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Harmonic functions on spaces with Ricci curvature bounded below

Banff International Research Station for Mathematical Innovation and Discovery

The so-called spaces with the Riemannian curvature-dimension condition (RCD spaces for short) are metric measure spaces which are non-necessarily smooth but admit a notion of "Ricci curvature bounded below and dimension bounded above". These arise naturally as Gromov-Hausdorff limits of Riemannian manifolds with these conditions and, in contrast to manifolds, RCD spaces may have topological or metric singularities. Nevertheless, several properties and results from Riemannian geometry can be extended to this non-smooth setting. In this talk I will present recent work, joint with Guido de Philippis, in which we show that the gradients of harmonic functions vanish at the singular points of the space. I will mention two consequences of this result on smooth manifolds: it implies that there does not exist an a priori estimate on the modulus of continuity of the gradient of harmonic functions depending only on lower bounds of the sectional curvature and that there is no a priori Calderón-Zygmund inequality for the Laplacian with bounds that depend only on lower bounds of the sectional curvature.

Mathematics; Global Analysis; Analysis On Manifolds; Differential Geometry; Geometric Analysis

video/mp4

Author affiliation: Centro de Investigación en Matemáticas

2023-10-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86343

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/