arXiv research

Study describes limits of surfaces in a mathematical space.

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

Close to complex projective spaces, Ricci shrinkers are rigid.

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs …

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.

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.

Study flat manifolds' collapsed limits as flat orbifolds.

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close…

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…

We show that for n dimensional manifolds whose the Ricci curvature is greater or equal to n-1 and for k in {1,...,n+1}, the k-th eigenvalue for the Laplacian is close to n if and only if the manifold contains a subset which is Gromov-Hausdorff close to the unit sphere of dimension k-1. For k=n+1, this gives a new proof…

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

Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance func…

Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.

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.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

Proves stability of convex disks close to round caps.

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

The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.

Proves torus sequences can't collapse to intervals under curvature bounds.

Let $(Y,d)$ be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, $Y$ is a smooth manifold satisfying a shrinking Ricci soliton equation.

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

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

The study proves compactness and structure of Ricci flow limits.

Study shows different fundamental groups for manifolds with same limit.

Space of hyperbolic surfaces is path-connected.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows

Let $(Y, d)$ be a Gromov-Hausdorff limit of $n$-dimensional closed shrinking Kähler-Ricci solitons with uniformly bounded volumes and Futaki invariants. We prove that off a closed subset of codimension at least 4, Y is a smooth manifold satisfying a shrinking Kähler-Ricci soliton equation. A similar convergence result …

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

The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.

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

The paper proves topological stability between RCD spaces and Riemannian manifolds.

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

Study Gromov-Hausdorff convergence of metric pairs and tuples.

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

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.

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

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.

We construct for every finite-dimensional Alexandrov space $A$ and every point $p \in A$ a $2$-convex function $f_p$ in a small neighborhood around $p$, which approximates $\operatorname{dist}_p^2$ up to second order. Moreover, the function $f_p$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dim…

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

Stability of Wasserstein spaces under various convergence types.

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

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

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…

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

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

Study shows dimension constraints for isometry groups in non-collapsed Riemannian manifolds.

Flow analysis leads to metric completion in Kähler geometry.