Bryant Surfaces with Smooth Ends

TL;DR

This paper studies Bryant surfaces with smooth ends in hyperbolic space, revealing their energy quantization, explicit examples, and their relation to soliton spheres and minimal surfaces.

Contribution

It introduces the concept of smooth ends for Bryant surfaces, establishes energy quantization, provides explicit examples, and links Bryant surfaces to soliton spheres and minimal surfaces.

Findings

Willmore energy quantized as 4Pi times pole order

Explicit Bryant spheres with arbitrary smooth ends

Characterization of surfaces with vanishing Bryant quartic Q

Abstract

A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured disc and has a pole at the puncture. The Willmore energy of compact Bryant surfaces with smooth ends is quantized. It equals 4Pi times the total pole order of its Bryant representation. The possible Willmore energies of Bryant spheres with smooth ends are 4Pi*d where d is a positive integer different from 2,3,5,7. Bryant spheres with smooth ends are examples of soliton spheres, a class of rational conformal immersions of the sphere which also includes Willmore spheres in the conformal 3-sphere. We give explicit examples of Bryant spheres with an arbitrary number of smooth ends. We conclude the paper by showing that Bryant's quartic differential Q vanishes identically for a compact surface in the 3-sphere if and only if it is the compactification of either a complete finite total curvature Euclidean minimal surface with planar ends or a compact Bryant surface with smooth ends.