Gromov-Hausdorff Limit Research Articles

Mahler-type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds

In this paper, we study a non-collapsed Gromov–Hausdorff limit of a sequence of compact Heisenberg manifolds with sub-Riemannian metrics. In the case of strictly sub-Riemannian case, we show that if a sequence has an upper bound of the diameter and a lower bound of Popp’s measure, then it has a convergent subsequence in the Gromov–Hausdorff topology, and the limit is isometric to a compact Heisenberg manifold of the same dimension. The same conclusion is also shown for Riemannian case with the additional assumption on Ricci curvature lower bounds.

Graphs with nonnegative curvature outside a finite subset, harmonic functions, and number of ends

AbstractWe study graphs with nonnegative Bakry–Émery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov–Hausdorff convergence, we prove that the space of bounded harmonic functions is finite dimensional and, as a corollary, the number of nonparabolic ends is finite.

Average general fractal dimensions of typical compact metric spaces

The main objective of this paper is to investigate the average general fractal dimensions of typical compact metric spaces within the Gromov-Hausdorff space, using the Gromov-Hausdorff metric. As an application of the main results, we demonstrate that a typical compact metric space Σ exhibits such irregularity that the lower and upper average general Hewitt-Stromberg dimensions, as well as the general box dimensions corresponding to all higher-order Hölder and Cesàro averages, diverge significantly.

Metric measure spaces with Ricci bounds from below

In this survey we introduce new results on metric measure spaces with Ricci curvature bounded below, so-called R C D \mathrm {RCD} spaces, including stability results of several energies with respect to measured Gromov-Hausdorff convergence, Weyl’s law, and a characterization of noncollapsed RCD spaces.

Horosymmetric limits of Kähler–Ricci flow on Fano $\mathbf{G}$-manifolds

We prove that on a Fano \mathbf{G} -manifold (M,J) , the Gromov–Hausdorff limit of the Kähler–Ricci flow with initial metric in 2\pi c_{1}(M) must be a \mathbb{Q} -Fano horosymmetric variety M_{\infty} which admits a singular Kähler–Ricci soliton. Moreover, M_{\infty} is the limit of a \mathbb{C}^{*} -degeneration of M induced by an element in the Lie algebra of the Cartan torus of \mathbf{G} . A similar result can be proved for Kähler–Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the existence of type II singularities of Kähler–Ricci flows on Fano \mathbf{G} -manifolds to Fano horosymmetric manifolds.

Eguchi–Hanson metrics arising from Kähler–Einstein edge metrics

Calabi–Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kähler-Einstein edge metrics singular along two disjoint divisors on the Calabi–Hirzebruch manifolds and study their Gromov–Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi–Hanson metric arises in this way naturally as a Gromov–Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov–Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov–Rubinstein. This gives a new interpretation of both the Eguchi–Hanson space and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces.

On the Gromov–Hausdorff limits of compact surfaces with boundary

In this work we investigate Gromov–Hausdorff limits of compact surfaces carrying length metrics. More precisely, we consider the case where all surfaces have the same Euler characteristic. We give a complete description of the limit spaces and study their topological properties. Our investigation builds on the results of a previous work which treats the case of closed surfaces.

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2 -manifold with boundary) with locally finite Hausdorff 2 -measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2 -sphere with finite Hausdorff 2 -measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

Gromov-Hausdorff limits of closed surfaces

Abstract We completely describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

Stability and equivariant Gromov–Hausdorff convergence

AbstractWe give applications of equivariant Gromov–Hausdorff convergence in various contexts. Namely, using equivariant Gromov–Hausdorff convergence, we prove a stability result in the setting of compact finite‐dimensional Alexandrov spaces. Moreover, we introduce the notion of an almost commutative diagram and show that its use simplifies both exposition and argument.

Gromov–Hausdorff convergence of metric pairs and metric tuples

We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.

Diameter estimates in Kähler geometry

AbstractDiameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long‐standing problem of uniform diameter bounds and Gromov–Hausdorff convergence of the Kähler–Ricci flow, for both finite‐time and long‐time solutions.

The asymptotic behavior of Bergman kernels

Let (X,d,p) be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds (Ml,ωl,Ll,hl,pl) with Ric(hl)=2πωl, Ric(ωl)≥−Λωl and Vol(B1(pl))≥v, ∀l∈N, where Λ,v>0 are constants. Then X is a normal complex space [33].In this paper, we discuss the convergence of the Hermitian line bundles (Ll,hl) and the Bergman kernels. In particular, we show that the Kähler forms ωl converge to a unique closed positive current ωX on Xreg. By establishing a version of L2 estimate on the limit line bundle on X, we give a convergence result of Fubini-Study currents on X. Then we prove that the convergence of Bergman kernels implies a uniform Lp asymptotic expansion of Bergman kernel on the collection of n-dimensional polarized Kähler manifolds (M,ω,L,h) with Ricci lower bound −Λ and non-collapsing condition Vol(B1(x))≥v>0. Under the additional orthogonal bisectional curvature lower bound, we will also give a uniform C0 asymptotic estimate of Bergman kernel for all sufficiently large m, which improves a theorem of Jiang [28]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [20].

From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds

From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds

An extreme limit with nonnegative scalar curvature

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the W1,p sense for p<2 to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov–Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.

Null distance and Gromov–Hausdorff convergence of warped product spacetimes

Null distance and Gromov–Hausdorff convergence of warped product spacetimes

Calabi symmetry and the continuity method

We study the convergence and curvature blow up of the continuity method on a generalized Hirzebruch surface. We show that the Gromov–Hausdorff convergence is similar to that of the Kähler–Ricci flow and obtain curvature estimates. We also show that a general solution to the continuity method either exists at all times, or the scalar curvature blows up. This behavior is known to be exhibited by the Kähler–Ricci flow.

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

AbstractIn this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows

Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

The Continuity Equation on Hopf and Inoue Surfaces

Abstract We study the continuity equation of La Nave–Tian, extended to the Hermitian setting by Sherman–Weinkove, on Hopf and Inoue surfaces. We prove a priori estimates for solutions in both cases and Gromov–Hausdorff convergence of Inoue surfaces to a circle.