Título: Harmonic functions on spaces with Ricci curvature bounded below
Palestrante: Jesús Núñez-Zimbrón (CIMAT)
Local: Transmissão online
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ID da reunião: 811 6291 5241
Resumo:The so-called spaces with the Riemannian curvature-dimension conditions (RCD spaces) are metric measure spaces which are not necessarily smooth but admit a notion of “Ricci curvature bounded below and dimension bounded above”. These spaces arise naturally as Gromov-Hausdorff limits of Riemannian manifolds with these conditions and, in contrast to manifolds, RCD spaces typically have topological or metric singularities. Nevertheless a considerable amount of Riemannian geometry can be recovered for these spaces. In this talk I will present recent work joint with Guido De Phillipis, in which we show that the gradients of harmonic functions vanish at the singular points of the space. I will mention two applications 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 there is no a priori Calderón-Zygmund inequality for the Laplacian with bounds depending only on the sectional curvature.
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