On the Lawson-Osserman conjecture

TL;DR

This paper proves that Lipschitz critical points of the area functional in the plane are smooth, confirming a long-standing conjecture by Lawson and Osserman from 1977.

Contribution

It establishes the smoothness of Lipschitz area-critical points in the planar case, resolving the Lawson-Osserman conjecture.

Findings

Lipschitz critical points are smooth in the planar case

Solves the Lawson-Osserman conjecture from 1977

Advances understanding of minimal surface regularity

Abstract

We prove that if $u : B_1 \subset \mathbb{R}^2 \rightarrow \mathbb{R}^n$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.