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Andrea Mondino

Publications and source records attributed to Andrea Mondino.

At least 73 records · Page 4Linked to original sources
arXiv2017

On quotients of spaces with Ricci curvature bounded below

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ 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 $\mathsf{RCD}^{*}(K,N)$-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An $\mathsf{RCD}^{*}(K,N)$-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 $K\in \mathbb{R}$ and dimension $N$, then the quotient space is an $\mathsf{RCD}^{*}(K,N)$-space. Additionally, we tackle the same problem for the $\mathsf{CD}/\mathsf{CD}^*$ and $\mathsf{MCP}$ curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to $\mathsf{RCD}^{*}(K,N)$-spaces with essential minimal dimension $n\leq N$; a structure theorem for $\mathsf{RCD}^{*}(K,N)$-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.

math.MG↗
arXiv2017

Optimal maps in essentially non-branching spaces

In this note we prove that in a metric measure space $(X, d, m)$ verifying the measure contraction property with parameters $K \in \mathbb{R}$ and $1< N< \infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $m$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

math.MG↗
arXiv2017

Nonlinear diffusion equations and curvature conditions in metric measure spaces

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD*(K,N) condition of Bacher-Sturm.

math.AP↗
arXiv2017

On the volume measure of non-smooth spaces with Ricci curvature bounded below

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.

math.MG↗
arXiv2017

Angles between curves in metric measure spaces

The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on $RCD^{*}(K,N)$ metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

math.MG↗
arXiv2016

Rigidity for critical points in the Levy-Gromov inequality

The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes.

math.DG↗
arXiv2016

On a isoperimetric-isodiametric inequality

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 $\mathbb{R}^n$. 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 $C^0$-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 $8$.

math.AP↗
arXiv2016

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.

math.MG↗
arXiv2016

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II: Morse Theory

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore functional on exponentiated small Clifford tori degenerating, under the action of the Möbius group, to small geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

math.DG↗
arXiv2016

On the universal cover and the fundamental group of an $RCD^*(K,N)$-space

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).

math.MG↗
arXiv2016

A non-existence result for minimal catenoids in asymptotically flat spaces

We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold for more general classes of asymptotically flat data.

math.DG↗
arXiv2016

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

For metric measure spaces verifying the reduced curvature-dimension condition $CD^*(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally $RCD^*(K,N)$-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, $p$-spectral gap (or equivalently $p$-Poincaré inequality) for any $p\in [1,\infty)$, log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching $CD^*(K,N)$ spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition $CD(K,N)$ in the sense of Bakry-Émery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$-spectral gap. Finally let us mention that for essentially non-branching metric measure spaces, the local curvature-dimension condition $CD_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $CD^*(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.

math.MG↗
arXiv2015

Measure rigidity of Ricci curvature lower bounds

The measure contraction property, $\mathsf{MCP}$ for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Radon measure $\mathsf{m}$ over $\mathbb{R}^{d}$ is such that $(\mathbb{R}^{d},|\cdot |, \mathsf{m})$ verifies a weaker variant of $\mathsf{MCP}$, then its support $\text{spt}(\mathsf{m})$ must be convex and $\mathsf{m}$ has to be absolutely continuous with respect to the relevant Hausdorff measure of $\text{spt}(\mathsf{m})$. This result is then used as a starting point to investigate the rigidity of $\mathsf{MCP}$ in the metric framework. We introduce the new notion of $reference \ measure$ for a metric space and prove that if $(X,\mathsf{d},\mathsf{m})$ is essentially non-branching and verifies $\mathsf{MCP}$, and $μ$ is an essentially non-branching $\mathsf{MCP}$ reference measure for $(\text{spt}(\mathsf{m}), \mathsf{d})$, then $\mathsf{m}$ is absolutely continuous with respect to $μ$, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching $\mathsf{MCP}$ reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov-Hausdorff convergence, provided an additional uniform bound holds. In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting.

math.MG↗
arXiv2014

A Gap Theorem for Willmore Tori and an application to the Willmore Flow

In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2π^2$ and attains this minimal value if and only if the torus is a Möbius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.

math.DG↗
arXiv2014

Existence of Integral $m$-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, $p>m$, in Riemannian Manifolds

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq m m$, under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$ involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.

math.DG↗
arXiv2013

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions $CD(K,\infty)$ and $RCD(K,\infty)$ are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the $L^2$-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted $Γ$-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the $RCD(K,\infty)$ condition with $K>0$. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

math.MG↗
arXiv2013

Concentration of small Willmore spheres in Riemannian 3-manifolds

Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(Φ_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 π$, and Hausdorff converging to a point $\bar{p}\in M$, then $Scal(\bar{p})=0$ and $\nabla Scal(\bar{p})=0$ (resp. $\nabla Scal(\bar{p})=0$). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in \cite{LM1}-\cite{LM2}. An application to the Hawking mass is also established.

math.DG↗