Translating solutions to the Gauss curvature flow with flat sides

2016/10/23 by Choi, Kyeongsu, Daskalopoulos, Panagiota, Lee, Ki-Ahm · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.07206

Abstract

We derive local C² estimates for complete non-compact translating solitons of the Gauss curvature flow in ℝ³ which are graphs over a convex domain Ω. This is closely is related to deriving local C^1,1 estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded domain Ω, we establish the existence of a C^1,1_loc translating soliton. In particular, when the boundary ∂ Ω has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.

Cited by

• Ancient Gauss Curvature Flows of Bounded Width
• Invariant λ-translators for the Gauss curvature flow in Euclidean space

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• On existence of hypersurfaces translating by powers of Gauss curvature