Uniqueness of self-similar shrinkers with asymptotically conical ends
Author:
Lu Wang
Journal:
J. Amer. Math. Soc. 27 (2014), 613-638
MSC (2010):
Primary 53C44, 53C24, 35J15; Secondary 35B60
DOI:
https://doi.org/10.1090/S0894-0347-2014-00792-X
Published electronically:
March 19, 2014
MathSciNet review:
3194490
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let $C\subset \mathbb {R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb {R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete properly embedded self-shrinker asymptotic to it.
- U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704
- 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
- Xu Cheng and Detang Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc. 141 (2013), no. 2, 687–696. MR 2996973, DOI https://doi.org/10.1090/S0002-9939-2012-11922-7
- David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1–15. MR 1302814
- 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 https://doi.org/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 https://doi.org/10.4171/CMH/260
- 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
- Celso J. Costa, Example of a complete minimal immersion in ${\bf R}^3$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), no. 1-2, 47–54. MR 794728, DOI https://doi.org/10.1007/BF02584707
- Q. Ding and Y.L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456.
- 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
- Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI https://doi.org/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 https://doi.org/10.1007/BF01232278
- Luis Escauriaza and Francisco Javier Fernández, Unique continuation for parabolic operators, Ark. Mat. 41 (2003), no. 1, 35–60. MR 1971939, DOI https://doi.org/10.1007/BF02384566
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Decay at infinity of caloric functions within characteristic hyperplanes, Math. Res. Lett. 13 (2006), no. 2-3, 441–453. MR 2231129, DOI https://doi.org/10.4310/MRL.2006.v13.n3.a8
- L. Escauriaza, G. Seregin, and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 147–157. MR 2005639, DOI https://doi.org/10.1007/s00205-003-0263-8
- L. Escauriaza, G. Seregin, and V. Šverák, On backward uniqueness for parabolic equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32, 100–103, 272 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4577–4579. MR 1923546, DOI https://doi.org/10.1023/B%3AJOTH.0000041475.11233.d8
- Nicola Garofalo and Fang-Hua Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347–366. MR 882069, DOI https://doi.org/10.1002/cpa.3160400305
- Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245–268. MR 833393, DOI https://doi.org/10.1512/iumj.1986.35.35015
- David Hoffman and William H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), no. 1, 1–34. MR 1038356, DOI https://doi.org/10.2307/1971506
- 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 https://doi.org/10.1090/pspum/054.1/1216584
- Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
- T. Ilmanen, Personal communication.
- T. Ilmanen, Lectures on mean curvature flow and related equations (1995). unpublished notes.
- T. Ilmanen, Singularities of mean curvature flow of surfaces (1995). preprint.
- David Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math. 62 (1986), no. 2, 118–134. MR 865834, DOI https://doi.org/10.1016/0001-8708%2886%2990096-4
- David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463–494. With an appendix by E. M. Stein. MR 794370, DOI https://doi.org/10.2307/1971205
- Nikolaos Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95–169. MR 1601434
- Nikolaos Kapouleas, S.J. Kleene, and N.M. Møller, Mean curvature self-shrinkers of high genus: non-compact examples, J. Reine Angew. Math.,. to appear, arXiv: 1106.5454.
- S.J. Kleene and N.M. Møller, Self-shrinkers with a rotational symmetry, Trans. Amer. Math. Soc.,. to appear, arXiv: 1008.1609.
- B. Kotschwar, Personal communication.
- B. Kotschwar, Ricci flow and the holonomy group. J. Reine Angew. Math., to appear, arXiv: 1105.3722.
- Brett L. Kotschwar, Backwards uniqueness for the Ricci flow, Int. Math. Res. Not. IMRN 21 (2010), 4064–4097. MR 2738351, DOI https://doi.org/10.1093/imrn/rnq022
- Lu Li and Vladimír Šverák, Backward uniqueness for the heat equation in cones, Comm. Partial Differential Equations 37 (2012), no. 8, 1414–1429. MR 2957545, DOI https://doi.org/10.1080/03605302.2011.635323
- Fang-Hua Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 43 (1990), no. 1, 127–136. MR 1024191, DOI https://doi.org/10.1002/cpa.3160430105
- S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half space, Port. Math. (N.S.) 58 (2001), no. 1, 1–24. MR 1820835
- N.M. Møller, Closed self-shrinking surfaces in $\mathbb {R}^3$ via the torus (2001). preprint.
- Tu A. Nguyen, On a question of Landis and Oleinik, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2875–2899. MR 2592940, DOI https://doi.org/10.1090/S0002-9947-10-04733-1
- Xuan Hien Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. III, Duke Math. J.,. to appear, arXiv: 1106.5272.
- 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, 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 https://doi.org/10.1090/S0002-9947-08-04748-X
- Yifei Pan and Thomas Wolff, A remark on unique continuation, J. Geom. Anal. 8 (1998), no. 4, 599–604. MR 1724207, DOI https://doi.org/10.1007/BF02921714
- Chi-Cheung Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), no. 3-4, 521–539. MR 1387458, DOI https://doi.org/10.1080/03605309608821195
- David L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math. 52 (1973), 189–211. MR 341256, DOI https://doi.org/10.1002/sapm1973523189
- Gregory Seregin and Vladimir Šverák, The Navier-Stokes equations and backward uniqueness, Nonlinear problems in mathematical physics and related topics, II, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 353–366. MR 1972005, DOI https://doi.org/10.1007/978-1-4615-0701-7_19
- C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat. 28 (1990), no. 1, 159–182. MR 1049649, DOI https://doi.org/10.1007/BF02387373
- L. Wang, Uniqueness of self-similar shrinkers with asymptotically cylindrical ends, J. Reine Angew. Math. (2013). in press.
- Lu Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303. MR 2780753, DOI https://doi.org/10.1007/s10711-010-9535-2
- Brian White, Partial regularity of mean-convex hypersurfaces flowing by mean curvature, Internat. Math. Res. Notices 4 (1994), 186 ff., approx. 8 pp.}, issn=1073-7928, review=\MR{1266114}, doi=10.1155/S1073792894000206,.
Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 53C44, 53C24, 35J15, 35B60
Retrieve articles in all journals with MSC (2010): 53C44, 53C24, 35J15, 35B60
Additional Information
Lu Wang
Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore
, Maryland 21218
Email: coral0426@gmail.com
Keywords: Self-shrinkers, mean curvature flow, backwards uniqueness
Received by editor(s): October 3, 2011
Received by editor(s) in revised form: June 27, 2013, and October 19, 2013
Published electronically: March 19, 2014
Article copyright: © Copyright 2014 American Mathematical Society