Alexandrov immersions, holonomy and minimal surfaces in $S^3$

TL;DR

This paper establishes a parametrization of certain 3-manifolds with minimal boundary surfaces using conformal boundary data, providing new proofs and generalizations in the context of constant curvature geometries.

Contribution

It introduces a new local parametrization of 3-manifolds with minimal boundary surfaces via conformal boundary data, extending to constant mean curvature and other geometries.

Findings

Parametrization of 3-manifolds by boundary conformal class for genus ≥ 2

New proof of Brendle's solution to the Lawson conjecture for genus 1

Generalizations to constant mean curvature and other 3-manifold geometries

Abstract

We prove that compact 3-manifolds $M$ of constant curvature +1 with boundary a minimal surface are locally naturally parametrized by the conformal class of the boundary metric $\gamma$ in the Teichmuller space of $\partial M$, when $genus(\partial M) \geq 2$. Stronger results are obtained in the case of genus 1 boundary, giving in particular a new proof of Brendle's solution of the Lawson conjecture. The results generalize to constant mean curvature surfaces, and surfaces in flat and hyperbolic 3-manifolds.