Recent progress on the Dirichlet problem for the minimal surface system and minimal cones

An Observation on the Dirichlet problem at infinity in Riemannian cones

In this short paper we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below).This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied tosmooth Riemannian manifolds with a special type of metrics (which generalise rotational symmetry) we obtain generalisations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODE's.

补遗：闭曲面的情况。（可平面图和具有整锥奇点的平坦曲面上的边值问题，I：狄利克雷问题）

Addendum: the case of closed surfaces. (Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem)

我们将平面$m$连通Jordan域的离散一致化定理[Journal für die reine und angewandte Mathematik 670（2012），65- 92]扩展到非正亏格的闭曲面。

Determinant of Friederichs Dirichlet Laplacians on $2$-dimensional hyperbolic cones

We explicitly express the spectral determinant of Friederichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary. The related results in the recent paper "Riemann-Roch isometries in the non-compact orbifold setting," J. Eur. Math. Soc. 22 (2020) by G. Freixas i Montplet and A. von Pippich turn out to be incorrect.

可平面图和平面上具有整锥奇点的边值问题，II：混合Dirichlet-Neumann问题

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

在本文中，我们继续在第一部分（张贴）开始的研究。考虑一个平面的有界$m$连通区域$Ω$，设$\bordΩ$为其边界.令$\mathcal{T}$是$Ω\cup\bordΩ$的胞元分解，其中每个2-胞元是三角形或四边形。从这些数据和电导函数，我们构建了一个典型的对$(S,f)$ $S$是一个特殊类型的（可能浸入）属$(m-1)$奇异平面，平铺矩形和$f$是一个能量保持映射到$S$从${\mathcal T}^{(1)}$。在第一部分中利用了定义在${\mathcal T}^{(0)}$上的Dirichlet问题的解，在本文中我们采用了混合Dirichlet-Neumann问题的解。

具有整锥奇点可平面图和平坦曲面上的边值问题，I：狄利克雷问题

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem

考虑一个平面的有界$m$连通区域$Ω$，设$\bordΩ$为其边界.令$\mathcal{T}$是$Ω\cup\bordΩ$的胞元分解，其中每个2-胞元是三角形或四边形。从这些数据和电导函数，我们构建了一个典型的对$(S,f)$，其中$S$是一个属$(m-1)$奇异平面平铺矩形和$f$是一个能量保持映射从${\mathcal T}^{(1)}$到$S$。

On instability of Type (II) Lawson-Osserman Cones

We obtain the instability of Type (II) Lawson-Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) unstable solutions with singularity to the Dirichlet problem for minimal graphs of high codimension versus smooth unstable ones by Lawson-Osserman through a min-max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs and unlikely detectible through the mean curvature flow or min-max theory.