Minimal surfaces and the new main inequality (Q6562479)

Let \(S\) be a Riemann surface and let \(\phi_1,\ldots, \phi_n\) be integrable holomorphic quadratic differentials on \(S\) whose sum is zero, and \(f_1, \ldots, f_n : S\to S'\) mutually homotopic quasiconformal maps to another Riemann surface with Beltrami forms \(\mu_1,\ldots, \mu_n\). The ``new main inequality'' holds if\N\[\N\mathrm{Re} \sum_{i=1}^{n}\int_S\phi_i.\frac{\mu_i}{1-\vert\mu_i\vert^2} \leq \sum_{i=1}^{n}\int_S\vert \phi \vert. \frac{\vert\mu_i\vert^2}{1-\vert\mu_i\vert^2} .\N\]\NThis was introduced by the first author as a tool to study minimal surfaces in products of hyperbolic surfaces in [Bull. Lond. Math. Soc. 53, No. 4, 1196--1204 (2021; Zbl 1478.53070); Geom. Funct. Anal. 32, No. 1, 31--52 (2022; Zbl 1492.53081)]. In the paper under review, the authors use the new main inequality as a minimizing criterion for minimal maps into products of \(\mathbb{R}\)-trees. The infinitesimal version of the new main inequality is established as a stability criterion for minimal maps to \(\mathbb{R}^n\). The authors also develop a new perspective on destabilizing minimal surfaces in \(\mathbb{R}^n\). As an application, they give a new proof of the instability of some classical minimal surfaces, including the most well known unstable minimal surface, namely the Enneper surface.