arXiv subjects

Fabio Cavalletti

Publications and source records attributed to Fabio Cavalletti.

At least 19 recordsLinked to original sources

arXiv2026

A singularity theorem in terms of asymptotic expansion

We prove a singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition $\mathsf{TCD}^e_p(0,N)$, yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.

math.DG↗

arXiv2026

On the geometry of synthetic null hypersurfaces

This paper develops a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple $(H, G, \mathfrak{m})$: $H$ is a closed achronal set in a topological causal space, $G$ is a gauge function encoding affine parametrizations along null generators, and $\mathfrak{m}$ is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. The central object is the synthetic null energy condition ($\mathsf{NC}^e(N)$), defined via the concavity of an entropy power functional along optimal transport, with parametrization given by the gauge $G$. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of the $\mathsf{NC}^e(N)$ condition is the stability under convergence of synthetic null hypersurfaces. The $\mathsf{NC}^e(N)$ condition is also remarkably fruitful for applications. First, it provides a framework for a synthetic version of Hawking's area theorem. Second, the celebrated Penrose's singularity theorem is proved for continuous spacetimes, and the existence of trapped regions is settled in the general setting of topological causal spaces satisfying the $\mathsf{NC}^e(N)$.

math.DG↗

arXiv2025

Optimal transport on null hypersurfaces and the null energy condition

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).

math.DG↗

arXiv2025

A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds

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 achronal hypersurfaces inside the interior of a black hole (original already in Schwarzschild) and an upper bound on the area of achronal hypersurfaces in cosmological spacetimes.

math.MG↗

arXiv2024

Rigidities of Isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature. The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general $\mathsf{RCD}$ spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

math.MG↗

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

Optimal transport between algebraic hypersurfaces

What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini-Study one. The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is Kähler and the corresponding metric is of Weil-Petersson type. To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own. We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.

math.DG↗

arXiv2023

A proof of the Hamiltonian Thom Isotopy Lemma

In this note we present a complete proof of the fact that all the submanifolds of a one parameter family of compact symplectic submanifolds inside a compact symplectic manifold are Hamiltonian isotopic.

math.SG↗

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

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

Isoperimetric inequality in noncompact MCP spaces

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds $MCP(0,N)$ and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for $MCP(0,N)$, and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localisation.

math.MG↗

arXiv2021

Geometry of Grassmannians and optimal transport of quantum states

Let $\mathsf{H}$ be a separable Hilbert space. We prove that the Grassmannian $\mathsf{P}_c(\mathsf{H})$ of the finite dimensional subspaces of $\mathsf{H}$ is an Alexandrov space of nonnegative curvature and we employ its metric geometry to develop the theory of optimal transport for the normal states of the von Neumann algebra of linear and bounded operators $\mathsf{B}(\mathsf{H})$. Seeing density matrices as discrete probability measures on $\mathsf{P}_c(\mathsf{H})$ (via the spectral theorem) we define an optimal transport cost and the Wasserstein distance for normal states. In particular we obtain a cost which induces the $w^*$-topology. Our construction is compatible with the quantum mechanics approach of composite systems as tensor products $\mathsf{H}\otimes \mathsf{H}$. We provide indeed an interpretation of the pure normal states of $\mathsf{B}(\mathsf{H}\otimes \mathsf{H})$ as families of transport maps. This also defines a Wasserstein cost for the pure normal states of $\mathsf{B}(\mathsf{H}\otimes \mathsf{H})$, reconciling with our proposal.

math.DG↗

arXiv2021

The Globalization Theorem for the Curvature Dimension Condition

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space $(X,{\mathsf d},{\mathfrak m})$ (so that $(\text{supp} \; {\mathfrak m},{\mathsf d})$ is a length-space and ${\mathfrak m}(X) < \infty$) verifying the local Curvature-Dimension condition $\mathsf{CD}_{loc}(K,N)$ with parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, also verifies the global Curvature-Dimension condition $\mathsf{CD}(K,N)$. In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between $L^1$ and $L^2$ optimal-transport-based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.

math.MG↗

arXiv2021

Indeterminacy estimates and the size of nodal sets in singular spaces

We obtain the sharp version of the uncertainty principle recently introduced in [47], and improved by [13], relating the size of the zero set of a continuous function having zero mean and the optimal transport cost between the mass of the positive part and the negative one. The result is actually valid for the wide family of metric measure spaces verifying a synthetic lower bound on the Ricci curvature, namely the MCP(K,N) or CD(K,N) condition, thus also extending the scope beyond the smooth setting of Riemannian manifolds. Applying the uncertainty principle to eigenfunctions of the Laplacian in possibly non-smooth spaces, we obtain new lower bounds on the size of their nodal sets in terms of the eigenvalues. Those cases where the Laplacian is possibly non-linear are also covered and applications to linear combinations of eigenfunctions of the Laplacian are derived. To the best of our knowledge, no previous results were known for non-smooth spaces.

math.DG↗

arXiv2020

Independence of synthetic Curvature Dimension conditions on transport distance exponent

The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $\mathrm{CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance function as transport cost does not influence the theory. By denoting with $\mathrm{CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathrm{CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ -- at least in spaces whose geodesics do not branch. We show that the trait d'union between all the seemingly unrelated $\mathrm{CD}_{p}(K,N)$ conditions is the needle decomposition or localization technique associated to the $L^{1}$-optimal transport problem. We also establish the local-to-global property of $\mathrm{CD}_{p}(K,N)$ spaces.

math.DG↗

arXiv2020

Displacement convexity of Entropy and the distance cost Optimal Transportation

During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott-Sturm-Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along $W_{2}$-geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of $L^1$-Optimal transport problems yielding a different curvature dimension $\mathsf{CD}^{1}(K,N)$ [8] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the $\mathsf{CD}^{1}(K,N)$ condition can be formulated in terms of displacement convexity along $W_{1}$-geodesics.

math.MG↗

arXiv2019

New formulas for the Laplacian of distance functions and applications

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).

math.MG↗

arXiv2018

Isoperimetric inequality under Measure-Contraction property

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.

math.MG↗