No abstract

Abstract. We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to Finsler manifolds.

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of L operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau [14].By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in R 3 obtained by Colding-Minicozzi under weaker conditions.

We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds.