Bryant surfaces with smooth ends

A smooth end of a Bryant surface is a conformally immersed punctured disc ofmean curvature $1$ in hyperbolic space that extends smoothly through theideal boundary. The Bryant representation of a smooth end is well defined onthe punctured disc and has a pole at the puncture. The Willmore energy ofcompact Bryant surfaces with smooth ends is quantized. It equals $4\pi$times the total pole order of its Bryant representation. The possibleWillmore energies of Bryant spheres with smooth ends are $4\pi(\N^*\setminus\{2,3,5,7\})$. Bryant spheres with smooth ends are examples of solitonspheres, a class of rational conformal immersions of the sphere which alsoincludes Willmore spheres in the conformal 3-sphere $S^3$. We give explicitexamples of Bryant spheres with an arbitrary number of smooth ends. Weconclude the paper by showing that Bryant's quartic differential $\mathcal Q$vanishes identically for a compact surface in $S^3$ if and only if it is thecompactification of either a complete finite total curvature Euclideanminimal surface with planar ends or a compact Bryant surface with smoothends.

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Published 1 January 2009