On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature

Abstract

The relation between curvature and topology is a fundamental problem in differential geometry. For example, the Gauss-Bonnet theorem says the sign of curvature could determine the genus of the surface. Brendle and Schoen [8] proved that if a compact manifold has sectional curvature between 1/4 and 1, then it is a space form. In the thesis, first, we classify complete noncompact three dimensional manifold with nonnegative Ricci curvature. As a corollary, we conrms a conjecture of Milnor in dimension three. Note that in the compact case, the classication was done by R. Hamilton by using the Ricci flow. Also, previously, there are some partial classications assuming additional conditions. Our proof will be based on the minimal surface theory developed by Schoen and Yau [74], Schoen and Fischer Colbrie [24].Next we study compact Kahler manifolds with nonpositive bisectional curvature. In particular, we confirm a conjecture of Yau which states that for there is a canonical fibration structure for these manifolds. More relating results will be proved. In the third part, we generalize the classical volume comparison theorem to the Kahler setting. We prove a few gap theorems which tells us some differences between Kahler geometry and Riemannian geometry. We also show that locally, the volume of a Kahler-Einstein manifold is no greater than that of the complex space forms. Note that when the bisectional curvature is bounded from below, the sharp volume comparison was obtained by Li and Wang.Then we prove a rigidity result for volume entropy. This was rst proved by Leddrapier and Wang. Our proof is much shorter and simpler. Finally, we study complete manifolds with nonnegative Bakry-Emery Ricci curvature. It turns out that when the potential f is bounded, geometrically these manifolds will be very similar with manifolds of nonnegative Ricci curvature. In particular, we partially classify complete three dimensional manifold with nonnegative Bakry-Emery Ricci curvature.