On existence of hypersurfaces translating by powers of Gauss curvature

TL;DR

This paper constructs complete convex hypersurfaces translating under Gauss curvature flow powers, revealing diverse asymptotic shapes and connecting to soliton solutions in geometric analysis.

Contribution

It introduces new convex hypersurfaces translating by powers of Gauss curvature and analyzes their asymptotic behavior and relation to solitons.

Findings

Existence of convex translators for powers of Gauss curvature in (0, 1/(n+2))

Asymptotic to shrinking solitons in alculus

Level sets include spheres, simplices, hypercubes

Abstract

In this paper we construct complete convex hypersurfaces in $\mathbb R^{n+1}$ which translate under the flow by powers $\alpha \in (0, \frac1{n+2})$ of the Gauss curvature. The level set of each solution is asymptotic to a shrinking soliton for the flow by power $\frac \alpha {1-\alpha}$ of the Gauss curvature in $\mathbb R^n$. For example, our construction reveals the existence of translators whose level set converges to the sphere, simplex, hypercube and so on. The translating solitons exist as a family whose parameters correspond to Jacobi fields, solutions to linearized equation around the asymptotic profile.