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We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
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In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters bounded by c. We prove that an n-dimensional Riemannian manifold M with at least n-k eigenvalues of the Ricci curvature bounded below by a positive constant (n-1)b has k-dimensional Uryson width bounded by a constant c >0. The constant c depends only on b. In particular, it follows that a Riemannian n-manifold M with scalar curvature S bounded below by a positive constant n (n-1) s has (n-1)-dimensional Uryson width bounded by a constant c >0 depending only on s. This result confirms a conjecture of M. Gromov.
Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $mathbb{R}P^2$. We then classify closed three-dimensional $mathsf{CD}^*(0,3)$-Alexandrov spaces.
In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.
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