Minimal surfaces and the new main inequality

TL;DR

This paper introduces the new main inequality as a criterion for minimal maps to product spaces and explores stability conditions for minimal surfaces in Euclidean spaces, providing new insights and reproofs of classical results.

Contribution

It establishes the new main inequality as a key criterion for minimal maps and offers a novel perspective on destabilizing minimal surfaces, including reproofs of classical examples.

Findings

New main inequality as a minimizing criterion for minimal maps

Infinitesimal new main inequality as a stability criterion

Reproof of the instability of classical minimal surfaces like Enneper surface

Abstract

We establish the new main inequality as a minimizing criterion for minimal maps to products of $\mathbb{R}$-trees, and the infinitesimal new main inequality as a stability criterion for minimal maps to $\mathbb{R}^n$. Along the way, we develop a new perspective on destabilizing minimal surfaces in $\mathbb{R}^n$, and as a consequence we reprove the instability of some classical minimal surfaces; for example, the Enneper surface.