21 pages, 2 figures; v2: 2 examples and 2 figures added, references added, minor changes in presentation; v3: 26 pages, 5 figures, new introduction, abstract, and examples. Additional section on a characterization in terms of Bryant's quartic differential
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.
Bryant \cite{Bryant84} classified all Willmore spheres in $3$-space to be given by minimal surfaces in $\mathbb R^3$ with embedded planar ends. This note provides new explicit formulas for genus 0 minimal surfaces in $\mathbb R^3$ with $2k+1$ embedded planar ends for all $k\geq4.$ Peng and Xiao claimed these examples to exist in \cite{PengXiao2000}, but in the same paper they also claimed the existence of a minimal surface with 7 embedded planar ends, which was falsified by Bryant \cite{Bryant88}.
In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-Émery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $ε=ε(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\leε$. When $\mathrm{Ric}_f\ge\frac 12$ and $f(x)\le\frac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion.
Hyperbolic ends with particles and grafting on singular surfaces
我们证明了任何双曲结束粒子（锥奇点沿无穷曲线的角度小于$π$）承认一个唯一的叶理常数高斯曲率曲面。利用带有粒子的双曲端点和带有粒子的凸全局双曲极大（GHM）de Sitter时空之间的对偶形式，可以得出任何带有粒子的凸GHM de Sitter时空也允许唯一的常高斯曲率面叶理。证明了Teichmüller空间与测度层空间的乘积到复射影结构空间的嫁接映射对于锥奇点的角度小于$π$的曲面是同胚的，并且当嫁接被“光滑嫁接”代替时也是类似的.
