Related papers: The Boundary between Compact and Noncompact Comple…

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms…

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with…

We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of…

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^\infty$-close (bi-Lipschitz) smooth metric with two-sided Ricci…

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 a complete biharmonic submanifold $\phi:(M,g)\rightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt…

We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic…

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below…

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some…

In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The…

We investigate the compact submanifolds in Riemannian space forms of nonnegative sectional curvature that satisfy a lower bound on the Ricci curvature, that bound depending solely on the length of the mean curvature vector of the immersion.…

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is…

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension…

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

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…

In this paper, we develop the infinitesimal geometry of the limit spaces of compact Riemannian manifolds with boundary, where we assume lower bounds on the sectional curvatures of manifolds and boundaries and the second fundamental forms of…

This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis…

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…