Hausdorff Limits 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.

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.

Hausdorff limits of external rays: The topological picture

AbstractWe study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree with connected Julia sets.

Hausdorff limits of submanifolds of symplectic and contact manifolds

We study sequences of immersions respecting bounds coming from Riemannian geometry and apply the ensuing results to the study of sequences of submanifolds of symplectic and contact manifolds. This allows us to study the subtle interaction between the Hausdorff metric and the Lagrangian Hofer and spectral metrics. In the process, we get proofs of metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm. We also get C0-rigidity results for a vast class of important submanifolds of symplectic and contact manifolds in the presence of Riemannian bounds. Likewise, we get a Lagrangian generalization of results of Hofer [19] and Viterbo [42] on simultaneous C0 and Hofer/spectral limits — even without any such bounds.

On Some Systems of Equations in Abelian Varieties

Abstract We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety $A$, which split as the product of a linear subspace of the Lie algebra of $A$ and an algebraic variety. This is motivated by work of Zilber and of Bays–Kirby, which establishes that a positive answer to the conjecture would imply quasiminimality of certain structures on the complex numbers. Our proofs use various techniques from homology (duality between cup product and intersection), differential topology (transversality), and o-minimality (definability of Hausdorff limits), hence we have tried to give a self-contained exposition.

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.

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

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.

A linear programming approach to approximating the infinite time reachable set of strictly stable linear control systems

The infinite time reachable set of a strictly stable linear control system is the Hausdorff limit of the finite time reachable set of the origin as time tends to infinity. By definition, it encodes useful information on the long-term behavior of the control system. Its characterization as a limit set gives rise to numerical methods for its computation that are based on forward iteration of approximate finite time reachable sets. These methods tend to be computationally expensive, because they essentially perform a Minkowski sum in every single forward step. We develop a new approach to computing the infinite time reachable set that is based on the invariance properties of the control system and the desired set. These allow us to characterize a polyhedral outer approximation as the unique solution to a linear program with constraints that incorporate the system dynamics. In particular, this approach does not rely on forward iteration of finite time reachable sets.

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.

An Arnold-type principle for non-smooth objects

In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in \(C^0\) settings: suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: \(C^0\) Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles.

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.

Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces

We give an analogue of triangle comparison for Kähler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov–Hausdorff limits. We discuss tangent cones and singular Kähler spaces.

Filled Julia Sets of Chebyshev Polynomials

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set \(K\subset {\mathbb C}\) and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.