In this paper, we study closed four-dimensional manifolds. particular, show that under various new pinching curvature conditions (for example, the sectional is no more than 5/6 of smallest Ricci eigenvalue) then manifold definite. If restricting to a metric with harmonic Weyl tensor, it must be self-dual or anti-self-dual same conditions. Similarly, if an Einstein metric, either complex project...

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n−1)-submanifold S . The main result is that the fundamental group of M \S is relatively hyperbolic, relative to fundamental groups of the ends of M \S , and M \S admits a complete finite volume A -regular Riemannian metric of negative sectional curvature. It fo...

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one e...