hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

We prove that the manifold Mn of minimal radial curvature Kmin o ≥ 1 is homeomorphic to the sphere Sn if its radius or volume is larger than half the radius or volume of the round sphere of constant curvature 1. These results are optimal and give a complete generalization of the corresponding results for manifolds of sectional curvature bounded from below. 0. Introduction and results 0.1. A sta...

In this paper the Maximal Diameter Theorem of Riemannian geometry is proven for Riemannian orbifolds. In particular, it is shown that a complete Riemannian orbifold with Ricci curvature bounded below by (n− 1) and diameter = π, must have constant sectional curvature 1, and must be a quotient of the sphere (S, can) of constant sectional curvature 1 by a subgroup of the orthogonal group O(n+1) ac...

In this paper, by supposing a natural comparison inequality on the positive r-th mean curvatures of the hypersurface, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces immersed in a semi-Riemannian warped product of constant sectional curvature. Generalizing the above results, under a restriction on the sectional curvature or the Ricci curvature tensor of the fiber...

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...

Let (M,ω) be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective has ample canonical bundle. In this paper, we show any irreducible subvariety of general type, thus confirming in particular case celebrated conjecture Lang. Moreover, can extend the theorem to quasinegative curvature building on earlier...