Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

We study the injectivity radius bound for 3-d Ricci flow with bounded curvature. As applications, we show the long time existence of the Ricci flow with positive Ricci curvature and with curvature decay condition at infinity. We partially settle a question of Chow-Lu-Ni [Hamilton’s Ricci Flow, p. 302].     

Geography of 4-manifolds with positive scalar curvature

We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov’s observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

New connected sums with positive Ricci curvature

We construct a new infinite family of Ricci positive manifolds, generalising a well-known result of Sha and Yang.      

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

Conformal vector fields on complete Finsler spaces of constant Ricci curvature

In this work, it is proved that if a complete Finsler manifold of positive constant Ricci curvature admits a solution to a certain ODE, then it is homeomorphic to the n-sphere. Next, a geometric meaning is obtained for solutions of this ODE, which is applicable to Einstein–Randers spaces. Moreover, some results on Finsler spaces admitting a special conformal vector field are obtained.     

Four-manifolds with positive isotropic curvature

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.     

Noncompact 4-manifolds with uniformly positive isotropic curvature

In this paper, we study Ricci flow on noncompact 4-manifolds with uniformly positive isotropic curvature and with no essential imcompressible space form. That means there is positive lower bound of isotropic curvature and bounded geometry. Then by Perelman's technique, we can analyze the structures of such manifolds.     

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

On the four-dimensional T<Superscript>2</Superscript>-manifolds of positive Ricci curvature

We prove that there is a T ²-invariant Riemannian metric of positive Ricci curvature on every four-dimensional simply connected T ²-manifold.     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.      

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Volume entropy based on integral Ricci curvature and volume ratio

For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.     

Greatest lower bounds on Ricci curvature for toric Fano manifolds

In this short note, based on the work of Wang and Zhu (2004) [8], we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.