arXiv research

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

Classifies gravitational instantons with quadratic volume growth.

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.

The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

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…

The paper proves unique characterization of gravitational instantons with specific volume growth.

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

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\to 0$ as $d(x)=d_g(x,p)\to \infty$, then $M^n$ has the quadratic curvature dec…

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

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 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…

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

The paper proves manifold isometries for certain gradient Ricci solitons.

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…

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

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 …

Study noncollapsed F-limit metric solitons, proving properties similar to smooth Ricci shrinkers.

We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…

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 prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.

Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.

The study provides volume growth estimates for specific types of manifolds.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n und…

Paper proves volume growth estimate for steady gradient Ricci solitons.

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

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.

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

Study on volume growth of horospheres in specific Heintze groups.

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…

Graphs with stronger curvature grow faster.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

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.

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo…

We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results al…

Study volume growth in Milnor fibers using real Lagrangians.

Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear 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…

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

We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

The study proves that certain manifolds can have metrics with specific volume growth.

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

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…