Título: Harmonic functions on spaces with Ricci curvature bounded below

Palestrante: Jesús Núñez-Zimbrón (CIMAT)

Data: 04/01/2022
Horário: 14:00h
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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Título: Gradient map for the action of a real reductive Lie group

Palestrante: Leonardo Biliotti (Università di Parma)
Data: 19/10/2021
Horário: 10:00h
Local: Transmissão online

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ID da reunião: 824 5727 0949

Resumo: We study the action of a real reductive group G on a real submanifold X of a Kahler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group G^\C and that with respect to a compatible maximal compact subgroup U of G^\C the action on Z is Hamiltonian. There is a corresponding gradient map μ : X → p where g = k⊕p is a Cartan decomposition of g. Using an Ad(K)-invariant inner product we obtain the norm square of the gradient map. In this talk we investigate convexity properties of the gradient map. We also describe compact orbits of a parabolic subgroup of G. Finally, we investigate the norm square of the gradient map. As an application we prove that a norm square of a two orbit variety M is Morse-Bott obtaining results on the cohomology and the K-invariant cohomology of M.  A part of this talk is a joint work with my PhD student Joshua Windare (arXiv:2106.13074, arXiv:2105.05765 and arXiv:2012.14858).

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Título: An isoperimetric interpretation for the renormalized volume of convex co-compact hyperbolic 3-manifolds

Palestrante: Celso Viana (UFMG)

Data: 26/10/2021
Horário: 14:00h
Local: Transmissão online

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ID da reunião: 878 0813 6599

Resumo: In this we will discuss an important class of hyperbolic 3-manifolds known as quasi-Fuchsian 3-manifolds and the notion of renormalized volume in these spaces. We will address some aspects of the isoperimetric problem in these manifolds and present a characterization of the renormalized volume in terms of isoperimetric data at infinity.

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Título: Sharp solvability criteria for Dirichlet problems of mean curvature type in

Palestrante: Yunelsy N. Alvarez (Universidade de São Paulo)
Data: 28/09/2021
Horário: 14:00h
Local: Transmissão online

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ID da reunião: 898 9580 4229

Resumo: In this talk, we investigate the existence of graphs with prescribed mean curvature in Riemannian manifolds. Specifically, we show that a condition - inherited from the Euclidean setting - is sharp for the solvability of the Dirichlet problem for prescribed mean curvature equations in a large class of manifolds.

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