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

Understand

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 $Y$.