arXiv subjects

Andrea Mondino

Publications and source records attributed to Andrea Mondino.

At least 37 records · Page 2Linked to original sources

arXiv2024

Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability

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.

math.DG↗

arXiv2024

Can you hear the Planck mass?

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.

hep-th↗

arXiv2024

Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures

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.

math.DG↗

arXiv2023

Gravity from thermodynamics: optimal transport and negative effective dimensions

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 $Λ$; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of $Λ$ 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.

hep-th↗

arXiv2023

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

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.

math.DG↗

arXiv2023

Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications

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.

math.MG↗

arXiv2023

Harmonic functions and gravity localization

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 Schrödinger 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.

hep-th↗

arXiv2023

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

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.

math.DG↗

arXiv2023

On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications

We show that several different interpretations of the inequality $Δf\leqη$ 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.

math.DG↗

arXiv2022

A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

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.

math.DG↗

arXiv2022

Moduli spaces of compact RCD(0,N)-structures

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.

math.MG↗

arXiv2022

Entropy-Transport distances between unbalanced metric measure spaces

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.

math.MG↗

arXiv2022

The metric measure boundary of spaces with Ricci curvature bounded below

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.

math.DG↗

arXiv2022

A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces

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-Hölder 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.

math.DG↗

arXiv2022

An upper bound on the revised first Betti number and a torus stability result for RCD spaces

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-Hölder homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-δ, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).

math.DG↗

arXiv2022

Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity

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 $Ω\subset M$ satisfies that every point has a neighbourhood $U\subset Ω$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $Ω$ 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.

math.DG↗

arXiv2022

Quantitative Obata's Theorem

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.

math.FA↗

arXiv2021

Quantization of the Willmore Energy in Riemannian Manifolds

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.

math.AP↗