arXiv research

This paper deals with global asymptotic stability of prolongations of flows induced by specific vector fields and their prolongations. The method used is based on various estimates of the flows.

A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.

Study on discrete Okounkov bodies and their applications.

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

Stability of singularity formation in Yang-Mills fields in higher dimensions.

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

New method creates vacuum data at minimal and borderline decay thresholds.

We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture o…

Study on stability of cylindrical singularities in MCF of finite codimensions.

New findings on Mabuchi energy and stability of manifolds.

Paper proves convergence of SA algorithm via martingale and converse Lyapunov methods.

Continuous-time distributed mirror descent with integral feedback converges to global optimum.

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this …

In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

We study constant mean curvature Lorentzian hypersurfaces of $\mathbb{R}^{1,d+1}$ from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric s…

Study stabilizes second-order systems to first-order dynamics.

We initiate the study of the spherically symmetric Einstein-Klein-Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial bou…

We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space…

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking $T$ to be trivial, we see that asymptotic Chow-stability follows from stron…

The paper proves stability of a Ricci flat metric on a product of Einstein homogeneous spaces.

The paper develops a new algorithm for RBMs using dynamical mean-field theory.

This paper addresses pure gauge questions in the study of (asymptotically) de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results which a…

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.

New proof of past stability for Kasner solutions in $(3+1)$-dimensional Einstein vacuum spacetime.

Study stability thresholds of big line bundles, proving bounds and generalizing results.

Study shows stability in X-ray transform on specific hyperbolic manifolds.

Global stability bounds for matrix frames in phase retrieval problems.

We establish the full global non-linear stability of the Kerr-de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extendin…

Stability of positive mass theorem for hyperbolic manifolds studied.

Extends Minkowski stability proof to minimal decay assumptions.

Extends global stability of Minkowski spacetime to minimal decay assumptions.

Abstract: Necessary conditions for stabilizing subsets in systems are found.

Self-similar solutions to geometric flows are stable under small perturbations.

This paper introduces Non-Autonomous Input-Output Stable Network(NAIS-Net), a very deep architecture where each stacked processing block is derived from a time-invariant non-autonomous dynamical system. Non-autonomy is implemented by skip connections from the block input to each of the unrolled processing stages and al…

Stability proved for open Milne spacetime, showing gravity's long-term behavior.

Proves stability of Minkowski space for specific initial data.

In [7] Klainerman introduced the hyperboloidal method to prove the global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in [14] for proving, under the ma…

Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and c…

Large learning rates lead to various implicit biases in nonconvex optimization.

In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.

The paper analyzes stability and asymptotic behavior of hedging strategies in binomial and trinomial models.

Efficient inference for adaptive data with directional stability condition.

Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.

The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $ω_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ will be stud…

New stability and isolation results for Einstein manifolds.