  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.
