New pinching estimates for Inverse curvature flows in space forms

Abstract

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-\alpha }$, where $\alpha \in (0,1]$ for $K_N=0,-1$ and $\alpha =1$ for $K_N=1$, $F$ is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual $F_{\ast }$ approaching zero on the boundary of the positive cone ${\Gamma }_{+}$. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flow.

15 pages,v2

References in corpus (3)

• Fully nonlinear parabolic equations in two space variables
• Surfaces expanding by non-concave curvature functions
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

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• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
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• Shifted inverse curvature flows in hyperbolic space
• Harmonic mean curvature flow and geometric inequalities
• Contracting axially symmetric hypersurfaces by powers of the ${\sigma }_k$-curvature
• Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows