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 C2 estimates for complete noncompact translating solitons of the Gauss curvature flow in 3 which are graphs over a convex domain Ω. This is closely is related to deriving local C1,1 estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain Ω, we establish the existence of a Cloc1,1 translating soliton. In particular, when the boundary Ω 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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Keywords
Gauss curvature flow, translator, regularity, free boundary
Mathematical Subject Classification 2010
Primary: 53C44
Milestones
Received: 20 November 2018
Revised: 9 September 2019
Accepted: 25 October 2019
Published: 20 March 2021
Authors
Kyeongsu Choi
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Korea Institute for Advanced Study
Seoul
South Korea
Panagiota Daskalopoulos
Department of Mathematics
Columbia University
New York, NY
United States
Ki-Ahm Lee
Department of Mathematical Sciences
Seoul National University
Seoul
South Korea
Korea Institute for Advanced Study
Seoul
South Korea