Ricci Curvature on Alexandrov spaces and Rigidity Theorems
By Hui-Chun Zhang, Xiping Zhu. 2009
124 citations recorded by the source corpus.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
Subfields: Applied Mathematics, Astronomy and Astrophysics
View Paper: https://www.semanticscholar.org/paper/38a1bedeeda59577aea71ec280d154ef18986557
How researchers see this paper
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In Applied Mathematics
- Luigi Ambrosio
- Andrea Mondino
- Nicola Gigli
- Fabio Cavalletti
- Shouhei Honda
- Matthias Erbar
- Karl‐Theodor Sturm
- Kazumasa Kuwada
In Astronomy and Astrophysics
- Andrea Mondino
- Luigi Ambrosio
- Fabio Cavalletti
- Nicola Gigli
- Shouhei Honda
- Matthias Erbar
- Karl‐Theodor Sturm
- Kazumasa Kuwada
Citations in the corpus
- Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, result): “Petrunin [58] proved that the lower curvature bound in the sense of comparison angles is compatible with the optimal transport type lower bound on the Ricci curvature given by Lott-Sturm-Villani (see also [67]).”
- Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “The final statement follows from the compatibility between Alexandrov spaces and RCD-spaces [45,56].”
- Yau's gradient estimates on Alexandrov spaces (2010, methodology): “By the conditionRic> −K and Laplacian comparison (see [51] or [22]), we have L|xxo |2 6 c(n,K, diamΩ) and L|yyo|2 6 c(n,K, diamΩ).”
- On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “The curvature-dimension condition CD(K,N) has been verified for Riemannian manifolds [39], Finsler spaces [31], Alexandrov spaces [35], [42], cones [8] and warped products of Riemannian manifolds [24].”
- The Splitting Theorem in Non-Smooth Context (2013, unknown): “• An isometric splitting for Alexandrov spaces with curvature bounded from below (possibly by a negative number) with non-negative Ricci curvature, by Zhang and Zhu in [68].”