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

Michael T Anderson

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arXiv · 1407.6925

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

Michael T Anderson

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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.

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Michael T Anderson. 2014-07-25. Alexandrov immersions, holonomy and minimal surfaces in $S^3$. https://arxiv.org/abs/1407.6925

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