On The Eigenvalue Estimates And Rigidity For Riemannian Manifolds

The eigenvalue estimates of f-Laplacian and the rigidity for different classes of Riemannian manifolds are investigated systematically in this thesis which focuses on the analytical and topological properties of manifolds. The main results of this paper are as follows:First of all, we obtain a universal lower bound for the first eigenvalue of the f-Laplacian on weighted manifolds. In the case of RicfN≥K(K∈R), this lower bound result not only covers the classical results of lower bound for the first eigenvalue of the Laplacian (set N=0for example), but also is suitable for more general spaces—N-quasi-Einstein manifolds and weighted manifolds. In the case of RicfN≥K(K<0), our result extends the Yang Conjecture to the first eigenvalue for the f-Laplacian. Moreover, a sharp lower bound for λ1on closed manifolds (which include gradient Ricci solitons) with Ricf≥K (K€R) is derived. Employing this bound, a universal lower bound for the diameter of compact nontrivial shrinking Ricci solitons is given.Secondly, we complete the estimate of sharp lower bound of the scalar curvature for gradient Yamabe solitons and achieve the corresponding examples whose scalar curvature attains the sharp bound. In addition, the estimates for the potential function and the rigidity theorems of Yamabe solitons are derived. Furthermore, we construct an example which is a steady quasi gradient Yamabe soliton and present the lower bound of the scalar curvature and the rigidity results for quasi gradient Yamabe solitons.Next, a rigidity theorem for n-dimensional non-compact Bach-flat Riemannian manifolds is proved, which extends the corresponding result for4-dimensional Bach-flat manifolds that attracted more attention, and is suitable for the general n- dimensional case.Fourthly, by constructing new test function, we arrive at a rigidity theorem for those Riemannian manifolds with harmonic curvature.