Accepted Paper

Inserted: 12 aug 2026
Last Updated: 12 aug 2026

Journal: Séminaires et Congrès, Société Mathématique de France
Year: 2026

Lecture notes for the mini-course given by the author at the Winter School in Geometric Measure Theory at Westlake University in January 2025. To appear in the series Séminaires et Congrès of the Société Mathématique de France

Abstract:

For several natural phenomena, the use of the surface area functional is a first approximation. In order to capture microstructures, numerous models in applied sciences employ directionally dependent functionals, known as anisotropic energies. Since anisotropic energies are not invariant under rigid motions, their critical points do not enjoy the same conservation laws as isotropic minimal surfaces. For instance, the monotonicity formula for the density ratio is not known to hold for minimizers of general anisotropic energies. Consequently, the study of anisotropic minimal surfaces is more challenging than the study of their isotropic counterparts. In this mini-course, we give an overview of the state of the art in the existence and regularity theory of anisotropic minimal surfaces. In particular, we first focus on solutions of the anisotropic Plateau problem and subsequently move to the investigation of the rectifiability and regularity theory for critical points of anisotropic energies. To conclude, we provide applications to the min-max theory for the construction of closed optimally regular anisotropic minimal hypersurfaces in closed Riemannian manifolds.

Tags: ANGEVA