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

Abstract: 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.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic

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Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

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