A note on the two-dimensional Lagrangian mean curvature equation

Arunima Bhattacharya

Vol. 318 (2022), No. 1, 43–50

DOI: 10.2140/pjm.2022.318.43

Abstract

We use Warren–Yuan’s (Warren and Yuan 2009) superisoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two-dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike in (Bhattacharya 2021), the simplified approach in this paper does not require the Michael–Simon mean value and Sobolev inequalities on generalized submanifolds of ${ℝ}^n$ (Michael and Simon 1973).

Keywords

Lagrangian mean curvature, critical phase, supercritical phase, Hessian estimates

Mathematical Subject Classification

Primary: 35B45

Secondary: 35J60

Milestones

Received: 19 October 2021

Revised: 14 April 2022

Accepted: 16 April 2022

Published: 1 August 2022