Related papers: Curvature bounds for regularized riemannian metric…
In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…
In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…
We show that a general class of singular K\"ahler metrics with Ricci curvature bounded below define K\"ahler currents. In particular the result applies to singular K\"ahler-Einstein metrics on klt pairs, and an analogous result holds for…
In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true…
On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact…
We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.
In this paper, we derive a general regularity estimate for any 4-d spacetime, in terms of a priori bounds of the Ricci curvature and Lie derivative of the Lorentzian metric with respect to an arbitrary timelike vector field.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is…
Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic K\"ahler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric…
We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…
We provide a general contractibility criterion for subsets of Riemannian metrics on the disc. For instance, this result applies to the space of metrics that have positive Gauss curvature and make the boundary circle convex (or geodesic).…
In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional K\"ahler Ricci flow. These integral estimates improve…
We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.
We use certain Morse functions to construct conformal metrics with negative sectional curvature on locally conformally flat manifolds with boundary. Moreover, without conformally flatness assumption, we also construct conformal metric of…
In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.
This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their…
We study the problem of prescribing the Ricci curvature in the class of naturally reductive metrics on a compact Lie group. We derive necessary as well as sufficient conditions for the solvability of the equations and provide a series of…
We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and…
We prove a canonical variation-type result for submersion metrics with positive intermediate Ricci curvatures. This can then be used in conjunction with surgery techniques to establish the existence of metrics with positive intermediate…
In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $\mu$ on $X$, then the magnitude of $\mu$ at any point of small injectivity radius is uniform bounded from…