arXiv subjects

Nicola Gigli

Publications and source records attributed to Nicola Gigli.

At least 19 recordsLinked to original sources

arXiv2026

Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch--Kronheimer--Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal--Szekeres spacetime in terms of its radial null geodesics.

math.DG↗

arXiv2026

Pyramids and Extended Metric Measure Spaces

A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkhäuser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio--Gigli--Savaré (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent. We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This shows, for the first time, that all pyramids can be realised by concrete geometric spaces. Based on this representation, we introduce a new notion, a concentrated emm-space. We then define the observable distance for concentrated emm-spaces and establish three equivalent characterisations of concentration in terms of pyramids, Lipschitz observables and the observable distance. Furthermore, by developing an fibration approach, we show the $Γ$-$\limsup$ inequality of Cheeger energies as well as the stability of the log-Sobolev inequality and the Poincaré inequality under the weak convergence of pyramids associated with emm-spaces. Our results provide a new geometric approach for studying both the convergence of emm-spaces and concentration-of-measure phenomena across a broad class of infinite-dimensional models that have so far remained largely beyond the reach of existing geometric methods. Many of the significant examples arise in probability theory, including the Wiener space, the configuration space, Gaussian fields such as massive Gaussian free fields, spatial white noise and massive bi-Laplacian fields.

math.MG↗

arXiv2026

Mean curvature and sharp Willmore inequalities in metric spaces

The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.

math.DG↗

arXiv2026

An elliptic proof of the splitting theorems from Lorentzian geometry

We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic $p$-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman Lorentzian splitting theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

math.DG↗

arXiv2026

The splitting theorem in non-smooth context

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

math.MG↗

arXiv2026

Some examples of use of transfinite induction in analysis

It is not uncommon in analysis that existence of extremal objects is obtained via an iterative procedure: we start from a given admissible object, then modify it, then modify again etc... If being extremal means maximimizing a real valued quantity and we are sure to approach the supremum fast enough, after a countable number of steps and a limiting procedure we are done. In this short note we want to advertise a slightly different line of thought, where rather than trying to approach the supremum fast enough, we: try to increase, if possible, the function to be maximized and, at the same time, index our recursive procedure over ordinals. Since there are no increasing functions from $ω_1$ to $\R$, the procedure must stop at some countable ordinal and existence is proved anyway. The advantage of this line of reasoning is that it can be helpful even in situations where it is not so evident how to measure `being maximal' via a real valued function. This is the case, for instance, for existence of a Maximal Globally Hyperbolic Development of an initial data set in General Relativity. Speaking of this particular example, we also show that such `real-valued quantification' of the size of a development is actually possible, thus existence of a maximal one can be obtained in a countable number of steps using the original argument in [2] together with the standard procedure depicted above. This provides a way alternative to the one given in [5] to `dezornify' the proof in [2].

math.DG↗

arXiv2026

PDE aspects of the dynamical optimal transport in the Lorentzian setting

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.

math.AP↗

arXiv2025

Hyperbolic Banach spaces

The standard theory of Banach spaces is built upon the notions of vector space, triangle inequality and Cauchy completeness. Here we propose a `hyperbolic' variant of this `elliptic' framework where general linear combinations are replaced by linear combinations with non-negative coefficients, triangle inequality is replaced by reverse triangle inequality and Cauchy completeness is replaced by the order-theoretic notion of directed completeness. The motivation for our investigation is in non-smooth Lorentzian geometry: we believe that to unlock the full potential of the field, and ultimately extract more informations about the smooth world, some version of `Lorentzian functional analysis' is needed, especially in relation to timelike lower Ricci curvature bounds. An example of structure we investigate is obtained by starting with a Banach space, multiplying it by $\mathbb R$ and considering the `future cone' in there. Because of this, some of the results in this manuscript might be read through the lenses of standard Banach spaces theory. From this perspective, the classical Hahn-Banach and Baire category theorems can be seen as consequences of statements obtained here. A different kind of example is that of $L^p$ spaces for $p\leq1$. Their structure and natural duality relations fit particularly well in our framework, to the extent that they have been an important source of inspiration for the axiomatization chosen in this paper. We also investigate the notion of directed completeness regardless of any algebraic structure, as we believe it is central even in the finite-dimensional non-smooth Lorentzian framework, for instance to achieve a compactness theorem à la Gromov. This study unveils connections between Geroch-Kronheimer-Penrose's concept of ideal point in a spacetime, Beppo Levi's monotone convergence theorem and certain aspects of domain theory.

math.FA↗

arXiv2025

A Lorentzian splitting theorem for continuously differentiable metrics and weights

We prove a splitting theorem for globally hyperbolic, weighted spacetimes with metrics and weights of regularity $C^1$ by combining elliptic techniques for the negative homogeneity $p$-d'Alembert operator from our recent work in the smooth setting with the concept of line-adapted curves introduced here. Our results extend the Lorentzian splitting theorem proved for smooth globally hyperbolic spacetimes by Galloway -- and variants of its weighted counterparts by Case and Woolgar--Wylie -- to this low regularity setting.

math.DG↗

arXiv2025

Non-collapsed spaces with Ricci curvature bounded from below

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces. In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence.

math.MG↗

arXiv2025

A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes

We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus.

math.DG↗

arXiv2024

Stability of the heat flow under convergence in concentration and consequences

We extend to the framework of convergence in concentration virtually all the results concerning stability of Sobolev functions and differential operators known to be in place under the stronger measured-Gromov-Hausdorff convergence. These include, in particular: i) A general $Γ$--$\varlimsup$ inequality for the Cheeger energy, ii) Convergence of the heat flow under a uniform ${\sf CD}(K,\infty)$ condition on the spaces. As we will show, building on ideas developed for mGH-convergence, out of this latter result we can obtain clean stability statements for differential, flows of vector fields, Hessian, eigenvalues of the Laplacian and related objects in presence of uniform lower Ricci bounds. At the technical level, one of the tools we develop to establish these results is the notion of convergence of maps between metric measure spaces converging in concentration. Previous results in the field concerned the stability of the ${\sf CD}$ (Funano-Shioya '13) and ${\sf RCD}$ (Ozawa-Yokota '19) conditions. The latter was obtained proving $Γ$-convergence for the Cheeger energies. This is not sufficient to pass to the limit in the heat flow; one of the byproducts of our work is the improvement of this to Mosco-convergence.

math.MG↗

arXiv2024

Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations

The aim of this paper is twofold. - In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We use these estimates to prove that, as the viscosity tends to zero, solutions of this equation converge to the evolution driven by the Hopf--Lax formula, in accordance with the smooth case. - We then use such convergence to study the small-time Large Deviation Principle for both the heat kernel and the Brownian motion: we obtain the expected behavior under the additional assumption that the space is proper. As an application of the latter point, we also discuss the $Γ$-convergence of the Schrödinger problem to the quadratic optimal transport problem in proper RCD(K,$\infty$) spaces.

math.PR↗

arXiv2024

Comments on the regularity of harmonic maps between singular spaces

In this work we are going to establish Hölder continuity of harmonic maps from an open set $Ω$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(κ)$ space, with the constraint that the image of $Ω$ via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.

math.AP↗

arXiv2024

Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces

The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the Gromov-Hausdorff distance. This fact was shown by Cheeger and Colding in an appendix of one of their celebrated works on Ricci limit spaces [8]. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in [8] and makes explicit all the relevant estimates and constructions. As a byproduct we also shows a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author.

math.MG↗

arXiv2023

A general splitting principle on RCD spaces and applications to spaces with positive spectrum

In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

math.MG↗

arXiv2023

Fine representation of Hessian of convex functions and Ricci tensor on RCD spaces

It is known that on $\mathrm{RCD}$ spaces one can define a distributional Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition $${\bf Ric}=ω\,|{\bf Ric}|$$ for a suitable non-negative measure $|{\bf Ric}|$ and unitary tensor field $ω$. The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.

math.MG↗

arXiv2023

On the regularity of harmonic maps from ${\sf RCD}(K,N)$ to ${\sf CAT}(0)$ spaces and related results

For an harmonic map $u$ from a domain $U\subset{\rm X}$ in an ${\sf RCD}(K,N)$ space ${\rm X}$ to a ${\sf CAT}(0)$ space ${\rm Y}$ we prove the Lipschitz estimate \[ {\rm Lip}(u|_B)\leq \frac {C(K^-R^2,N)}r\inf_{{\sf o}\in {\rm Y}}\,\sqrt{\frac1{{\mathfrak m}(2B)}\int_{2B}{\sf d}_{\rm Y}^2(u(\cdot),{\sf o})\, {\rm d}{\mathfrak m}}, \qquad \forall 2B\subset U \] where $r\in(0,R)$ is the radius of $B$. This is obtained by combining classical Moser's iteration, a Bochner-type inequality that we derive (guided by recent works of Zhang-Zhu) together with a reverse Poincaré inequality that is also established here. A direct consequence of our estimate is a Lioville-Yau type theorem in the case $K=0$. Among the ingredients we develop for the proof, a variational principle valid in general ${\sf RCD}$ spaces is particularly relevant. It can be roughly stated as: if $({\rm X},{\sf d},{\mathfrak m})$ is ${\sf RCD}(K,\infty)$ and $f\in C_b({\rm X})$ is so that $Δf\leq C$ for some constant $C>0$, then for every $t>0$ and ${\mathfrak m}$-a.e.\ $x\in{\rm X}$ there is a unique minimizer $F_t(x)$ for $ y\ \mapsto\ f(y)+\frac{{\sf d}^2(x,y)}{2t} $ and the map $F_t$ satisfies \[ (F_t)_*{\mathfrak m}\leq e^{t(C+2K^-{\sf Osc}(f))}{\mathfrak m},\qquad\text{where}\qquad {\sf Osc}(f):=\sup f-\inf f. \] Here existence is in place without any sort of compactness assumption and uniqueness should be intended in a sense analogue to that in place for Regular Lagrangian Flows and Optimal Maps (and is related to both these concepts). Finally, we also obtain a Rademacher-type result for Lipschitz maps between spaces as above.

math.MG↗