The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains

On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves

For a smooth closed embedded planar curve $Γ$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\geq1$ having the curve $Γ$ as boundary, without any prescription on the conormal. By general lower bound estimates, in case $Γ$ is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and the infimum equals $β_\mathfrak{g}-4π$, where $β_\mathfrak{g}$ is the energy of the closed minimizing surface of genus $\mathfrak{g}$. We also prove that the same result also hol…

面积最小化流的奇点分析，第三部分：平面频率≠2的分支点、高阶渐近性和局部拓扑

Analysis of singularities of area-minimizing currents, Part III: branch points of planar frequency $\neq$ 2, higher order asymptotics, and the local topology

该论文为研究n维面积最小化可求长流T的局部结构，分析平面频率≠2的分支点，得到高阶渐近性、分支集分解及拓扑控制结果，避免使用中心流形，基于平面频率单调性公式论证，为后续研究奠定基础。

What are the shapes of embedded minimal surfaces and why?

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The study is divided into three cases, depending on the topology of the surface. Case one is where the surface is a disk, in case two the surface is a planar domain (genus zero), and the third case is that of finite (non-zero) genus. The complete understanding of the disk case is appli…

高亏格曲面上的调和函数计算

Computation of harmonic functions on higher genus surfaces

本文将Bernstein-Walsch平面调和函数逼近定理推广至Riemann曲面，精确刻画逼近速率，并基于双曲多边形粘合表示提出高效数值求解Laplace边界值问题的方法。