Related papers: A lower bound for the curvature integral under an …

This article surveys results for Riemannian manifolds of positive and non-negative sectional curvature with symmetries.

We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded…

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when…

We prove a laplacian comparison theorem in the barrier sense for the function distance to the boundary of Riemannian manifolds with nonnegative Ricci curvature, area and mean curvature of the boundary bounded above. As an application we get…

Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound.…

We consider the following generalisation of a well-known problem in Riemannian geometry: When is a smooth real-valued function s on a given compact n-dimensional manifold M (with or without boundary) the scalar curvature of some smooth…

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…

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami…

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature…

Following ideas of Gromov we prove scalar and mean curvature comparison results for Riemannian bands with lower scalar curvature bounds in dimension $n\leq7$. The model spaces we use are warped products over scalar-flat manifolds with…

We show that any Riemannian metric conformal to the round metric on $S^n$, for $n\geq 4$, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on $S^n$ in the sense of uniform convergence of Riemannian…

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed…

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result…

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci…

By making use of the classification of real simple Lie algebra, we get the maximum of the squared length of restricted roots case by case, thus we get the upper bounds of sectional curvature for irreducible Riemannian symmetric spaces of…

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 establish Sobolev and Moser-Trudinger inequalities with best constants on noncompact Riemannan manifolds with Ricci curvature bounded below, and positive injectivity radius.

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with…

We study some preservation phenomena for lower bound of total scalar curvatures on a smooth manifold. In particular, we prove that the lower bound of the weighted total scalar curvature (which is known as Perelman's…

The objet of this paper is the study of the variations of a functional whose integrant is the r-th weighted curvature on the hypersurface of a closed Riemannian manifold. Some applications to hypersurfaces of the Euclidean space and the…