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

## TL;DR

This paper establishes convergence of certain curvature flows in Euclidean and hyperbolic spaces, showing that under specific conditions, hypersurfaces evolve smoothly to spheres, with some flows losing convexity without initial pinching.

## Contribution

It proves convergence results for inverse curvature flows with high powers in Euclidean and hyperbolic spaces, including conditions for preserving convexity and initial pinching.

## Key findings

- Flow speeds of the form F^{-p} with p>1 lead to smooth convergence to spheres.
- A pinching condition is necessary to maintain convexity during the flow.
- Some flows can lose convexity if initial conditions are not properly constrained.

## 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.