Surfaces expanding by the power of the Gauss curvature flow

Author:
Qi-Rui Li

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4089-4102

MSC (2010):
Primary 53C44, 35K55

DOI:
https://doi.org/10.1090/S0002-9939-2010-10431-8

Published electronically:
June 16, 2010

MathSciNet review:
2679630

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we describe the flow of 2-surfaces in for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity , when , converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for by Urbas and Huisken and for by Schnürer.

**1.**B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138(1999), no. 1, 151-161. MR**1714339 (2000i:53097)****2.**B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2(1994), no. 2, 151-171. MR**1385524 (97b:53012)****3.**B. Chow, Deforming convex hypersurfaces by the n-th root of the Gaussian curvature, J. Differential Geom. 22(1985) 117-138. MR**826427 (87f:58155)****4.**B. Chow and Peng Lu, The maximum principle for systems of parabolic equations subject to an avoidance set, Pacific J. Math. 214(2004), no. 2, 201-222. MR**2042930 (2004m:35027)****5.**B. 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 (97f:53064)****6.**W. Firey, Shapes of worn stones, Mathmatika 21(1974), 1-11. MR**0362045 (50:14487)****7.**C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom. 43(1996), no. 3, 612-641. MR**1412678 (97g:53067)****8.**R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17(1982), no. 2, 255-306. MR**664497 (84a:53050)****9.**G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1984), no. 1, 237-266. MR**772132 (86j:53097)****10.**G. Huisken and Alexander Polden, Geometric evolution equations for hypersurfaces, Calculus of variations and geometric problems (Cetraro, 1996), Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, pp. 45-84. MR**1731639 (2000j:53090)****11.**G. Huisken, On the expansion of convex hypersurfaces by the inverse of symmetric curvature functions (unpublished).**12.**J. A. McCoy, The surface area preserving mean curvature flow, Asian J. Math. 7(2003), no. 1, 7-30. MR**2015239 (2004m:53117)****13.**Oliver C. Schnürer, Surfaces expanding by the inverse Gauss curvature flow, J. Reine Angew. Math. 600(2006), 117-134. MR**2283800 (2007j:53074)****14.**Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38(1985), no. 6, 867-882. MR**812353 (87e:53009)****15.**J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows II. Negative powers, Advances in Differential Equations 4(1999), no. 3, 323-346. MR**1671253 (2000a:53117)****16.**J. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33(1991), no. 1, 91-125. MR**1085136 (91j:58155)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
53C44,
35K55

Retrieve articles in all journals with MSC (2010): 53C44, 35K55

Additional Information

**Qi-Rui Li**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Email:
85lqr@163.com

DOI:
https://doi.org/10.1090/S0002-9939-2010-10431-8

Keywords:
Surface,
expanding curvature flow,
velocity function $K^{-\alpha}$

Received by editor(s):
June 2, 2009

Received by editor(s) in revised form:
October 9, 2009, November 26, 2009, December 19, 2009, December 31, 2009, and February 6, 2010

Published electronically:
June 16, 2010

Communicated by:
Richard A. Wentworth

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.