Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds

Abstract

If $Y$ is a properly embedded minimal surface in a convex cocompact hyperbolic 3-manifold $M$ with boundary at infinity an embedded curve $\gamma$, then Graham and Witten showed how to define a renormalized area $\calA$ of $Y$ via Hadamard regularization. We study renormalized area as a functional on the space of all such minimal surfaces. This requires a closer examination of these moduli spaces; following White and Coskunuzer, we prove these are Banach manifolds and that the natural map taking $Y$ to $\gamma$ is Fredholm of index zero and proper, which leads to the existence of a $\ZZ$-valued degree theory for this mapping. We show that $\calA(Y)$ can be expressed as a sum of the Euler characteristic of $Y$ and the total integral of norm squared of the trace-free second fundamental form of $Y$. An extension of renormalized area to a wider class of nonminimal surfaces has a similar formula also involving the integral of mean curvature squared. We prove a formula for the first variation of renormalized area, and characterize the critical points when $M = \HH^3$ and $\gamma$ has a single component. All of these results have analogues for 4-dimensional Poincar\'e-Einstein metrics. We conclude by discussing the relationship of $\calA$ to the Willmore functional.