This thesis is primarily devoted to the study of analytic and geometric properties of metric measure spaces with a Ricci curvature bounded from below. The first result concerns the study of how a hypothesis on the Hodge cohomology affects the rigidity of a metric measure space with non negative Ricci curvature and finite dimension: we prove that if the dimension of the first cohomology group of a RCD∗(0,N) space is N, then the space is a flat torus. This generalizes a classical result in Riemannian geometry due to Bochner to the non-smooth setting of RCD spaces. The second result provides a direct proof of the strong maximum principle on finite dimen- sional RCD spaces mainly based on the Laplacian comparison of the squared distance.

Cohomology and other analytical aspects of RCD spaces / Rigoni, Chiara. - (2017 Sep 29).

Cohomology and other analytical aspects of RCD spaces

Rigoni, Chiara
2017

Abstract

This thesis is primarily devoted to the study of analytic and geometric properties of metric measure spaces with a Ricci curvature bounded from below. The first result concerns the study of how a hypothesis on the Hodge cohomology affects the rigidity of a metric measure space with non negative Ricci curvature and finite dimension: we prove that if the dimension of the first cohomology group of a RCD∗(0,N) space is N, then the space is a flat torus. This generalizes a classical result in Riemannian geometry due to Bochner to the non-smooth setting of RCD spaces. The second result provides a direct proof of the strong maximum principle on finite dimen- sional RCD spaces mainly based on the Laplacian comparison of the squared distance.
Gigli, Nicola
Rigoni, Chiara
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/76180
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