Handlebodies with constant curvature metrics and minimal surface boundary.
- Rio de Janeiro: IMPA, 2015.
- video online
We study the moduli space of constant curvature metrics g on a 3-d handlebody with boundary having mean curvature 0, so minimal surface boundary (or more generally CMC boundary). This is a generalization of Alexandrov immersed minimal surfaces in 3-d space forms. We prove that this moduli space is a smooth manifold, locally diffeomorphic to the Teichmuller space of the boundary surface, when the genus of the boundary is at least 2. We conjecture that the spaces are in fact diffeomorphic (on each component). This result is false per se for genus 1 boundaries, but the method of proof gives rise to a new proof of Brendle's solution of the Lawson conjecture on embedded minimal tori in S^3. The talk will discuss the context and basic ideas of the proof. We hope to discuss relations and/or questions with hyperbolic 3-manifolds .