arXiv research

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

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

Study shows uniform decay rate for singular mean curvature flows.

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

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.

New Sobolev inequalities found for curved spaces.

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

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.

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

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.

New findings on flatness of certain metrics with fast decay.

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

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

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

The paper establishes curvature estimates for solitons in higher dimensions.

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.

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.

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…

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

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.

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…

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 …

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

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.

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

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

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…

The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.

Flat Ricci-flat manifolds with bounded gradient of Green function are flat.

We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity.

Study gap phenomenon in flat manifolds with Ricci curvature.

The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible comple…

Polynomial decay of correlations shown for curved surfaces.

The study confirms positivity of Q-curvatures for specific conformal metrics.

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundam…

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-di…

New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.

This paper constructs a class of complete Kähler metrics of positive holomorphic sectional curvature on ${\bf C}^n$ and finds that the constructed metrics satisfy the following properties: As the geodesic distance $ρ\to\infty,$ the volume of geodesic balls grows like $O(ρ^{\frac{2(β+1)n}{β+2}})$ and the Riemannian scal…

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström solution arising from regular initial data remain globally bounded…

Study finite curvature solutions on surfaces with nonnegative Gauss curvature.

This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instant…