DEFORMING PINCHED HYPERSURFACES OF THE HYPERBOLIC SPACE BY POWERS OF THE MEAN CURVATURE INTO SPHERES

Title & Authors
DEFORMING PINCHED HYPERSURFACES OF THE HYPERBOLIC SPACE BY POWERS OF THE MEAN CURVATURE INTO SPHERES
Guo, Shunzi; Li, Guanghan; Wu, Chuanxi;

Abstract
This paper concerns closed hypersurfaces of dimension $\small{n{\geq}2}$ in the hyperbolic space $\small{{\mathbb{H}}_{\kappa}^{n+1}}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power $\small{{\beta}{\geq}1}$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and $\small{{\beta}}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in $\small{{\mathbb{H}}_{\kappa}^{n+1}}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of $\small{{\mathbb{H}}_{\kappa}^{n+1}}$.
Keywords
powers of the mean curvature;horosphere;convex hypersurface;hyperbolic space;normalization;
Language
English
Cited by
References
1.
B. H. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151-171.

2.
B. H. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407-431.

3.
B. H. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151-161.

4.
B. H. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1-34.

5.
B. H. Andrews, Moving surfaces by non-concave curvature functions, preprint (2004), available at arXiv:math.DG/0402273.

6.
A. Borisenko and V. Miquel, Total curvatures of convex hypersurfaces in hyperbolic space, Illinois J. Math. 43 (1999), no. 1, 61-78.

7.
E. Cabezas-Rivas and V. Miquel, Volume preserving mean curvature flow in the hyperbolic space, Indiana Univ. Math. J. 56 (2007), no. 5, 2061-2086.

8.
E. Cabezas-Rivas and C. Sinestrari, Volume-preserving flow by powers of the m-th mean curvature, Calc. Var. Partial Differential Equations 38 (2009), no. 3-4, 441-469.

9.
B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117-138.

10.
B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63-82.

11.
E. DiBenedetto and A. Friedman, Holder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22.

12.
C. Gerhardt, Curvature Problems, Series in Geometry and Topology, 39, International Press, Somerville, MA, Series in Geometry and Topology, 2006.

13.
S. Z. Guo, G. H. Li, and C. X. Wu, Contraction of horosphere-convex hypersurfaces by powers of the mean curvature in the hyperbolic state, J. Korean Math. Soc. 50 (2013), no. 6, 1311-1332.

14.
G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266.

15.
G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463-480.

16.
G. H. Li, L. J. Yu, and C. X. Wu, Curvature flow with a general forcing term in Euclidean spaces, J. Math. Anal. Appl. 353 (2009), no. 2, 508-520.

17.
M. Makowski, Mixed volume preserving curvature flows in hyperbolic space, preprint, arxiv:12308.1898v1, [math.DG] 9 Aug 2012.

18.
J. A. McCoy, Mixed volume preserving curvature flows, Calc. Var. Partial Differential Equations 24 (2005), no. 2, 131-154.

19.
O. C. Schnurer, Surfaces contracting with speed ${\mid}A{\mid}^2$, J. Differential Geom. 71 (2005), no. 4, 347-363.

20.
R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275-288.

21.
F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), no. 4, 721-733.

22.
F. Schulze, Convexity estimates for flows by powers of the mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 261-277.

23.
J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1, 62-105.

24.
K. Tso, Deforming a Hypersurface by Its Gauss-Kronecker Curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882.