Translating solutions to the Gauss curvature flow with flat sides

Choi, Kyeongsu; Daskalopoulos, Panagiota; Lee, Ki-Ahm

doi:10.2140/apde.2021.14.595

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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

DOI: 10.2140/apde.2021.14.595

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 $Ω$ . 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 $Ω$ , we establish the existence of a $C_{loc}^{1, 1}$ translating soliton. In particular, when the boundary $\partial Ω$ 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

Vol. 14, No. 2, 2021