The paper extends vector bundle theory to non-Hausdorff manifolds.
problem Generalizing vector bundle theory to non-Hausdorff manifolds.
method Using Čech cohomology to classify real non-Hausdorff line bundles.
result Vector bundles over non-Hausdorff manifolds can be constructed as colimits of standard vector bundles.
Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.
problem Distance between Lorentzian manifolds.
method Intrinsic timed Hausdorff convergence.
result Intrinsic timed Hausdorff convergence implies Gromov-Hausdorff and big bang convergence.
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
Study of de Rham cohomology on non-Hausdorff manifolds.
problem De Rham cohomology on non-Hausdorff manifolds.
method Careful discussion of non-Hausdorff differential forms, Mayer-Vietoris sequences.
result Proved de Rham's Theorem and Gauss-Bonnet theorem for non-Hausdorff manifolds, including counterterms.
We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.
Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.
problem Understanding Calabi-Yau metrics on converging manifolds.
method Analysis of Gromov-Hausdorff limits of metrics on Calabi-Yau fibrations.
result Gromov-Hausdorff limit is homeomorphic to the base of the fibration and discriminant locus has high Hausdorff codimension.
New Hausdorff integrations for Lie algebroids and symplectic groupoids.
problem Integrating Lie algebroids and symplectic groupoids.
method Hausdorff versions of Lie Integration Theorems 1 and 2, Lie equivalences, and algebraic approach to holonomy.
result Generalization of integration of subalgebroids to non-wide cases and detailed exploration of foliation groupoids.
This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distributi…
We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.
The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.
problem Proving the existence of isoperimetric regions in Riemannian manifolds.
method Gromov-Hausdorff asymptotic analysis to study perimeter-minimizing sequences.
result Existence of isoperimetric regions in noncollapsed Riemannian manifolds with Ricci curvature bound.
In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.
The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.
problem Understanding the Gromov-Hausdorff limits of tori with Ricci conditions.
method Constructing metrics on Rn and analyzing their limits. result The Gromov-Hausdorff limit of tori with Ricci bounds is not always a topological manifold.
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact Ck-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where k≥1. The most important fact in this discussion is as follows. The Hausdorff distance fun…
Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.
problem Computing cohomology for foliated manifolds, especially when non-Hausdorff.
method Develops a Künneth formula for specific cases of Hausdorff foliated cohomology and finite-dimensional cohomology.
result Valid Künneth formula for certain foliated cohomology spaces, with counterexamples for others.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold.
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
Study the metric geometry of Cauchy hypersurfaces in spacetimes.
problem Properties of the space of Cauchy hypersurfaces.
method Equipped with a Hausdorff-type metric, studied completeness and local compactness.
result Generalized completeness results for spacetimes.
Study of submanifolds in symplectic and contact manifolds using Hausdorff metrics.
problem Understanding the subtle interactions between submanifolds and metrics in symplectic and contact geometry.
method Applying Hausdorff metric to study sequences of submanifolds and proving metric versions of conjectures.
result Proves metric versions of the nearby Lagrangian conjecture and Viterbo conjecture on spectral norm.
Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.
problem Characterizing the Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature.
method Analysis of the Gromov-Hausdorff limit space of orthonormal frame bundles equipped with an almost canonical metric.
result The singular set of the limit space has codimension ≥4 and the complement contains an open and dense C1,α-Riemannian manifold. We discuss the behavior of (λ1.p(M))1/p with respect to the Gromov-Hausdorff topology and the variable p, where λ1,p(M) is the first positive eigenvalue of the p-Laplacian on a compact Riemannian manifold M. Applications include new estimates for the first eigenvalues of the p-Laplacian on Rieman…
Study collapsing Calabi-Yau metrics and flows on fiber spaces.
problem Understanding the behavior of Calabi-Yau metrics and flows during collapsing.
method Analyzing the collapsing of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces.
result Identify the collapsed Gromov-Hausdorff limit and bounds for Hausdorff measure.
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.
problem Understanding structure of limits of manifolds with Ricci curvature bounds.
method Mosco convergence of Dirichlet energies to Cheeger energy, introduction of monotone quantities, volume convergence.
result Volume convergence to Hausdorff n-measure in limits of manifolds.
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curv…
The paper characterizes limits of manifolds using Gromov-Hausdorff metric.
problem Characterizing limits of generalized manifolds using Gromov-Hausdorff metric.
method Using Gromov-Hausdorff metric dG, the paper proves that manifold-like generalized n-manifolds are limits of topological n-manifolds under certain conditions. result Manifold-like generalized n-manifolds are limits of topological n-manifolds under specific conditions. The paper improves estimates on singular sets in manifolds with integral curvature bounds.
problem Estimating the singular set of manifolds with integral curvature bounds.
method Using Gromov-Hausdorff limits and Cheeger-Naber methods.
result Improved Minkowski dimension estimate for singular sets.
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.
problem The study of limits of sub-Riemannian Heisenberg manifolds and their properties.
method Analysis of Gromov-Hausdorff limits with upper and lower bounds on diameter and Popp's measure.
result The non-collapsed limits of sub-Riemannian Heisenberg manifolds converge to a compact Heisenberg manifold.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
This paper studies limits of aspherical manifolds with specific curvature conditions.
problem Understanding the Gromov-Hausdorff limits of aspherical manifolds with given curvature constraints.
method Analyzing sequences of compact manifolds with Ricci curvature or sectional curvature conditions, and using diffeomorphism or homeomorphism properties.
result If the manifolds are diffeomorphic or homeomorphic to nilmanifolds, their limits are also diffeomorphic or homeomorphic to nilmanifolds.
We prove that if Y is the Gromov-Hausdorff limit of a sequence of complete manifolds, Min, with a uniform lower bound on Ricci curvature then Y has a universal cover.
Newly discovered Eguchi-Hanson metric arises from edge metrics.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler ma…
The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.
problem Understanding the geometry and dimensions of Lorentzian spaces.
method Construction of Gromov-Hausdorff metrics, calculation of dimensions, and analysis of Lorentzian spaces.
result Dushnik-Miller dimension of Minkowski spaces is countably infinite.
New principles prove precompactness of domains with lower Ricci curvature bound.
problem Proving precompactness of domains with lower Ricci curvature bound.
method Quantitative Hopf-Rinow theorem and doubling property.
result New precompactness principles applicable to incomplete Riemannian manifolds.
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.
Given a sequence of complete(compact or noncompact) Kähler manifolds Min with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …
We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the Lk/2 norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension k singularity.
Global existence and convergence of pluriclosed flow on Oeljeklaus-Toma manifolds.
problem Global existence and convergence of pluriclosed flow on specific complex manifolds.
method Established global existence with arbitrary initial data and Gromov-Hausdorff convergence of blowdown limits.
result Gromov-Hausdorff convergence of blowdown limits to a torus under conjectural bounds.
New findings on leafwise quasi-geodesic foliations in 3-manifolds.
problem Understanding leafwise quasi-geodesic foliations in 3-manifolds.
method Analyzing intersections of transverse foliations in 3-manifolds with Gromov hyperbolic leaves.
result Hausdorff leafspace condition for leafwise quasi-geodesic foliations.
The topology of the Hausdorff leaf spaces (HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transversal and tangential …
Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.
problem Understanding the limiting behavior and symmetry properties of Ricci flow on homogeneous spaces.
method Introducing a novel normalization for the homogeneous Ricci flow and characterizing Gromov-Hausdorff limits.
result Full classification of Gromov-Hausdorff limits and detailed phase portraits for three-isotropy-summands flag manifolds.
Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.
problem Characterizing hyperconvex representations of hyperbolic groups.
method Geometric and topological analysis of representations in mPSL(d,C). result Virtual isomorphism to Kleinian groups and flag manifold Hausdorff dimension restriction.
Study limits of curved spaces with boundaries.
problem Understanding geometric structures of curved spaces with boundary constraints.
method Gromov-Hausdorff convergence and collapsing analysis of compact Riemannian manifolds with boundary.
result Describe local geometric structure of limit spaces and establish stability results.