Tori in $S^3$ minimizing locally the conformal volume

TL;DR

This paper characterizes conformal immersions of complex tori into the 3-sphere that locally minimize conformal volume, showing they satisfy specific elliptic PDEs and are either minimal, CMC flat, or conformally constrained minimal tori.

Contribution

It classifies all conformal immersions of tori into S^3 that minimize conformal volume, identifying their geometric and PDE properties within a unified weak immersion framework.

Findings

Conformal volume minimizers satisfy certain elliptic PDEs.

Critical points of area within a conformal class are either minimal or flat CMC tori.

Degenerate conformal points are characterized as minimal or flat CMC surfaces.

Abstract

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fixed conformally congruent area. On the way to establish this result we prove that tori which are critical points of the area for perturbations within a given conformal class and which are degenerate points of the conformal class mapping - i.e. isothermic - are either minimal surfaces or flat CMC tori. These results are all proved in the general framework of weak immersions.