New pinching estimates for Inverse curvature flows in space forms

TL;DR

This paper establishes new pinching estimates for inverse curvature flows of convex hypersurfaces in space forms, ensuring curvature ratios remain controlled, which leads to smooth convergence of the flow.

Contribution

It introduces novel pinching estimates for inverse curvature flows in space forms with specific speed functions, extending previous results and enabling convergence proofs.

Findings

Curvature ratios are controlled by initial data.

Flow hypersurfaces exhibit smooth convergence.

New estimates apply to various 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 $F_*$ 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.