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
2017-09-29

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: http://hdl.handle.net/20.500.11767/76180
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