arXiv research

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

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

Study Gromov-Hausdorff convergence of metric pairs and tuples.

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

Proves compactness for timed-metric spaces using new distance and maps.

Stability of Wasserstein spaces under various convergence types.

The paper explores null distance convergence for warped product spacetimes.

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

Compactness theorem for timed-metric spaces established.

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 limits of manifolds with Kato bound Ricci curvature, proving volume convergence.

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 …

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

Paper uses Gromov-Hausdorff convergence to re-examine surface classification.

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…

Topology of non-orientable spaces without boundary is studied.

Study limits of Kähler submanifolds and prove no holomorphic isometries.

The paper connects different convergence concepts in geometric analysis.

Research shows surfaces close to planes in Hausdorff distance.

Paper introduces a new convergence for Lorentzian spaces using causal diamonds.

Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.

In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact $C^k$-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$. The most important fact in this discussion is as follows. The Hausdorff distance fun…

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…

Study shows convergence of cscK surfaces in Hilbert scheme.

Global existence and convergence of pluriclosed flow on Oeljeklaus-Toma manifolds.

Study shows tori metrics converging to flat under specific conditions.

In this paper, we will show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets.

Defines new metrics for Lorentzian spaces and their convergence.

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

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…

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…

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

In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that t…

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prov…

Flow analysis leads to metric completion in Kähler geometry.

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

Study on curvature blow-up and convergence of continuity method on Hirzebruch surface.

The study proves compactness and structure of Ricci flow limits.

The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.

Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.

New geometric measure simplifies complex analysis.

Study of irreversible metric-measure spaces, proving convergence and stability results.

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

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…

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

Two singular mean curvature flows converge if Hausdorff distance decreases faster than inverse time.

Compactness theorems for $G_2$-solitons established with scalar curvature and potential function constraints.

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…