https://neckpinch.com/authors/andrea-mondino/New arXiv papers by Andrea Mondino, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1704.05428/arxiv:1704.05428Tue, 18 Apr 2017 00:00:00 GMTFernando Galaz-García, Martin Kell, Andrea Mondino, Gerardo Sosa — Let (M,g) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional curvature of (M,g) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of RCD^(*)(K,N)-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An RCD^(*)(K,N)-space is a metric measure space with an upper dimension bound N and weighted Ricci curvature bounded below by K in a generalized sense. In particular, this shows that if (M,g) has Ricci curvature bounded below by K∈ ℝ and dimension N, then the quotient space is an RCD^(*)(K,N)-space. Additionally, we tackle the same problem for the CD/CD^* and MCP curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to RCD^(*)(K,N)-spaces with essential minimal dimension n≤ N; a structure theorem for RCD^(*)(K,N)-spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows.Ricci flowWeak & singular flowsLimit spaces & RCDHomogeneous spacesStability & uniqueness https://neckpinch.com/papers/1703.02119/arxiv:1703.02119Mon, 06 Mar 2017 00:00:00 GMTFabio Cavalletti, Andrea Mondino — Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.Ricci flowWeak & singular flowsLimit spaces & RCDGeometric inequalities https://neckpinch.com/papers/1603.05263/arxiv:1603.05263Wed, 16 Mar 2016 00:00:00 GMTAndrea Mondino, Emanuele Spadaro — The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan-Hadamard spaces as well as on minimal submanifolds of ℝⁿ. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with non-negative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean. We then pass to consider the problem of the existence and the regularity of optimal shapes (i.e. regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. We give examples of spaces where there exists no isoperimetric-isodiametric region and we prove that on the other hand on compact spaces and on C⁰-locally-asymptotic Euclidean manifolds with non-negative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this last class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so called Hawking gravitational instantons and the Bryant-type Ricci solitons). We finally prove that the boundary of the isoperimetric-isodiametric regions satisfy the optimal C^(1,1) regularity outside of a closed subset of Hausdorff co-dimension 8.Ricci flowRicci solitonsRicci-flat & Calabi–YauFree boundary & capillaryGeometric inequalitiesMinimal & CMC surfacesPhysics