Gromov-Hausdorff Limit Research Articles

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

For metric measure spaces verifying the reduced curvature-dimension condition $CD^*(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally $RCD^*(K,N)$-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, $p$-spectral gap (or equivalently $p$-Poincar\'e inequality) for any $p\in [1,\infty)$, log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching $CD^*(K,N)$ spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition $CD(K,N)$ in the sense of Bakry-\'Emery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$-spectral gap. Finally let us mention that for essentially non-branching metric measure spaces, the local curvature-dimension condition $CD_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $CD^*(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic L\'evy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.

The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each two compact metric spaces) can be connected by a geodesic. For finite metric spaces a geodesic is constructed explicitly.

Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the GromovHausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kahler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kahler manifolds with nonnegative bisectional curvature.

Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.

On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.

On curvature pinching of conic 2-spheres

We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a generalization of the previous results of Chen-Lin and Bartolucci for 2-spheres with one or two conic points.

On the Kähler Ricci Flow on Projective Manifolds of General Type

We consider the K\"ahler Ricci flow on a smooth minimal model of general type, we show that if the Ricci curvature is uniformly bounded below along the K\"ahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model. Moreover, we can give a purely analytic proof of a recent result of Tosatti-Zhang (\cite{TZ}) that if the canonical line bundle $K_X$ is big and nef, but not ample, then the flow is of Type IIb.

Gromov–Hausdorff distance for pointed metric spaces

We establish an embedding theorem for the Gromov–Hausdorff limit of a sequence of proper pointed metric spaces.

On diameter controls and smooth convergence away from singularities

We prove that if a family of metrics, gi, on a compact Riemannian manifold, Mn, have a uniform lower Ricci curvature bound and converge to g∞ smoothly away from a singular set, S, with Hausdorff measure, Hn−1(S)=0, and if there exists connected precompact exhaustion, Wj, of Mn∖S satisfying diamgi(Mn)≤D0, Volgi(∂Wj)≤A0 and Volgi(Mn∖Wj)≤Vj where limj→∞⁡Vj=0 then the Gromov–Hausdorff limit exists and agrees with the metric completion of (Mn∖S,g∞). This is a strong improvement over prior work of the author with Sormani that had the additional assumption that the singular set had to be a smooth submanifold of codimension two. We have a second main theorem in which the Hausdorff measure condition on S is replaced by diameter estimates on the connected components of the boundary of the exhaustion, ∂Wj. This second theorem allows for singular sets which are open subregions of the manifold. In addition, we show that the uniform lower Ricci curvature bounds in these theorems can be replaced by the existence of a uniform linear contractibility function. If this condition is removed altogether, then we prove that limj→∞⁡dF(Mj′,N′)=0, in which Mj′ and N′ are the settled completions of (M,gj) and (M∞∖S,g∞) respectively and dF is the Sormani–Wenger Intrinsic Flat distance. We present examples demonstrating the necessity of many of the hypotheses in our theorems.

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Let $M$ be a complete K\"ahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the proof: The Cheeger-Colding theory on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions.

Bi-Lipschitz pieces between manifolds

A well-known class of questions asks the following: if X and Y are metric measure spaces and f : X \to Y is a Lipschitz mapping whose image has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors s -regular, topological d -manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov–Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds.

Essential circles and Gromov–Hausdorff convergence of covers

The [Formula: see text]-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of covering maps of compact geodesic spaces called “circle covers” that are “closed” with respect to Gromov–Hausdorff convergence and include [Formula: see text]-covers. In fact, we use circle covers to completely understand the limiting behavior of [Formula: see text]-covers. The proofs use the descrete homotopy methods developed by Berestovskii, Plaut, and Wilkins, and in fact we show that when [Formula: see text], the Sormani–Wei [Formula: see text]-cover is isometric to the Berestovskii–Plaut–Wilkins [Formula: see text]-cover. Of possible independent interest, our arguments involve showing that “almost isometries” between compact geodesic spaces result in explicitly controlled quasi-isometries between their [Formula: see text]-covers. Finally, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.

Convergence of isometries, with semicontinuity of symmetry of Alexandrov spaces

The equivariant Gromov–Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.

On low-dimensional Ricci limit spaces

We call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space . Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

Minimizing closed geodesics via critical points of the uniform energy

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy persist under the Gromov-Hausdorff convergence of Riemannian manifolds. Additionally, we relate half-geodesics (1/2-geodesics) to the Grove-Shiohama critical points of the distance function. This relationship affords us the ability to study the behavior of half-geodesics via the well developed field of critical point theory. Along the way we provide a complete characterization of the differentiability of the Riemannian distance function.

A remark on regular points of Ricci limit spaces

Let $Y$ be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in $Y$ is called $k$-regular, if its tangent is unique and is isometric to an $k$-dimensional Euclidean space. By \cite{B5}, there is $k>0$ such that the set of all $k$-regular point $\mathcal{R}_k$ has a full renormalized measure. An open problem is if $\mathcal{R}_l=\emptyset$ for all $l<k$? The main result in this paper asserts that if $\mathcal{R}_1\ne \emptyset$, then $Y$ is a one dimensional topological manifold. Our result improves the Handa's result \cite{Honda} that under the assumption that $1\leq \mathrm{dim}_H(Y)<2$.

Measure rigidity of Ricci curvature lower bounds

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space.We start our investigation from the Euclidean case by proving that if a positive Radon measure m over Rd is such that (Rd,|⋅|,m) verifies a weaker variant of MCP, then its support spt(m) must be convex and m has to be absolutely continuous with respect to the relevant Hausdorff measure of spt(m). This result is then used as a starting point to investigate the rigidity of MCP in the metric framework.We introduce the new notion of reference measure for a metric space and prove that if (X,d,m) is essentially non-branching and verifies MCP, and μ is an essentially non-branching MCP reference measure for (spt(m),d), then m is absolutely continuous with respect to μ, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching MCP reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov–Hausdorff convergence, provided an additional uniform bound holds.In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting. The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group.

Gromov–Hausdorff stability of linkage-based hierarchical clustering methods

A hierarchical clustering method is stable if small perturbations on the data set produce small perturbations in the result. These perturbations are measured using the Gromov–Hausdorff metric. We study the problem of stability on linkage-based hierarchical clustering methods. We obtain that, under some basic conditions, standard linkage-based methods are semi-stable. We also prove that, avoiding exotic examples, introducing any unchaining condition in the algorithm always produces unstable methods.

Inverse Limit Spaces Satisfying a Poincaré Inequality

Abstract We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].