Optimal Transport, weak Laplacian bounds and minimal boundaries in non-smooth spaces with Lower Ricci Curvature bounds
Andrea Mondino (University of Oxford)
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.
mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control
Audience: researchers in the topic
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
Organizers: | Simon Blatt*, Philipp Reiter*, Armin Schikorra*, Guofang Wang |
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