# Contraction of surfaces in hyperbolic space and in sphere

## TL;DR

This paper studies curvature flows of smooth closed surfaces in hyperbolic space and the sphere, proving finite-time contraction to points and spherical shapes under certain curvature conditions and flow powers.

## Contribution

It establishes new results on the contraction and shape evolution of surfaces under curvature flows in hyperbolic space and the sphere, with specific conditions on the flow powers.

## Key findings

- Surfaces with positive scalar curvature remain positive under the flow in hyperbolic space.
- Surfaces contract to points and become spherical in finite time for certain flow powers.
- Convex surfaces in the sphere contract to round points under specified curvature flows.

## Abstract

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $\alpha$ of the mean curvature $H$, the evolving surface $M_t$ has positive scalar curvature for $t>0$. By assuming $\alpha\in [1,4]$, we can further prove that $M_t$ contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power $\alpha\in [1/2,1]$. In the sphere case, we show that the flow by a positive power $\alpha$ of mean curvature contracts strictly convex surface in $\mathbb{S}^3$ to a round point in finite time if $\alpha\in [1,5]$. The same conclusion also holds for the flow by powers of Gauss curvature provided that the power $\alpha\in [1/2,1]$.