The goal of the seminar is to report on recent joint work with Daniele Semola, motivated by a question of Gromov to establish a “synthetic regularity theory" for minimal surfaces in non-smooth ambient spaces.
In the setting of non-smooth spaces with lower Ricci Curvature bounds:
We establish a new principle relating lower Ricci Curvature bounds to the preservation of Laplacian bounds under the evolution via the Hopf-Lax semigroup;
We develop an intrinsic viscosity theory of Laplacian bounds and prove equivalence with other weak notions of Laplacian bounds;
We prove sharp Laplacian bounds on the distance function from a set (locally) minimizing the perimeter: this corresponds to vanishing mean curvature in the smooth setting;
We study the regularity of boundaries of sets (locally) minimizing the perimeter, obtaining sharp bounds on the Hausdorff co-dimension of the singular set plus content estimates and topological regularity of the regular set.
Optimal transport plays the role of underlying technical tool for addressing various points.