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

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 $genus(\partial M)\ge 2$. 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.

arXiv commentwithdrawn for reconstruction. Error in the "stability argument" on p.22

References in corpus (6)

• Unique continuation results for Ricci curvature and applications
• Embedded minimal tori in S^3 and the Lawson conjecture
• New minimal surfaces in S^3 desingularizing the Clifford tori
• Continuation unique a partir de l'infini conforme pour les metriques d'Einstein
• Extension of symmetries on Einstein manifolds with boundary
• Harmonic deformations of hyperbolic 3-manifolds