Related papers: On the weighted orthogonal Ricci curvature

We prove the existence of a smooth complete conformal metric with prescribed kth elementary symmetric function of negative Ricci curvature under certain condition on general domain in Euclidean space. We then formulate this problem for more…

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci…

We construct a geometric framework for cosmological large-scale structure based on optimal transport theory and Wasserstein geometry. In this framework, Ricci curvature on the probability measure space $\mathcal{P}_2(M)$ is characterized by…

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional…

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our…

We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results…

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some…

In this paper, we establish a K\"ahlerian or projectivity criterion for a class of compact Hermitian surfaces with non-positive second Chern-Ricci curvature.

We record two remarks. First, for a compact K\"ahler manifold with semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration is exactly the number of non-truly-flat directions. Second, for compact K\"ahler…

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact K\"ahler manifolds. In this short note, we give anothe proof of…

It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric…

In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci…

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the…

We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this…

In this paper, the concept of isotropic projective Ricci curvature has been investigated. By classification of Randers metric of isotropic projective Ricci curvature, it is shown that Randers metric of projective Ricci curvature is…

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of…

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target…

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler…

We develop geometric analysis on weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted…

We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field $T$, this problem asks for solutions to the equation $\mathrm{Ric}(g)=cT$ for some constant $c$. Our approach is based on examining global…