On existence of hypersurfaces translating by powers of Gauss curvature

2022/04/19 by Beomjun Choi, Choi, Beomjun
Mathematics · Physics and Astronomy · #35J96 #53E99 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2204.09002

openalex publication_date 2022/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we construct complete convex hypersurfaces in \mathbb Rⁿ⁺¹ which translate under the flow by powers α∈ (0, \frac1n+2) of the Gauss curvature. The level set of each solution is asymptotic to a shrinking soliton for the flow by power \frac α1-α of the Gauss curvature in \mathbb Rⁿ. 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.

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• Translating surfaces under flows by sub-affine-critical powers of Gauss curvature
• Gauss map of translating solitons of mean curvature flow
• Convexity and the Dirichlet problem of translating mean curvature flows
• Continuous family of surfaces translating by powers of Gauss curvature