Gromov-Hausdorff Limit Research Articles

The Grushin hemisphere as a Ricci limit space with curvature ≥1

The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere S m + n S^{m+n} with R i c ≥ 1 Ric\ge 1 such that its Gromov-Hausdorff limit is the n n -dimensional Grushin hemisphere.

Correction: New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Correction: New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Limits of Manifolds in the Gromov–Hausdorff Metric Space

We apply the Gromov-Hausdorff metric $d_G$ for characterization of certain generalized manifolds. Previously, we have proved that with respect to the metric $d_G,$ generalized $n$-manifolds are limits of spaces which are obtained by gluing two topological $n$-manifolds by a controlled homotopy equivalence (the so-called $2$-patch spaces). In the present paper, we consider the so-called {\sl manifold-like} generalized $n$-manifolds $X^{n},$ introduced in 1966 by Marde\v{s}i\'{c} and Segal, which are characterized by the existence of $\delta$-mappings $f_{\delta}$ of $X^n$ onto closed manifolds $M^{n}_{\delta},$ for arbitrary small $\delta>0$, i.e. there exist onto maps $f_{\delta}\colon X^{n}\to M^{n}_{\delta}$ such that for every $u\in M^{n}_{\delta}$, $f^{-1}_{\delta}(u)$ has diameter less than $\delta$. We prove that with respect to the metric $d_G,$ manifold-like generalized $n$-manifolds $X^{n}$ are limits of topological $n$-manifolds $M^{n}_{i}$. Moreover, if topological $n$-manifolds $M^{n}_{i}$ satisfy a certain local contractibility condition $\mathcal{M}(\varrho, n)$, we prove that generalized $n$-manifold $X^{n}$ is resolvable.

The rigidity of sharp spectral gap in non-negatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap underRic≥0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD0,N spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry–Émery manifolds along with products, certain quotients and measured Gromov–Hausdorff limits of such spaces. In precise terms, we show in such spaces,λ1=π2diam2if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular 1D-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.

New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Let \(\{X_i\}\) be a sequence of compact n-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov–Hausdorff sense to a compact Alexandrov space X. The paper (Alesker in Arnold Math J 4(1):1–17, 2018) outlined (without a proof) a construction of an integer-valued function on X; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of the \(X_i\). In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.

The isoperimetric problem on Riemannian manifolds via Gromov–Hausdorff asymptotic analysis

In this paper, we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some (possibly nonsmooth) Gromov–Hausdorff limits at infinity. The Gromov–Hausdorff asymptotic analysis allows us to recover and extend different previous existence theorems. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.

Time Zero Regularity of Ricci Flow

Abstract We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails that was given in [26]. We prove a positive result in the case that the flow satisfies a lower ${ {\textrm {K}_{\textrm {IC}_1}}}$ curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov–Hausdorff limits of WPIC1 manifolds are WPIC1.

Subspace foliations and collapse of closed flat manifolds

AbstractWe study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.

Noncollapsed degeneration of Einstein 4–manifolds, II

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted H\"older spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we extend the obstruction to the desingularization of Einstein orbifolds found by Biquard, and prove that it holds for any desingularization by trees of quotients of gravitational instantons only assuming a mere Gromov-Hausdorff convergence instead of specific weighted H\"older spaces. This is conjecturally the general case, but it can at least be ensured by topological assumptions such as a spin structure on the degenerating manifolds. We also identify an obstruction to desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE spaces.

Noncollapsed degeneration of Einstein 4–manifolds, I

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove that Einstein manifolds sufficiently close, in the Gromov-Hausdorff sense, to an orbifold are actually close to a gluing of model spaces in suitable weighted H\"older spaces. The proof consists in controlling the metric in the neck regions thanks to the construction of optimal coordinates. This refined convergence is the cornerstone of our subsequent work on the degeneration of Einstein metrics or, equivalently, on the desingularization of Einstein orbifolds in which we show that all Einstein metrics Gromov-Hausdorff close to an Einstein orbifold are the result of a gluing-perturbation procedure. This procedure turns out to be generally obstructed, and this provides the first obstructions to a Gromov-Hausdorff desingularization of Einstein orbifolds.

Some topological results of Ricci limit spaces

We study the topology of a Ricci limit space ( X , p ) (X,p) , which is the Gromov-Hausdorff limit of a sequence of complete n n -manifolds ( M i , p i ) (M_i, p_i) with R i c ≥ − ( n − 1 ) \mathrm {Ric}\ge -(n-1) . Our first result shows that, if M i M_i has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then X X is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of X X if M i M_i has a uniform diameter bound; this improves a result by Ennis and Wei [Differential Geom. Appl. 24 (2006), pp. 554-562].

Compact quantum metric spaces from free graph algebras

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel.We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.

A first-order condition for the independence on p of weak gradients

It is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend on p.In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, guarantees that in fact such dependence is not present.We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions.

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

We prove that if M is a closed n-dimensional Riemannian manifold, n ge 3, with mathrm{Ric}ge n-1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere mathbb {S}^n, then M is isometric to mathbb {S}^n. An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov–Hausdorff sense to a spherical suspension. These statements are obtained in the mathrm {RCD}-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact mathrm {CD} space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of mathrm {RCD} spaces and on a Pólya–Szegő inequality of Euclidean-type in mathrm {CD} spaces. As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov–Hausdorff convergence, in the mathrm {RCD}-setting.

On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $\Delta_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.

Null Distance and Convergence of Lorentzian Length Spaces

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov–Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.

A metric stability result for the very strict CD condition

Schultz (2018) generalized the work of Rajala and Sturm (2014), proving that a weak non-branching condition holds in the more general setting of very strict CD spaces. Anyway, similar to what happens for the strong CD condition, the very strict CD condition seems not to be stable with respect to the measured Gromov Hausdorff convergence (cf. Magnabosco, 2022).In this article I prove a stability result for the very strict CD condition, assuming some metric requirements on the converging sequence and on the limit space. The proof relies on the notions of consistent geodesic flow and consistent plan selection, which allow to treat separately the static and the dynamic part of a Wasserstein geodesic. As an application, I prove that the metric measure space RN equipped with a crystalline norm and with the Lebesgue measure satisfies the very strict CD(0,∞) condition.

Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kähler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, and a rigidity theorem for Kähler manifolds with almost Euclidean volume growth.

Small curvature concentration and Ricci flow smoothing

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times which only depends on the initial almost Euclidean structure. As applications, we use the Ricci flows to study the diffeomorphism type of manifolds and the regularity of Gromov-Hausdorff limit of manifolds with small curvature concentration.

Average Hewitt-Stromberg and box dimensions of typical compact metric spaces

The main aim of this paper is to study average Hewitt-Stromberg and box dimensions of typical compact metric spaces belonging to the Gromov-Hausdorff space equipped with the Gromov-Hausdorff metric.