arXiv research

The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…

Study compares spectral properties of a specific tensor in geometry.

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^…

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

Researchers developed volume comparison theorems in Finsler spacetimes.

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

Survey on open manifolds with nonnegative Ricci curvature and open questions.

We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate…

Ancient Ricci flows with nonnegative curvature operator have bounded entropy.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.

Paper studies fundamental groups of certain Ricci solitons.

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation $e^Φ = \det D^2 Φ$ on proper convex cones. We…

Complete Ricci flow from singular 3D manifold with pseudolocality.

Proves inequalities on curved spaces with positive curvature.

Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.

Flat open manifolds with full first Betti number have zero curvature.

The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.

Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to fo…

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

New topological restrictions found for spaces with nonnegative Ricci curvature.

Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.

The paper proves a theorem about splitting manifolds with specific curvature properties.

Uniformly Euclidean metrics with isolated singularities on certain manifolds are Ricci flat and have nonnegative synthetic Ricci curvature.

In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the $(λ_1, λ_2)$-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigid…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotica…

The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.

The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.

Proves Sobolev inequality on manifolds with specific curvature properties.

The study classifies graphs with specific curvature and maximum degree.

Extends a result on manifolds with specific curvature properties.

In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($I\ge5$) Riemannian m…

Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.

New theorem shows curvature concentration depends linearly on volume ratio.

We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E …

Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.

New curvature defined via graph resistances leads to Ricci flow.

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…

In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that $f$ is …

Ancient solutions to Kähler Ricci flow classified completely.

Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.