Principal Investigator: Antonio De Rosa
The focus of this project is to advance the theory of anisotropic geometric variational problems. A vast literature is devoted to the study
of critical points of the area functional, referred to as minimal surfaces. However, minimizing the surface area is often an idealization
in physics. In order to account for preferred inhomogeneous and directionally dependent configurations and to capture microstructures,
more general anisotropic energies are often utilized in several important models. Relevant examples include crystal structures, capillarity
problems, gravitational fields and homogenization problems. Motivated by these applications, anisotropic energies have attracted an
increasing interest in the geometric analysis community. Moreover in differential geometry they lead to the study of Finsler manifolds.
Unlike the rich theory for the area functional, very little is understood in the anisotropic setting, as many of the essential techniques do
not remain valid. This project aims to develop the tools to prove existence, regularity and uniqueness properties of the critical points of
anisotropic functionals, referred to as anisotropic minimal surfaces. In order to show their existence in general Riemannian manifolds,
it will be crucial to generalize the min-max theory. This theory plays a crucial role in proving a number of conjectures in geometry and
topology. In order to determine the regularity of anisotropic minimal surfaces, I will study the associated geometric nonlinear elliptic
partial differential equations (PDEs). Finally, in addition to the stationary configurations, this research will shed light on geometric flows,
through the analysis of the related parabolic PDEs. The new methods developed in this project will provide new insights and results even
for the isotropic theory: in solving the size minimization problem, in the vectorial Allen-Cahn approximation of the general codimension
Brakke flow, and in the Almgren-Pitts min-max construction.