Communications in Analysis and Geometry

Volume 27 (2019)

Number 4

A minimum principle for Lagrangian graphs

Pages: 857 – 876

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n4.a4

Authors

Tamás Darvas (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Yanir A. Rubinstein (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Abstract

The classical minimum principle is foundational in convex and complex analysis and plays an important rôle in the study of the real and complex Monge–Ampère equations. This note establishes a minimum principle in Lagrangian geometry. This principle relates the classical Lagrangian angle of Harvey–Lawson and the space-time Lagrangian angle introduced recently by Rubinstein–Solomon. As an application, this gives a new formula for solutions of the degenerate special Lagrangian equation in space-time in terms of the (time) partial Legendre transform of a family of solutions of obstacle problems for the (space) non-degenerate special Lagrangian equation.

Received 29 June 2016

Accepted 7 April 2017

Published 8 October 2019