Uniqueness of closed self-similar solutions to the Gauss curvature flow

TL;DR

This paper proves the uniqueness of strictly convex closed smooth self-similar solutions to the -Gauss curvature flow within a specific  range, showing such flows shrink to round spheres.

Contribution

It introduces a Pogorelov type computation and applies the strong maximum principle to establish solution uniqueness for the -Gauss curvature flow.

Findings

Self-similar solutions are unique within the specified  range.

Such flows shrink convex hypersurfaces to round spheres.

The method combines a novel computation with maximum principle techniques.

Abstract

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.