Abstract:In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD * (K, N) spaces) with non-empty one dimensional regular sets.In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdor dimension N and the class of RCD * (K, N) spaces coincide for N < (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any RCD spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such RCD * (K, N ) spaces are exactly N . We also prove that every tangent cone at any point on such RCD spaces is a metric cone.

In this paper, we study regular sets in metric measure spaces with bounded Ricci curvature. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also we define the dimension of metric measure spaces and prove the lower semicontinuity of that under the Gromov-Hausdorff convergence.2010 Mathematics Subject Classification. 51F99(primary), and 53C20(secondary).

In the present paper, we introduce a concept of Ricci curvature on hypergraphs for a nonlinear Laplacian. We prove that our definition of the Ricci curvature is a generalization of Lin-Lu-Yau's coarse Ricci curvature for graphs to hypergraphs. We also show a lower bound of nonzero eigenvalues of Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understanding how nonlinearity of Laplacian causes complexity of curvatures.

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