Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. De Rosa

The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

created by derosa on 23 Apr 2022
modified on 24 May 2022


Submitted Paper

Inserted: 23 apr 2022
Last Updated: 24 may 2022

Year: 2022


We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.