# Uniqueness Theorems of Self-Conformal Solutions to Inverse Curvature Flows

## TL;DR

This paper proves that under certain natural conditions, the round sphere is the unique closed self-conformal solution to inverse curvature flows in Euclidean space, extending known uniqueness results.

## Contribution

It establishes the rigidity of the sphere as the only closed self-conformal solution under natural geometric conditions for inverse curvature flows.

## Key findings

- Round sphere is the only closed solution under certain conditions.
- Uniqueness holds for flows evolving by conformal Killing fields.
- Results extend previous known uniqueness theorems.

## Abstract

It has been known in that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree -1 homogeneous functions of principle curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in much stronger sense, that under some natural conditions such as star-shapedness, it is the only closed solution to the inverse mean curavture flow and the above-mentioned flows in the Euclidean space which evolves by diffeomorphisms generated by conformal Killing fields.