arXiv research

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Upper bound on geodesic ball volume in Riemannian manifolds.

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…

Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.

Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.

The paper proves conditions for isoperimetric regions in curved spaces.

Study shows expanding Ricci solitons from specific metric cones.

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.

Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…

Survey on 4-manifolds with specific curvature properties.

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

Flat space for manifolds with tiny curvature.

The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

Proves inequalities on curved spaces with positive curvature.

In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Study on minimal submanifolds with finite curvature in Euclidean space.

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.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…

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

A 3-manifold's Ricci pinching condition implies it's flat if it has Euclidean volume growth.

Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.

The paper examines properties and rigidity of self-expanders in Euclidean space.

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

The study classifies complete self-shrinkers in Euclidean space.

Minimal submanifolds confined in space are highly restricted.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

The study examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff…

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete…

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

Study bubbling Kahler metrics using algebraic geometry.

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.

Study on higher-dimensional quasigeodesics in metric spaces.

The paper proves a Minkowski inequality on specific Riemannian manifolds.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclide…

We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.