arXiv research

Sharp convergence theorem for sphere submanifolds proved.

Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.

The paper proves a convergence theorem for Wiener measures on holonomy groups.

Survey of recent Kleinian representation convergence results.

Survey on mean curvature flow with sphere theorems and Yau rigidity theory.

Study Gromov-Hausdorff convergence of metric pairs and tuples.

Study on surfaces pinched by curvature in space forms converging under specific conditions.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.

The paper proves stability of the positive mass theorem using intrinsic flat convergence.

We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem und…

Proves convergence of normal forms for infinite-dimensional Lie pseudo-group actions.

In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].

This paper studies neural network operators and their convergence properties.

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

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem li…

The paper explores null distance convergence for warped product spacetimes.

New theorem proves convergence of various discrete conformal structures to conformal maps.

Study on PDEs in Heston model with unique solution and convergence proof.

This paper is devoted to the study of convergence of sequences of solutions to the constant mean curvature H equation. The convergence domain is defined. The main Theorem characterizes the complement of this convergence domain: it shows that circle arcs of curvature 2H compose this complement. We then give results whic…

The paper reconstructs Lorentzian spacetimes from causal sets.

In this paper, we mainly study the compactness and local structure of immersing surfaces in $\mathbb{R}^n$ with local uniform bounded area and small total curvature $\int_{Σ\cap B_1(0)} |A|^2$. A key ingredient is a new quantity which we call isothermal radius. Using the estimate of the isothermal radius we establish a…

This paper formalizes $Q$-learning and linear TD convergence using Lean 4.

Paper proves CLTs for Q-learning with asynchronous updates.

The paper calculates how fast optimal investment strategies approach CRRA strategies in stochastic factor models.

Study sequences of static spacetimes using null distance convergence.

New proof shows local wealth condensation in economic models with biases.

Compactness results for Hermitian manifolds help understand Type IIB flow.

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorf…

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdo…

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theo…

In this paper, we prove a convergence theorem for sequences of Einstein Yang-Mills systems on $U(1)$-bundles over closed $n$-manifolds with some bounds for volumes, diameters, $L^{2}$-norms of bundle curvatures and $L^{\frac{n}{2}}$-norms of curvature tensors. This result is a generalization of earlier compactness the…

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to Hélein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in $\mathbb{R}^{n}$ w…

In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Usin…

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

Compactness theorem for timed-metric spaces established.

Study on prescribing positive curvature with conical singularities on a sphere.

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which cover all elliptic, overdetermined elliptic, subelliptic and parabolic…

In this paper we prove a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds.

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion o…

We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $S\subset M$, to a metric, $g_\infty$, on $M\setminus S$. We prove theorems which describe when $M_j=(M, g_j)$ converge in the Gromov-Hausdorff sense to the metric completion, $(M_\in…

Study geodesic Lie groups' convergence to limits with quantitative estimates.

Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence $\{S_n \rightarrow S\}$ of finite Galois covers of a hyperbolic Riemann Surface $S$, converging to the universal cover. The theorem states that the sequence of for…

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}\mathbb{P}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as $…

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.

This paper strengthens the central limit theorem for order statistics using relative entropy.