Some Geometry And Analysis About Ricci Solitons

In this paper, we consider the following five problems:The classification of locally conformally flat gradient shrinking Ricci solitons; Gradient estimates for a parabolic equation; Gradient estimates for positive smooth f-harmonic functions; Zero points of vector fields on Ricci solitons; Eigenvalue comparison theorem for N-Bakry-Emery Ricci tensor.In Chapter two, we remove the nonnegative curvature assumption for a clas-sification theorem on locally conformally flat noncompact shrinking Ricci solitons given by X.D.Cao and B.Wang.In Chapter three, we derive a local gradient estimate for the positive solution to the following parabolic equation ut = Au+au log u+bu in M, where a, b are real constants, M is a complete non-compact Riemannian manifold. As a corollary, we give a local gradient estimate and a bound for the positive solution to the corresponding elliptic equationΔn+au log u+bu= 0 in M.In Chapter four, For Riemannian manifolds with a measure (M, g, e-f dvolg) we study the gradient estimates for positive smooth f-harmonic functions when the oo-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, gen-eralizing the classical ones of Yau(i.e. when f is constant)In Chapter five, Let (M, g) be a complete Riemannian manifold satisfying Ric+1/2 Lxg≥λg with some constantλ> 0 and |Ric|≤C. We prove that the zero points of X are accumulated in some geodesic ball.In Chapter six, In this paper, we prove the first eigenvalue comparison theorem of f-Laplacian for N-Bakry-Emery Ricci tensor by the Laplace curva-ture comparison. Therefore, we generalize the eigenvalue comparison theorem of S.Y.Cheng.