Critical points of the solution to the $H_R=H_L$ surface equation

TL;DR

This paper investigates the critical points of solutions to the elliptic $H_R=H_L$ surface equation, providing geometric conditions, uniqueness results, and bounds on the domain inradius related to the equation's solutions.

Contribution

It introduces a necessary geometric condition for critical points, establishes a new uniqueness theorem for the Dirichlet problem, and improves bounds on the inradius of solution domains.

Findings

Necessary geometric condition for critical points

New uniqueness result for the Dirichlet problem

Improved Heinz-type inradius bound

Abstract

Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and $\mathbb{L}^3$ are locally described as the graph of the solutions to the $H_R=H_L$ surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the $H_R=H_L$ surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class $\mathcal{C}^2$ in terms of the curvature of its level curves.