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In the paper we discuss gap phenomena of three different types related to Ricci (and sectional) curvature. The first type is about spectral gaps. The second type is about sharp gap metric-rigidity, originally due to Anderson. The third is about sharp gap topological-rigidity. We also propose open problems along these directions., Comment: 18 pages

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality for the principal frequency of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995]. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N is close enough to 2 or 3. The inequality contains the so-called ``asymptotic volume ratio", and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the RCD(0,N) space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied., Comment: 30 pages

The goal of the present work is to study optimal transport on null hypersurfaces inside Lorentzian manifolds. The challenge here is that optimal transport along a null hypersurface is completely degenerate, as the cost takes only the two values $0$ and $+\infty$. The tools developed in the manuscript enable to give an optimal transport characterization of the null energy condition (namely, non-negative Ricci curvature in the null directions) for Lorentzian manifolds in terms of convexity properties of the Boltzmann--Shannon entropy along null-geodesics of probability measures. We obtain as applications: a stability result under convergence of spacetimes, a comparison result for null-cones, and the Hawking area theorem (both in sharp form, for possibly weighted measures, and with apparently new rigidity statements)., Comment: 52 pages

We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, Duke Math. J. (1992)], but instead of asking that one boundary component is compact and mean convex, we require that it is parabolic and convex. The parabolicity assumption cannot be dropped as, otherwise, the catenoid in ambient dimension four would give a counterexample. As an application, we deduce a half-space theorem for mean convex sets in product manifolds (resp. for sets whose boundary has mean curvature bounded below by a definite constant, in warped products with negative curvature). The results are obtained by combining glueing techniques for manifolds and optimal transport tools from synthetic Ricci curvature bounds., Comment: 18 pages. Improvements with respect to version 1: the main results (Theorem 1 and 2) have been sharpened by allowing multiple boundary components, moreover we added some applications to the half-space property (see Corollary 1)

For the Laplacian of an $n$-Riemannian manifold $X$, the Weyl law states that the $k$-th eigenvalue is asymptotically proportional to $(k/V)^{2/n}$, where $V$ is the volume of $X$. We show that this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on $X$ behaves in the expected $(4+n)$-dimensional way at short distances. In simple product compactifications, when particle motion on $X$ is ergodic, for large $k$ the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 Kaluza--Klein modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume $V$, while the Planck mass famously depends on a weighted volume obtained as an integral of the warping function. We resolve this tension by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, a property that we call weighted quantum ergodicity. This has implications for the problem of gravity localization, which we discuss. We show that for spaces with D$p$-brane singularities the spectrum is discrete only for $p =6,7,8$, and for these cases we rigorously prove the Weyl law by applying modern techniques from RCD theory., Comment: 35 pages, 1 figure

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci curvature. This seems to be the first non-trivial result giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov-Hausdorff convergence, via a degeneration result to orbifolds by Anderson., Comment: 7 pages

The goal of this note is to prove the Half Space Property for $RCD(0,N)$ spaces, namely that if $(X,d,m)$ is a parabolic $RCD(0,N)$ space and $ C \subset X \times \mathbb{R}$ is locally the boundary of a locally perimeter minimizing set and it is contained in a half space, then $C$ is a locally finite union of horizontal slices. If the assumption of local perimeter minimizing is strengthened into global perimeter minimizing, then the conclusion can be strengthened into uniqueness of the horizontal slice. As a consequence, we obtain oscillation estimates and a Half Space Theorem for minimal hypersurfaces in products $M \times \mathbb{R}$, where $M$ is a parabolic smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature. On the way of proving the main results, we also obtain some properties of Green's functions on $RCD(K,N)$ spaces that are of independent interest., Comment: 39 pages. Added Corollary 4.29 and improved exposition

The goal of the paper is to prove the equivalence of distributional and synthetic Ricci curvature lower bounds for a weighted Riemannian manifold with continuous metric tensor having Christoffel symbols in $L^2_{{\rm loc}}$, and with weight in $C^0\cap W^{1,2}_{{\rm loc}}$. The regularity assumptions are sharp, in the sense that they are minimal in order to define the distributional Ricci curvature tensor., Comment: 53 pages

The paper establishes a sharp and rigid isoperimetric-type inequality in Lorentzian signature under the assumption of Ricci curvature bounded below in the timelike directions. The inequality is proved in the high generality of Lorentzian pre-length spaces satisfying timelike Ricci lower bounds in a synthetic sense via optimal transport, the so-called $\mathsf{TCD}^e_p(K,N)$ spaces. The results are new already for smooth Lorentzian manifolds. Applications include an upper bound on the area of Cauchy hypersurfaces inside the interior of a black hole (original already in Schwarzschild) and an upper bound on the area of Cauchy hypersurfaces in cosmological space-times.

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non collapsed RCD spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth., Comment: 37 pages

In models with extra dimensions, matter particles can be easily localized to a 'brane world', but gravitational attraction tends to spread out in the extra dimensions unless they are small. Strong warping gradients can help localize gravity closer to the brane. In this note we give a mathematically rigorous proof that the internal wave-function of the massless graviton is constant as an eigenfunction of the weighted Laplacian, and hence is a power of the warping as a bound state in an analogue Schr\"odinger potential. This holds even in presence of singularities induced by thin branes. We also reassess the status of AdS vacuum solutions where the graviton is massive. We prove a bound on scale separation for such models, as an application of our recent results on KK masses. We also use them to estimate the scale at which gravity is localized, without having to compute the spectrum explicitly. For example, we point out that localization can be obtained at least up to the cosmological scale in string/M-theory solutions with infinite-volume Riemann surfaces; and in a known class of N = 4 models, when the number of NS5- and D5-branes is roughly equal., Comment: 43 pages, 2 figures

We show that several different interpretations of the inequality $\Delta f\leq\eta$ are equivalent in the setting of $\mathrm{RCD}(K,N)$ spaces. With respect to previously available results in this direction, we improve both on the generality of the underlying space and in terms of regularity to be assumed on the function $f$. Applications are presented., Comment: Comments are welcome!

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant $\Lambda$; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of $\Lambda$ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy., Comment: 57 pages + 4 appendices, 4 figures. v2: expanded introduction and comments on the Casimir vacuum. v3: clarification on the agreement of the Casimir vacuum with conjectured bounds

Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces., Comment: 153 pages. v2: new Section 10.2 on the Grushin plane. v3: minor cosmetic changes. Accepted version, to appear on Memoirs of the AMS

We solve a conjecture raised by Kapovitch, Lytchak, and Petrunin by showing that the metric measure boundary is vanishing on any ${\rm RCD}(K,N)$ space without boundary. Our result, combined with [Kapovitch-Lytchak-Petrunin '21], settles an open question about the existence of infinite geodesics on Alexandrov spaces without boundary raised by Perelman and Petrunin in 1996.

The scope of this survey is to give a self-contained introduction to synthetic timelike Ricci curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport $\&$ entropy tools, including a synthetic version of Hawking's singularity theorem and a synthetic characterisation of Einstein's vacuum equations. We will also discuss some motivations arising from the smooth world and some possible directions for future research., Comment: Final version, published in "General Relativity and Gravitation", special issue in honour of Roger Penrose. arXiv admin note: text overlap with arXiv:2004.08934

We establish Lipschitz regularity of harmonic maps from $\mathrm{RCD}(K,N)$ metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into $\mathrm{CAT}(0)$ metric spaces with non-positive sectional curvature. Under the same assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-term. This gives a fairly complete generalization of the classical theory for smooth source and target spaces to their natural synthetic counterparts and an affirmative answer to a question raised several times in the recent literature. The proofs build on a new interpretation of the interplay between Optimal Transport and the Heat Flow on the source space and on an original perturbation argument in the spirit of the viscosity theory of PDEs., Comment: Final version accepted for publication in American Journal of Mathematics

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of RCD(0,N)-structures. First, we relate the convergence of RCD(0,N)-structures on a space to the associated lifts' equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of RCD(0,N)-structures that have non-trivial rational homotopy groups., Comment: 40 pages. Final version to appear in Math. Annalen

We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay in a compact domain of the moduli space., Comment: 82 pages

In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-H\"older inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical "Chiti Comparison Theorem". We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds., Comment: 14 pages. Final Version, to appear in the Proc. Amer. Math. Soc

We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry--\'Emery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdS$_d$ vacua with a bridge admitting an AdS$_{d+1}$ interpretation, the holographic dual is a CFT$_d$ with two CFT$_{d-1}$ boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for $d=4$. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it., Comment: 61 pages, 4 figures

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces): - we develop an intrinsic theory of Laplacian bounds in viscosity sense and in a pointwise, heat flow related, sense, showing their equivalence also with Laplacian bounds in distributional sense; - relying on these tools, we establish a PDE principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the $p$-Hopf-Lax semigroup, for general exponents $p\in[1,\infty)$. This principle admits a broad range of applications, going much beyond the topic of the present paper; - we prove sharp Laplacian bounds on the distance function from a set (locally) minimizing the perimeter with a flexible technique, not involving any regularity theory; this corresponds to vanishing mean curvature in the smooth setting and encodes also information about the second variation of the area; - we initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter, obtaining sharp dimension estimates for their singular sets, quantitative estimates of independent interest even in the smooth setting and topological regularity away from the singular set. The class of RCD(K,N) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks. Moreover, the tools that we develop here have applications to classical questions in Geometric Analysis on smooth, non compact Riemannian manifolds with lower Ricci curvature bounds., Comment: 96 pages. Revised version. Accepted by Mem. Amer. Math. Soc

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main result states that if an open subset $\Omega\subset M$ satisfies that every point has a neighbourhood $U\subset \Omega$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $\Omega$ is locally isometric to Euclidean ${\mathbb R}^3$ (resp. locally isometric to the Hyperbolic 3-space ${\mathbb H}^3$). Under mild asymptotic conditions on the manifold $(M,g)$ (which encompass as special cases the standard "asymptotically flat" or, respectively, "asymptotically hyperbolic" assumptions) the previous quasi-local rigidity statement implies a \emph{global rigidity}: if every point in $M$ has a neighbourhood $U$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $(M,g)$ is globally isometric to Euclidean ${\mathbb R}^3$ (resp. globally isometric to the Hyperbolic 3-space ${\mathbb H}^3$). Also, if the space is not flat (resp. not of constant sectional curvature $-1$), the methods give a small yet explicit and strictly positive lower bound on the Hawking mass of suitable spherical surfaces. We infer a small yet explicit and strictly positive lower bound on the Bartnik mass of open sets (non-locally isometric to Euclidean ${\mathbb R}^{3}$) in terms of curvature tensors. Inspired by these results, in the appendix we propose a notion of "sup-Hawking mass" which satisfies some natural properties of a quasi-local mass., Comment: 39 pages. Final version, to appear in the Journal of the London Mathematical Society

We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H\"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-\delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding)., Comment: 38 pages. Final version, to appear in Commentarii Mathematici Helvetici

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8\pi$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes., Comment: 16 pages. Final version to appear in Advances in Calculus of Variations

Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" $\mathbf{D}$-distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense., Comment: 39 pages. Added a section with a comparison with the conic Gromov-Wasserstein distance. Some other minor improvements. Final version, to appear on "Probability Theory and Related Fields"

We prove pointwise and $L^{p}$-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an $\mathrm{RCD}(K,N)$ metric measure space, with $K>0$ and $N\in (1,\infty)$). The obtained Talenti-type comparison is sharp, rigid and stable with respect to $L^{2}$/measured-Gromov-Hausdorff topology; moreover, several aspects seem new even for smooth Riemannian manifolds. As applications of such Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an $\mathrm{RCD}$ version of the St.~Venant-P\'olya torsional rigidity comparison theorem (with associated rigidity and stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open subset for the Brownian motion., Comment: 35 pages. Final version to appear in Calculus of Variations and Partial Differential Equations

We prove that the sharp Buser's inequality obtained in the framework of $\mathsf{RCD}(1,\infty)$ spaces by the first two authors is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of $\mathsf{RCD}(K,\infty)$ spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained., Comment: Added new results: the discussion on Cheeger's inequality now fits into the study of a family of inequalities relating eigenvalues of the p-Laplacian. To appear on Journal of Functional Analysis

The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of ``timelike Ricci curvature bounded below and dimension bounded above'' for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity., Comment: 74 pages. Improved exposition, in particular in Sections 2.5 and 3.3. Final version accepted in Cambridge Journal of Mathematics

We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the general framework of (possibly non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method., Comment: 38 pages. Final version to appear in Analysis and PDEs

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence., Comment: 32 pages. Final version to appear in "Geometry & Topology"

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions., Comment: 29 pages. Final version published in Archive for Rational Mechanics and Analysis

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian. A celebrated lower bound of $\lambda_{1}$ in terms of $h$, $\lambda_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $\lambda_{1}$ in terms of $h$ was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is two fold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-\'Emery weighted) Ricci curvature bounded below by $K\in {\mathbb R}$ (the inequality is sharp for $K>0$ as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called $ \mathsf{RCD}(K,\infty)$ spaces., Comment: 19 pages. Final version published in The Journal of Geometric Analysis

The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years., Comment: 60 pages. Improved overall exposition, added a non-smooth example in the introduction and an appendix about a synthetic non-smooth framework for Einstein's equations. Final version accepted by the Journal of the European Mathematical Society (JEMS)

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of B\'erard-Meyer and Matei; remarkable examples of spaces fitting out framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci$\geq K>0$, finite dimensional Alexandrov spaces with curvature$\geq K>0$, Finsler manifolds with Ricci$\geq K>0$. In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of $RCD(K,N)$ spaces, which seem original even for smooth Riemannian manifolds with Ricci$\geq K>0$., Comment: 33 pages. Final version published in Journal de Math\'ematiques Pures et Appliqu\'ees

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32\pi$, moreover it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the first multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass., Comment: 23 pages

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N)., Comment: Final version to appear in Analysis and PDE

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., Comment: 51 pages. Comments welcome!

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., Comment: 22 pages. Final version published by Int. Math. Res. Not. (IMRN)

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., Comment: 21 pages

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., Comment: 5 pages

We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger formulation in terms of the perimeter., Comment: 13 pages, final version to appear in Rend. Lincei Mat. Appl. arXiv admin note: text overlap with arXiv:1505.02061

The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called $p$-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role of reference measure is played by the $p$-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving $p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure., Comment: Final version, published by Advances in Mathematics

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., Comment: Final version to appear in Communications in Contemporary Mathematics

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., Comment: Final version to appear in the Annali della Scuola Normale Superiore Classe di Scienze

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., Comment: Final version, to appear in Journal of the London Mathematical Society