Abstract
We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature. If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.
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Barles G., Biton S., Bourgoing M., Ley O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differ. Equ. 18(2), 159–179 (2003)
Bayard P., Schnürer O.C.: Entire spacelike hypersurfaces of constant Gauß curvature in Minkowski space. J. Reine Angew. Math. 627, 1–29 (2009)
Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)
Caffarelli L., Hardt R., Simon L.: Minimal surfaces with isolated singularities. Manuscr. Math. 48(1–3), 1–18 (1984)
Chau A., Schnürer O.C.: Stability of gradient Kähler-Ricci solitons. Comm. Anal. Geom. 13(4), 769–800 (2005)
Clutterbuck J., Schnürer O.C., Schulze F.: Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differ. Equ. 29(3), 281–293 (2007)
Clutterbuck, J.: Parabolic equations with continuous initial data. Ph.D. thesis, Australian National University (2004). arXiv:math.AP/0504455
Colding T.H., Minicozzi W.P. II: Sharp estimates for mean curvature flow of graphs. J. Reine Angew. Math. 574, 187–195 (2004)
Ecker, K.: Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser Boston Inc., Boston (2004)
Ecker K., Huisken G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105(3), 547–569 (1991)
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
Gerhardt, C.: Curvature problems. Series in Geometry and Topology, vol. 39. International Press, Somerville (2006)
Giusti, E.: Minimal surfaces and functions of bounded variation, vol. 80. Monogr. Math. Brikhäuser, Boston (1984)
Huisken G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
Huisken G., Sinestrari C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175, 137–221 (2009)
Koch, H., Lamm, T.: Geometric flows with rough initial data, arXiv: 0902.1488v1
Schnürer O.C., Schulze F., Simon M.: Stability of Euclidean space under Ricci flow. Comm. Anal. Geom. 16(1), 127–158 (2008)
Schnürer O.C.: Surfaces contracting with speed |A|2. J. Differ. Geom. 71(3), 347–363 (2005)
Stavrou N.: Selfsimilar solutions to the mean curvature flow. J. Reine Angew. Math. 499, 189–198 (1998)
White B.: The size of the singular set in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 13(3), 665–695 (2000)
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Clutterbuck, J., Schnürer, O.C. Stability of mean convex cones under mean curvature flow. Math. Z. 267, 535–547 (2011). https://doi.org/10.1007/s00209-009-0634-4
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DOI: https://doi.org/10.1007/s00209-009-0634-4