. MR 3263514
Mariel Sáez-Trumper, Relaxation of the flow of triods by curve shortening flow via the vector-valued parabolic Allen-Cahn equation, J. Reine Angew. Math. 634 (2009), 143–168. MR 2560408, DOI 10.1515/CRELLE.2009.071
Mariel Sáez Trumper, Uniqueness of self-similar solutions to the network flow in a given topological class, Comm. Partial Differential Equations 36 (2011), no. 2, 185–204. MR 2763337, DOI 10.1080/03605302.2010.539892
L. A. Santaló, An affine invariant for convex bodies of $n$-dimensional space, Portugal. Math. 8 (1949), 155–161 (Spanish). MR 39293
Guillermo Sapiro and Allen Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), no. 1, 79–120. MR 1255274, DOI 10.1006/jfan.1994.1004
Andreas Savas-Halilaj and Knut Smoczyk, Lagrangian mean curvature flow of Whitney spheres, Geom. Topol. 23 (2019), no. 2, 1057–1084. MR 3939057, DOI 10.2140/gt.2019.23.1057
Julian Scheuer, Pinching and asymptotical roundness for inverse curvature flows in Euclidean space, J. Geom. Anal. 26 (2016), no. 3, 2265–2281. MR 3511477, DOI 10.1007/s12220-015-9627-1
Julian Scheuer, Isotropic functions revisited, Arch. Math. (Basel) 110 (2018), no. 6, 591–604. MR 3803748, DOI 10.1007/s00013-018-1162-4
Julian Scheuer, Inverse curvature flows in Riemannian warped products, J. Funct. Anal. 276 (2019), no. 4, 1097–1144. MR 3906301, DOI 10.1016/j.jfa.2018.08.021
Saul Schleimer, Waldhausen’s theorem, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 299–317. MR 2408252, DOI 10.2140/gtm.2007.12.299
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
Oliver C. Schnürer, Surfaces contracting with speed $|A|^2$, J. Differential Geom. 71 (2005), no. 3, 347–363. MR 2198805
Oliver C. Schnürer, Surfaces expanding by the inverse Gaußcurvature flow, J. Reine Angew. Math. 600 (2006), 117–134. MR 2283800, DOI 10.1515/CRELLE.2006.088
Oliver C. Schnürer, Abderrahim Azouani, Marc Georgi, Juliette Hell, Nihar Jangle, Amos Koeller, Tobias Marxen, Sandra Ritthaler, Mariel Sáez, Felix Schulze, and Brian Smith, Evolution of convex lens-shaped networks under the curve shortening flow, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2265–2294. MR 2763716, DOI 10.1090/S0002-9947-2010-04820-2
Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231
Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928
Richard M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320. MR 1173050
Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR 634285, DOI 10.1002/cpa.3160340603
Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. MR 526976
A. Schoenflies, Beiträge zur Theorie der Punktmengen. III, Math. Ann. 62 (1906), no. 2, 286–328 (German). MR 1511377, DOI 10.1007/BF01449982
Felix Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), no. 4, 721–733. MR 2190140, DOI 10.1007/s00209-004-0721-5
Felix Schulze, Convexity estimates for flows by powers of the mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 261–277. MR 2244700
Felix Schulze, Nonlinear evolution by mean curvature and isoperimetric inequalities, J. Differential Geom. 79 (2008), no. 2, 197–241. MR 2420018
Felix Schulze, Uniqueness of compact tangent flows in mean curvature flow, J. Reine Angew. Math. 690 (2014), 163–172. MR 3200339, DOI 10.1515/crelle-2012-0070
Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
Natasa Sesum, Rate of convergence of the mean curvature flow, Comm. Pure Appl. Math. 61 (2008), no. 4, 464–485. MR 2383930, DOI 10.1002/cpa.20209
J. A. Sethian, Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws, J. Differential Geom. 31 (1990), no. 1, 131–161. MR 1030668
Leili Shahriyari, Translating graphs by mean curvature flow, Geom. Dedicata 175 (2015), 57–64. MR 3323629, DOI 10.1007/s10711-014-0028-6
Jason J. Sharples, Linear and quasilinear parabolic equations in Sobolev space, J. Differential Equations 202 (2004), no. 1, 111–142. MR 2060534, DOI 10.1016/j.jde.2004.03.020
Weimin Sheng and Xu-Jia Wang, Singularity profile in the mean curvature flow, Methods Appl. Anal. 16 (2009), no. 2, 139–155. MR 2563745, DOI 10.4310/MAA.2009.v16.n2.a1
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Carlo Sinestrari, Convex hypersurfaces evolving by volume preserving curvature flows, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1985–1993. MR 3396440, DOI 10.1007/s00526-015-0852-z
Stephen Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), 281–290. MR 104227, DOI 10.1090/S0002-9947-1959-0104227-9
G. Smith, On complete embedded translating solitons of the mean curvature flow that are of finite genus, arXiv:1501.04149v2, 2015.
Knut Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math. 3 (1997), 103–118. MR 1480081
Knut Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscripta Math. 95 (1998), no. 2, 225–236. MR 1603325, DOI 10.1007/s002290050025
Knut Smoczyk, Mean curvature flow in higher codimension: introduction and survey, Global differential geometry, Springer Proc. Math., vol. 17, Springer, Heidelberg, 2012, pp. 231–274. MR 3289845, DOI 10.1007/978-3-642-22842-1_{9}
Halil Mete Soner, Motion of a set by the curvature of its boundary, J. Differential Equations 101 (1993), no. 2, 313–372. MR 1204331, DOI 10.1006/jdeq.1993.1015
K. Sonnanburg, Blow-up continuity for type-I, mean-convex mean curvature flow, arXiv:1703.02619v1, 2017.
K. Sonnanburg, A Liouville theorem for mean curvature flow, arXiv:1711.02261v1, 2017.
Philippe Souplet and Qi S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38 (2006), no. 6, 1045–1053. MR 2285258, DOI 10.1112/S0024609306018947
J. Spruck and L. Sun, Convexity of $2$-convex translating solitons to the mean curvature flow in $\mathbb {R}^{n+1}$, arXiv:1910.02195v1, 2019.
J. Spruck and L. Xiao, Complete translating solitons to the mean curvature flow in $\mathbb {R}^3$ with nonnegative mean curvature, arXiv:1703.01003v3, 2017.
Axel Stahl, Convergence of solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations 4 (1996), no. 5, 421–441. MR 1402731, DOI 10.1007/BF01246150
Axel Stahl, Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations 4 (1996), no. 4, 385–407. MR 1393271, DOI 10.1007/BF01190825
J. J. Stoker, Über die Gestalt der positiv gekrümmten offenen Flächen im dreidimensionalen Raume, Compositio Math. 3 (1936), 55–88 (German). MR 1556933
Andrew Stone, A density function and the structure of singularities of the mean curvature flow, Calc. Var. Partial Differential Equations 2 (1994), no. 4, 443–480. MR 1383918, DOI 10.1007/BF01192093
Andrew Guy Stone, Singular and boundary behaviour in the mean curvature flow of hypersurfaces, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–Stanford University. MR 2691040
C. F. Sturm, Mémoire sur une classe d’équations à différences partielles, Collected Works of Charles François Sturm, Birkhäuser Basel, 2009, pp. 505–576.
A. Sun, Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces, arXiv:1810.08114v2, 2018.
Yoshihiro Tonegawa, Brakke’s mean curvature flow, SpringerBriefs in Mathematics, Springer, Singapore, 2019. An introduction. MR 3930606, DOI 10.1007/978-981-13-7075-5
Martin Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1385–1442 (French, with English and French summaries). MR 1427131
Dong-Ho Tsai, Geometric expansion of starshaped plane curves, Comm. Anal. Geom. 4 (1996), no. 3, 459–480. MR 1415752, DOI 10.4310/CAG.1996.v4.n3.a5
Dong-Ho Tsai, Blowup and convergence of expanding immersed convex plane curves, Comm. Anal. Geom. 8 (2000), no. 4, 761–794. MR 1792373, DOI 10.4310/CAG.2000.v8.n4.a3
Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. MR 812353, DOI 10.1002/cpa.3160380615
John Urbas, Complete noncompact self-similar solutions of Gauss curvature flows. I. Positive powers, Math. Ann. 311 (1998), no. 2, 251–274. MR 1625754, DOI 10.1007/s002080050187
John Urbas, Complete noncompact self-similar solutions of Gauss curvature flows. II. Negative powers, Adv. Differential Equations 4 (1999), no. 3, 323–346. MR 1671253
John Urbas, Convex curves moving homothetically by negative powers of their curvature, Asian J. Math. 3 (1999), no. 3, 635–656. MR 1793674, DOI 10.4310/AJM.1999.v3.n3.a4
John I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), no. 3, 355–372. MR 1082861, DOI 10.1007/BF02571249
John I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), no. 1, 91–125. MR 1085136
John Van Heijenoort, On locally convex manifolds, Comm. Pure Appl. Math. 5 (1952), 223–242. MR 52131, DOI 10.1002/cpa.3160050302
J. J. L. Velázquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 4, 595–628. MR 1318773
J. von Neumann, Discussion: Shape of metal grains, Metal Interfaces, American Society for Testing Materials, Cleveland, 1952, p. 108.
Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X
Guofang Wang and Chao Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532–556. MR 3197666, DOI 10.1016/j.aim.2014.01.024
Lu Wang, Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014), no. 3, 613–638. MR 3194490, DOI 10.1090/S0894-0347-2014-00792-X
Lu Wang, Uniqueness of self-similar shrinkers with asymptotically cylindrical ends, J. Reine Angew. Math. 715 (2016), 207–230. MR 3507924, DOI 10.1515/crelle-2014-0006
Mu-Tao Wang, Some recent developments in Lagrangian mean curvature flows, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 333–347. MR 2488942, DOI 10.4310/SDG.2007.v12.n1.a9
Xuefeng Wang, A remark on strong maximum principle for parabolic and elliptic systems, Proc. Amer. Math. Soc. 109 (1990), no. 2, 343–348. MR 1019284, DOI 10.1090/S0002-9939-1990-1019284-8
Xu-Jia Wang, Convex solutions to the mean curvature flow, Ann. of Math. (2) 173 (2011), no. 3, 1185–1239. MR 2800714, DOI 10.4007/annals.2011.173.3.1
Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
Yong Wei and Changwei Xiong, Inequalities of Alexandrov-Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere, Pacific J. Math. 277 (2015), no. 1, 219–239. MR 3393689, DOI 10.2140/pjm.2015.277.219
Hans F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. (6) 8 (1975), 295–310 (English, with Italian summary). MR 397126
Glen Wheeler and Valentina-Mira Wheeler, Mean curvature flow with free boundary outside a hypersphere, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8319–8342. MR 3710626, DOI 10.1090/tran/7305
Valentina Mira Wheeler, Mean curvature flow of entire graphs in a half-space with a free boundary, J. Reine Angew. Math. 690 (2014), 115–131. MR 3200337, DOI 10.1515/crelle-2012-0028
Valentina-Mira Wheeler, Non-parametric radially symmetric mean curvature flow with a free boundary, Math. Z. 276 (2014), no. 1-2, 281–298. MR 3150205, DOI 10.1007/s00209-013-1200-7
Valentina-Mira Wheeler, Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces, Geom. Dedicata 190 (2017), 157–183. MR 3704818, DOI 10.1007/s10711-017-0236-y
Brian White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), no. 3, 665–695. MR 1758759, DOI 10.1090/S0894-0347-00-00338-6
Brian White, Evolution of curves and surfaces by mean curvature, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 525–538. MR 1989203
Brian White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003), no. 1, 123–138. MR 1937202, DOI 10.1090/S0894-0347-02-00406-X
Brian White, A local regularity theorem for mean curvature flow, Ann. of Math. (2) 161 (2005), no. 3, 1487–1519. MR 2180405, DOI 10.4007/annals.2005.161.1487
Brian White, Currents and flat chains associated to varifolds, with an application to mean curvature flow, Duke Math. J. 148 (2009), no. 1, 41–62. MR 2515099, DOI 10.1215/00127094-2009-019
Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
D. V. Widder, The role of the Appell transformation in the theory of heat conduction, Trans. Amer. Math. Soc. 109 (1963), 121–134. MR 154068, DOI 10.1090/S0002-9947-1963-0154068-2
D. V. Widder, The heat equation, Pure and Applied Mathematics, Vol. 67, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0466967
Hiroki Yagisita, Asymptotic behaviors of star-shaped curves expanding by $V=1-K$, Differential Integral Equations 18 (2005), no. 2, 225–232. MR 2106103
Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203

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