arXiv research

Study describes limits of surfaces in a mathematical space.

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

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.

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

Study on metric spaces with properties (ETR), (LBD) and their convergence.

Newly discovered Eguchi-Hanson metric arises from edge metrics.

Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.

This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.

Study shows local topologies of certain geometric spaces.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ 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 …

Study flat manifolds' collapsed limits as flat orbifolds.

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…

Triangle comparison for Kaehler manifolds with curvature bounds.

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

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…

Extends arguments to limit structure in Calabi-Yau degenerations.

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…

This paper studies limits of aspherical manifolds with specific curvature conditions.

Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…

We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the $L^{k/2}$ norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension $k$ singularity.

Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.

Flow analysis leads to metric completion in Kähler geometry.

I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…

Study limits of curved spaces with boundaries.

Compactifies metrics on K3 surfaces with algebraic description.

Study shows different fundamental groups for manifolds with same limit.

New examples show strong Kato limits can be branching and not satisfy known conditions.

The paper characterizes limits of manifolds using Gromov-Hausdorff metric.

Study limits of Kähler submanifolds and prove no holomorphic isometries.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ 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…

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of the…

Defines new metrics for Lorentzian spaces and their convergence.

The study proves compactness and structure of Ricci flow limits.

Study of convergence of point-object configurations to a charged dust continuum.

We investigate the case of the Kahler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(-k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. In addition, we establish the result that the Gromov-Hausdorff limit coincides wi…

New theorem on flat tori stability using harmonic maps and Ricci flow.

Paper shows limits of Heisenberg manifolds are flat tori.

The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.

The paper proves a conjecture about manifold limits and characterizes their structure.

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler 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…

The paper explores null distance convergence for warped product spacetimes.

Magnitude is not continuous but may be stable for most finite metric spaces.