arXiv research

The paper classifies certain types of Ricci solitons with specific curvature properties.

We prove that any noncompact $κ$-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.

Study on curvature decay in steady Ricci solitons, proving dichotomy.

The paper establishes curvature estimates for solitons in higher dimensions.

Proves stability of charged black holes with small charge.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady …

Proves boundedness and decay for spin 2 Teukolsky system on Reissner-Nordström spacetime.

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null inf…

Proves boundedness and decay for spin 1 Teukolsky equation on Reissner-Nordström spacetime.

Study on scalar curvature decay on non-compact manifolds linked at infinity.

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…

Study on scalar curvature decay in four-dimensional steady solitons.

Let $(M^n, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that $M$ is holomorphically covered by a pseudoconvex domain in $\C^n$ which is homeomorphic to $\R^{2n}$, provided $(M^n, g)$ has uniform linear average quadratic curvature decay.

Study shows uniform decay rate for singular mean curvature flows.

We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the pr…

We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery's $CD(K,n)$ condition for linear graphs and prove the triviality of edge w…

New Sobolev inequalities found for curved spaces.

The paper studies steady solitons with curvature decay and proves their smoothness.

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on…

The study uses Ricci flow to prove flatness of certain Riemannian manifolds.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.

In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex…

Sharp decay constant for positive scalar curvature metrics on manifolds.

In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvature $R(x)$ satisfies $\frac{C_0^{-1}}{ρ(x)}\le R(x)\le \frac{C_0}{ρ(x)}$ for some constant $C_0>0$, where $ρ(x)$ denotes the distance from a fixed point $x_0$. Our result doesn't assume that the soliton is $…

New findings on flatness of certain metrics with fast decay.

The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.

New decay estimates for scalar curvature of steady gradient Ricci solitons.

New rigidity result for metrics with positive scalar curvature and specific decay.

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

The study proves manifolds with positive scalar curvature can be decomposed into spherical and toroidal pieces.

Last SGD iterate bounds for overparameterized linear regression.

In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curva…

We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

New proof shows 3-manifolds with specific curvature properties are either flat or compact.

We show that there are topological obstructions for a noncompact manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of Euclidean space of the same dimension.

Paper proves Gromov's conjecture on manifolds with certain group properties.

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

Study provides obstructions for Q-curvature on complete metrics in n-space.

We present a method in nonlinear elliptic systems to study curvature decays on asymptotically locally Euclidean (ALE) manifolds. In particular, we show that scalar flat Kahler and harmonic ALE metrics of real dimension n are of order n-2.

The paper proves existence and growth estimates for inverse mean curvature flow and related $p$-Laplacian Green kernel decay.

Ridgeless ReLU networks interpolate datasets and extrapolate based on curvature signs.

We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay like one over time squared. In order to do this, we prove a new Li-Yau type differential Harnack inequality for Ricci flow on surfaces.

We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature de…

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when …