Alexandrov immersions, holonomy and minimal surfaces in S³

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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 γ in the Teichmuller space of ∂ M, when genus(∂ M) ≥ 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.