Translating solutions to the Gauss curvature flow with flat sides

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

doi:10.2140/apde.2021.14.595

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1948-206X (e-only)

ISSN: 2157-5045 (print)

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/apde

We have not been able to recognize your IP address 44.192.65.228 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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