On the index of constant mean curvature 1 surfaces in hyperbolic space

TL;DR

This paper establishes a complete determination of the index of constant mean curvature 1 surfaces in hyperbolic space using their Riemann surface and Gauss map, with applications to explicit index computation and bounds.

Contribution

It introduces a method to compute and estimate the index of such surfaces based on their Bryant representation, providing new insights and proofs.

Findings

Explicit index calculations for catenoid cousins and examples

Lower bounds for the index of certain CMC 1 surfaces

A direct proof that finite total curvature implies finite index

Abstract

We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some other examples. Secondly, it allows us to be able to apply a method similar to that of Choe (using Killing vector fields on minimal surfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a constant mean curvature 1 surface in hyperbolic 3-space has finite total curvature, then it has finite index. Finally, we show that for any constant mean curvature 1 surface in hyperbolic 3-space that has been constructed via a correspondence to a minimal surface in Euclidean 3-space, we can take advantage of this correspondence to find a lower bound for its index.