Almost Monotonicity Formula for H-minimal Legendrian Surfaces in the Heisenberg Group

TL;DR

This paper establishes an almost monotonicity formula for H-minimal Legendrian surfaces in the Heisenberg group, leading to a Bernstein-Liouville theorem that supports the analysis in the proof of the Willmore conjecture.

Contribution

It introduces an almost monotonicity formula for H-minimal Legendrian surfaces in the Heisenberg group, enabling new analytical tools for geometric analysis.

Findings

Proved an almost monotonicity formula for H-minimal Legendrian surfaces.

Derived a Bernstein-Liouville type theorem from the formula.

Supported the analysis in the proof of the Willmore conjecture.

Abstract

We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467