论文标题
具有规定平均曲率的图表的最新刚度结果
Recent rigidity results for graphs with prescribed mean curvature
论文作者
论文摘要
这项调查描述了作者在图$ u上为规定的平均曲率问题获得的一些最新刚度结果：m \ rightarrow \ mathbb {r} $。在扭曲的产品环境空间中，重点放在最小，CMC和毛细管图以及平均曲率流的图形孤子上。给出了平均曲率算子的详细分析，重点是无限，liouville属性，梯度估算的最大原则。在几何应用中，我们提到了伯恩斯坦定理，用于具有非负RICCI曲率的歧管上的正整体最小图，以及在没有结合的域$ω\ subset m $上的毛细管图的分裂定理，即CMC图形满足过度确定的边界条件。
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $Ω\subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.
