We describe how some differential geometric bifurcation problems can be treated with the MATLAB continuation and bifurcation toolbox pde2path. The basic setup consists in solving the PDEs for the normal displacement of an immersed surface $X\subset\mathbb{R}^3$ and subsequent update of $X$ in each continuation step, combined with bifurcation detection and localization, followed by possible branch switching. Examples treated include some minimal surfaces such as Enneper's surface and a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid bridges and nodoids, and some 4th order biomembrane models. In all of these we find interesting symmetry breaking bifurcations. Some of these are (semi)analytically known and thus are used as benchmarks.

翻译：本文描述如何利用MATLAB延拓与分支工具箱pde2path处理若干微分几何分支问题。基本设置包括求解浸入曲面$X\subset\mathbb{R}^3$的法向位移偏微分方程，并在每个延拓步中更新$X$，同时进行分支检测与定位，随后实现可能的支路切换。处理的示例包括若干极小曲面（如恩内珀曲面和Schwarz-P族）、若干非零常平均曲率曲面（如液桥和结环曲面），以及若干四阶生物膜模型。在这些问题中，我们均发现了有趣的对称性破缺分支现象。其中部分分支具有（半）解析已知形式，因此被用作基准测试。