Some compact conformally flat manifolds with non-negative scalar curvature

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.     

An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci \ge 0

By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades. Received: 11 November 1998 / Revised: 7 April 1999     

Manifolds with almost nonnegative curvature operator and principal bundles

We study closed manifolds with almost nonnegative curvature operator (ANCO) and derive necessary and/or sufficient conditions for the total spaces of principal bundles over (A)NCO manifolds to admit ANCO connection metrics. In particular, we provide first examples of closed simply connected ANCO manifolds which do not admit metrics with nonnegative curvature operator.     

Sobolev Inequalities in Manifolds with Nonnegative Curvature

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

Poisson Equation on Some Complete Noncompact Manifolds

We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact manifolds with nonnegative curvature.     

A splitting theorem for manifolds of almost nonnegative Ricci curvature

Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature.     

Ricci curvature,radial curvature and large volume growth

In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.     

Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

Science China Mathematics - In a previous paper (Jiang and Yang (2021)), we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and...     

Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.     

Harmonic and conformally Killing forms on complete Riemannian manifold

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.     

Ricci flatness of asymptotically locally Euclidean metrics

In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.

     

The entropy formula for linear heat equation

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

Local gradient estimate for harmonic functions on Finsler manifolds

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorems on noncompact Finsler manifolds with nonnegative Ricci curvature.     

On the rigidity of Riemannian–Penrose inequality for asymptotically flat 3-manifolds with corners

Mathematische Zeitschrift - In this paper we prove a rigidity result for the equality case of the Penrose inequality on 3-dimensional asymptotically flat manifolds with nonnegative scalar curvature...     

Weighted Sobolev Inequalities and Ricci Flat Manifolds

In this paper, we prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying a reverse volume doubling condition. It enables us to obtain rigidity results for Ricci flat manifolds. Received: November 2006, Revision: April 2007, Accepted: April 2007     

Eigenvalues of $$(-\triangle + \frac{R}{2})$$ on manifolds with nonnegative curvature operator

In this paper, we show that the eigenvalues of are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.     

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré’s result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.     

Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds

In the first part of this paper, we prove the sharp global Li‐Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman‐type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vázquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd.