Calculus of Variations and Geometric Measure Theory

B. Bianchini - G. Colombo - M. Magliaro - L. Mari - P. Pucci - M. Rigoli

Recent rigidity results for graphs with prescribed mean curvature

created by mari1 on 25 Nov 2020
modified on 30 Aug 2021



Inserted: 25 nov 2020
Last Updated: 30 aug 2021

Journal: Math. Eng.
Volume: 3
Number: 5
Pages: 48
Year: 2021

ArXiv: 2007.07194 PDF


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 $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.