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

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.

References in corpus (1)

• Flow by the power of the Gauss curvature

Cited by in corpus (7)

• Ancient solutions of geometric flows with curvature pinching
• The evolution of complete non-compact graphs by powers of Gauss curvature
• Uniqueness of self-similar solutions to flows by quotient curvatures
• Translating surfaces under flows by sub-affine-critical powers of Gauss curvature
• Classification of ancient flows by sub-affine-critical powers of curvature in ${\mathbb{R}}^2$
• Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1
• Self-similar solutions of curvature flows in warped products