Related papers: Complete gradient shrinking Ricci solitons have fi…

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci…

Suppose $(M, g, f)$ is a 5-dimensional complete shrinking gradient Ricci soliton with $R=1$. If it has bounded curvature, we prove that it is a finite quotient of $\mathbb{R}^3\times \mathbb{S}^2$.

It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci…

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $\infty$-Bakry--Emery Ricci tensor is bounded by some positive…

We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the…

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or…

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in…

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and…

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…

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null…

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite…

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we…

In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and…

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of…

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many…

We investigate the topology of the compact submanifolds in round spheres that satisfy a lower bound on the Ricci curvature depending only on the length of the mean curvature vector of the immersion. Just in special cases, the limited…

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp…

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds…

A classical theorem of Bochner asserts that the isometry group of a compact Riemannian manifold with negative Ricci curvature is finite. In this paper we give several extensions of Bochner's theorem by allowing "small" positive Ricci…