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arXiv · 1610.07206

Translating solutions to the Gauss curvature flow with flat sides

Kyeongsu Choi·Panagiota Daskalopoulos·Ki-Ahm Lee
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Abstract

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ 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 $C^{1,1}_{\text{loc}}$ translating soliton. In particular, when the boundary $\partial Ω$ 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.

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BibTeXRIS

Kyeongsu Choi, Panagiota Daskalopoulos, Ki-Ahm Lee. 2018-10-04. Translating solutions to the Gauss curvature flow with flat sides. https://arxiv.org/abs/1610.07206

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