arXiv research

Study uses equivariant topology to measure distances between G metric spaces.

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

Topology of non-orientable spaces without boundary is studied.

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

Study shows local topologies of certain geometric spaces.

The paper connects geometric and topological concepts to bound distances between metric spaces.

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.

Schwarzschild 3-manifold stability proven for 3D Penrose inequality.

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

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…

New principles prove precompactness of domains with lower Ricci curvature bound.

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

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.

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…

We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_…

New method uses cohomology to quantify molecular similarity.

Flow analysis leads to metric completion in Kähler geometry.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

Study shows continuity of non-Kähler Calabi-Yau conifold transitions.

Space of hyperbolic surfaces is path-connected.

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

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…

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…

Circle's metric is at least π/4 away from any simply connected geodesic space.

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…

The paper extends Perelman's theorems on Ricci flow entropy.

Ancient solutions to Ricci flow on torus bundles have additional symmetries.

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of …

Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.

The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of …

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

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

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.

Stable solution found for manifold topology from boundary data.

Proves stability of convex disks close to round caps.

Solves weighted bi-colored plane tree enumeration and applies to geometric problems.

Study Ricci flow on spaces with conical singularities, proving existence and curvature estimates.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

Defines new metrics for Lorentzian spaces and their convergence.

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…

In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.

We establish a regularity result for the metric on any 4-dimensional extremal Kähler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise n…

Round cylinders are rigid in Ricci shrinkers close to the standard product.

Denote by $\mathcal{A}(κ)$ the set of all compact Alexandrov surfaces with curvature bounded below by $κ$ without boundary, endowed with the topology induced by the Gromov-Hausdorff metric. We determine the connected components of $\mathcal{A}(κ)$ and of its closure.

Study shows stability of Schrödinger operator spectral data on a manifold.

In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.