SYMMETRY OF TRANSLATING SOLUTIONS TO MEAN CURVATURE FLOWS

doi:10.1016/S0252-9602(10)60187-7

Abstract

Abstract:

First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.

Key words: mean curvature flow, symmetry, fully nonlinear, elliptic equation

CLC Number:

35J65

JIAN Huai-Yu, JU Hong-Jie, LIU Yan-Nan, SUN Wei. SYMMETRY OF TRANSLATING SOLUTIONS TO MEAN CURVATURE FLOWS[J].Acta mathematica scientia,Series B, 2010, 30(6): 2006-2016.

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References

[1] Huisken G, Sinestrari C. Mean curvature flow singularities for mean convex surfaces. Calc Var Partial Differ Equ, 1999, 8: 1--14

[2] Huisken G, Sinestrari C. Convexity estimates for mean curvature flow and singularities of mean convex surfaces.
Acta Math, 1999, 183: 45--70

[3] Wang X J. Convex solutions to the mean curvature flow. arXiv: math.DG/0404326v1 (submitted in Ann Math, 2003

[4] Sheng W M, Wang X J. Singularity profile in the mean curvature flow. Methods Appl Anal, 2009, 16: 139--155

[5] Liu Y N, Jian H Y. Evolution of hypersurfaces by mean curvature minus an externa force field. Sci China Ser A, 2007, 50(2): 231--239

[6] Jian H Y, Liu Y N. Long-time existence of mean curvature with externa force fields. Pacific J Math, 2008, 234: 311--325

[7] Liu Y N, Jian H Y. A curve flow evolved by a fourth order parabolic equation. Sci China Ser A, 2009, 52(10): 2177--2184

[8] Schulze F. Evolution of convex hypersurfaces by powers of the mean curvature. Math Z, 2005, 251: 721--733

[9] Schulze F. Nonlinear Evolution by mean curvature and isoperimetric inequalities. J Differ Geom, 2008, 79: 197--241

[10] White B. The nature of singularities in mean curvature flow of mean-convex sets. J Amer Math Soc, 2003, 16: 123--138; 197--241

[11] Gui C F, Jian H Y, Ju H J. Properties of translating solutions to mean curvature flow. Discrete Contin Dyn Syst, 2010, 28: 441--453

[12] Jian H Y, Liu Q H, Chen X Q. Convexity and symmetry of translating solitons in mean curvature flows. Chin Ann
Math, 2005, 26B: 413--422

[13] Altschuler S, Angenent S B, Giga Y. Mean curvature flow through singularities for surfaces of rotation. J Geom Anal, 1995, 5(3): 293--358

[14] Wang X J. Interior gradient estimates for mean curvature equations. Math Z, 1998, 228: 73--81

[15] Jian H Y, Ju H J. Existence of translating solutions to the flow by powers of mean curvature on unbounded domains. Preprint, 2010

[16] Jian H Y, Ju H J, Sun W. Traveling fronts of curvature flow with external force field. Commun Pure Appl Anal,
2010, 9: 975--986

[17] Aarons M. Mean curvature flow with a forcing term in Minkowski space. Calc Var Partial Differ Equ, 2005, 25: 205--246

[18] Ecker K. Interior estimates and longtime solutions for mean curvature flow of noncompact spaceike hypersurfaces in Minkowski space. J Differ Geom, 1997, 45: 481--498

[19] Ecker K. On mean curvature flow of spacelike hypersurfaces in asymptotical flat spacetimes. J Austral Math Soc Ser A, 1993, 55: 41--59

[20] Ecker K, Huisken G. Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Comm Math Phys, 1991, 135: 595--613

[21] Huisken G, Yau S T. Definition of center of mass for isolated physical system and unique foliations by stable spheres with constant curvature. Invent Math, 1996, 124: 281--311

[22] Liu Y N, Jian H Y. Evolution of spacelike hypersurfaces by mean curvature minus external force field in Minkowski space. Advanced Nonlinear Studies, 2009, 9: 513--522

[23] Jian H Y. Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J Differ Equ, 2006, 220: 147--162

[24] Ju H J, Lu J, Jian H Y. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Commun Pure Appl Anal, 2010, 9: 967--973

[25] Guan B, Jian H, Schoen R. Entire spacelike hypersurfaces of constant Gauss curvature in Minkowski space. J Rene Angew Math, 2006, 595: 167--188

[26] Treibergs A E. Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent Math, 1982, 66: 39--56

[27] Li C. Monotonocity and symmetry of solutions of nonlinear elliptic equations on unbounded domains. Commu Partial Differ Equ, 1991, 16: 585--612

[28] Li Y, Ni W M. Radial symmetry of positive solutions of nonlinear elliptic equations in Rⁿ. Comm Part Diff Equ, 1993, 18: 1043--1054

[29] Jian H Y, Wang X J. Continuity estimates for the Monge-Ampère equation. SIAM J Math Anal, 2007, 39: 608--626

[30] Jian H Y, Wang X J. Bernsterin theorem and regularity for a class of Monge Ampère equations. preprint, 2010

[31] Guan B, Jian H Y. The Monge-Ampère equation with infinite boundary value. Pacific J Math, 2004, 216: 77--94

[32] Jian H Y. Hessian equations with infinite Dirichlet boundary value. Indiana Univ Math J, 2006, 55: 1045--1062

[33] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 1983