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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

Cosmic Microwave Background (CMB) photons can undergo resonant conversion into axions in the presence of magnetized plasma distributed inside non-linear large-scale structure (LSS). This process leads to axion-induced patchy screening: secondary temperature and polarization anisotropies with a characteristic non-blackbody frequency dependence that are strongly correlated with the distribution of LSS along our past light cone. We compute the axion-induced patchy screening contribution to two- and three- point correlation functions that include CMB anisotropies and tracers of LSS within the halo model. We use these results to forecast the sensitivity of existing and future surveys to photon-axion couplings for axion masses between $2\times 10^{-13}$ eV and $3\times 10^{-12}$ eV, using a combination of empirical estimates from Planck data of the contribution from instrumental noise and foregrounds as well as modeled contributions on angular scales only accessible with future datasets. We demonstrate that an analysis using Planck and the unWISE galaxy catalogue would be complementary to the most sensitive existing astrophysical axion searches, probing couplings as small as $3\times 10^{-12} \, {\rm GeV}^{-1}$, while observations from a future survey such as CMB-S4 could extend this reach by almost an additional order of magnitude., Comment: 37 pages, 10 figures

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.

Small-scale dark matter structures lighter than a billion solar masses are an important probe of primordial density fluctuations and dark matter microphysics. Due to their lack of starlight emission, their only guaranteed signatures are gravitational in nature. We report on results of a search for astrometric weak lensing by compact dark matter subhalos in the Milky Way with Gaia DR3 data. Using a matched-filter analysis to look for correlated imprints of time-domain lensing on the proper motions of background stars in the Magellanic Clouds, we exclude order-unity substructure fractions in halos with masses $M_{l}$ between $10^{7} \, M_{\odot}$ and $10^{9} \, M_{\odot}$ and sizes of one parsec or smaller. We forecast that a similar approach based on proper accelerations across the entire sky with data from Gaia DR4 may be sensitive to substructure fractions of $f_{l} \gtrsim 10^{-3}$ in the much lower mass range of $10 \, M_{\odot} \lesssim M_{l} \lesssim 3 \times 10^{3} \, M_{\odot}$. We further propose an analogous technique for stacked star-star lensing events in the regime of large impact parameters. Our first implementation is not yet sufficiently sensitive but serves as a useful diagnostic and calibration tool; future data releases should enable average stellar mass measurements using this stacking method., Comment: 16 pages, 8 figures; v2: published version, references and minor comments added

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