Related papers: A note on two-positive Ricci curvature and positiv…

We show how to lift positive Ricci and almost non-negative curvatures from an orbit space $M/G$ to the corresponding $G$-manifold, $M$. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions…

Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the…

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are…

We prove the equivalence between the several notions of generalized Ricci curvature found in the literature. As an application, we characterize when the total generalized Ricci tensor is symmetric.

This article reports recent developments of the research on Hamilton's Ricci flow and its applications.

Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also, the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally a derivation of Newtonian Gravity…

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some…

This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are…

We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the…

We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds.

It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded above in terms of the Kato constant and the total Ricci…

We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.

Let (M, g) be a compact Einstein Riemannian manifold with boundary. We show that under certain conditions, the map that associates to a metric on M its Ricci curvature, its induced conformal class on the boundary, and its mean curvature on…

A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of…

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded…

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian…