Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

By Luigi Ambrosio, Nicola Gigli, Andrea Mondino, T. Rajala. 2012

253 citations recorded by the source corpus.

In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/cd479ae666aba1c7bd40c0de7ae95aed517ee6b8

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Citations in the corpus

The globalization theorem for the Curvature-Dimension condition (2016, background): “In the infinite dimensional case N = ∞, it was introduced in [7] for finite measures m and in [4] for σ-finite ones.”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, background): “Afterwards, Ambrosio-Gigli-MondinoRajala extended this notion to the non-compact metric spaces with σ-finite measures [3].”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “We remark that in [37] the authors refer to [27], [3], [6] and [51] for the various calculus rules and that in these latter references no properness assumption is made.”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, methodology): “After the introduction of the so-called curvature-dimension condition CD(K,N) in the seminal and independent works [47, 48] and [39], the notion of RCD(K,N) space was proposed in [28] after the study of its infinite-dimensional counterpart RCD(K,∞) in [5] (see also [4] for the case of σ-finite refer”
On quotients of spaces with Ricci curvature bounded below (2017, background): “In order to isolate Riemannian from Finslerian structures, Ambrosio-Gigli-Savaré [4] proposed to strengthen the classical CD conditions with the requirement that the heat flow is linear (see [6] for the present simplified axiomatization, and [8, 9, 28, 33] for the finite dimensional refinements).”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “We also mention the more recent Riemannian curvature dimension condition RCD introduced in the infinite dimensional case in [3, 5, 2] and then investigated by various authors in the finite dimensional refinement.”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “spaces, which were introduced and first studied in [9] (see also [6], which deals with the case of -finite reference measures).”