Curvature decay estimates of graphical mean curvature flow in higher codimensions
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- by Knut Smoczyk, Mao-Pei Tsui and Mu-Tao Wang PDF
- Trans. Amer. Math. Soc. 368 (2016), 7763-7775 Request permission
Abstract:
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions for a flat ambient space. To the best of our knowledge, these are the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat Riemann surfaces.References
- Li An-Min and Li Jimin, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel) 58 (1992), no. 6, 582–594. MR 1161925, DOI 10.1007/BF01193528
- Albert Chau, Jingyi Chen, and Weiyong He, Lagrangian mean curvature flow for entire Lipschitz graphs, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 199–220. MR 2898776, DOI 10.1007/s00526-011-0431-x
- Albert Chau, Jingyi Chen, and Yu Yuan, Lagrangian mean curvature flow for entire Lipschitz graphs II, Math. Ann. 357 (2013), no. 1, 165–183. MR 3084345, DOI 10.1007/s00208-013-0897-2
- 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 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
- Richard S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215–226. MR 1316556
- Tom Ilmanen, Singularities of mean curvature flow of surfaces, preprint (1997).
- Kuo-Wei Lee and Yng-Ing Lee, Mean curvature flow of the graphs of maps between compact manifolds, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5745–5759. MR 2817407, DOI 10.1090/S0002-9947-2011-05204-9
- Richard M. Schoen, The role of harmonic mappings in rigidity and deformation problems, Complex geometry (Osaka, 1990) Lecture Notes in Pure and Appl. Math., vol. 143, Dekker, New York, 1993, pp. 179–200. MR 1201611
- Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703, DOI 10.2307/2006981
- Andreas Savas-Halilaj and Knut Smoczyk, Homotopy of area decreasing maps by mean curvature flow, Adv. Math. 255 (2014), 455–473. MR 3167489, DOI 10.1016/j.aim.2014.01.014
- Andreas Savas-Halilaj and Knut Smoczyk, Evolution of contractions by mean curvature flow, Math. Ann. 361 (2015), no. 3-4, 725–740. MR 3319546, DOI 10.1007/s00208-014-1090-y
- Knut Smoczyk, A canonical way to deform a Lagrangian submanifold, arXiv: dg-ga/9605005 (1996).
- 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}
- Knut Smoczyk, Longtime existence of the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 20 (2004), no. 1, 25–46. MR 2047144, DOI 10.1007/s00526-003-0226-9
- Knut Smoczyk and Mu-Tao Wang, Mean curvature flows of Lagrangians submanifolds with convex potentials, J. Differential Geom. 62 (2002), no. 2, 243–257. MR 1988504
- Mao-Pei Tsui and Mu-Tao Wang, Mean curvature flows and isotopy of maps between spheres, Comm. Pure Appl. Math. 57 (2004), no. 8, 1110–1126. MR 2053760, DOI 10.1002/cpa.20022
- Mu-Tao Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential Geom. 57 (2001), no. 2, 301–338. MR 1879229
- Mu-Tao Wang, Deforming area preserving diffeomorphism of surfaces by mean curvature flow, Math. Res. Lett. 8 (2001), no. 5-6, 651–661. MR 1879809, DOI 10.4310/MRL.2001.v8.n5.a7
- Mu-Tao Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (2002), no. 3, 525–543. MR 1908059, DOI 10.1007/s002220100201
- Mu-Tao Wang, Gauss maps of the mean curvature flow, Math. Res. Lett. 10 (2003), no. 2-3, 287–299. MR 1981905, DOI 10.4310/MRL.2003.v10.n3.a2
- Mu-Tao Wang, Subsets of Grassmannians preserved by mean curvature flows, Comm. Anal. Geom. 13 (2005), no. 5, 981–998. MR 2216149
- Mu-Tao Wang, Remarks on a class of solutions to the minimal surface system, Geometric evolution equations, Contemp. Math., vol. 367, Amer. Math. Soc., Providence, RI, 2005, pp. 229–235. MR 2115762, DOI 10.1090/conm/367/06758
Additional Information
- Knut Smoczyk
- Affiliation: Institut für Differentialgeometrie and Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
- Email: smoczyk@math.uni-hannover.de
- Mao-Pei Tsui
- Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan – and– Department of Mathematics and Statistics, University of Toledo, 2801 W. Bancroft Street, Toledo, Ohio 43606-3390
- MR Author ID: 278086
- Email: mao-pei.tsui@utoledo.edu
- Mu-Tao Wang
- Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
- MR Author ID: 626881
- Email: mtwang@math.columbia.edu
- Received by editor(s): January 23, 2014
- Received by editor(s) in revised form: November 28, 2014
- Published electronically: January 26, 2016
- Additional Notes: The first author was supported by the DFG (German Research Foundation)
The second author was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation, #239677.
The third author was partially supported by National Science Foundation grants DMS 1105483 and DMS 1405152. - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7763-7775
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/tran/6624
- MathSciNet review: 3546783