Vol. 14, No. 2, 2021

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Translating solutions to the Gauss curvature flow with flat sides

Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee

Vol. 14 (2021), No. 2, 595–616
Abstract

We derive local ${C}^{2}$ estimates for complete noncompact translating solitons of the Gauss curvature flow in ${ℝ}^{3}$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local ${C}^{1,1}$ estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain $\Omega$, we establish the existence of a ${C}_{loc}^{1,1}$ translating soliton. In particular, when the boundary $\partial \Omega$ has line segments, we show the existence of flat sides of the translator from a local a priori nondegeneracy estimate near the free boundary.

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