Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

Abstract

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power $p>1$ loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.

18 pages. We included an example for the loss of convexity and pinching. In the third version we dropped the concavity assumption on F. Comments are welcome

References in corpus (7)

• Fully nonlinear parabolic equations in two space variables
• New pinching estimates for Inverse curvature flows in space forms
• Surfaces expanding by non-concave curvature functions
• Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold
• Inverse mean curvature flows in warped product manifolds
• Isotropic functions revisited
• Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

Cited by in corpus (8)

• Deforming a hypersurface by principal radii of curvature and support function
• New pinching estimates for Inverse curvature flows in space forms
• Surfaces expanding by non-concave curvature functions
• Orlicz-Minkowski flows
• Inverse curvature flows in Riemannian warped products
• Shifted inverse curvature flows in hyperbolic space
• Contraction of surfaces in hyperbolic space and in sphere
• Flowing the leaves of a foliation with normal speed given by the logarithm of general curvature functions