Andrea Mondino - Optimal Transport, weak Laplacian bounds and minimal boundaries in non-smooth spaces with Lower Ricci Curvature bounds

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.

Tuesday, October 19, 2021 - 13:00
Speaker Information
Andrea Mondino

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