Complete Riemann Flow Shape On The Laplace Operator And Its Associated Problems

The Laplacian on Riemannian manifolds is an essential linear operator, and it is also the main object to be studied of Geometric Analysis on manifolds. The article stated here will give some remarks to the following equation in two cases:For the case > 0, the equation expresses the eigenvalue of the Laplacian while for the case = 0, it is the existence of nontriv-ial bounded harmonic functions on complete noncompact manifolds. These are two interesting problems in Geometric Analysis on manifolds and they have attracted a lot of attentions.Here, we take the Moser iteration technique and study the two kinds of problems submitted above. In chapter 3, we will estimate the first eigenvalue of Laplacian from below on manifolds with a little negative curvature. In chapter 4, we will prove the existence of bounded nontrivial harmonic functions on some classes of complete manifolds which will generalize the results of S. Y. Cheng's.