New Results In Curvature Flows And Eigenvalue Problems

The curvature flow, such as the Ricci flow, the mean curvature flow, the crosscurvature flow, the Gaussian curvature flow, plays an increasingly important role inmodern diflerential geometry. It is also a powerful tool.The curvature flow on manifolds with boundary is always a hot topic. In thisthesis, we first consider the cross curvature flow on 3-manifolds with boundary. Byusing the DeTurck method, we give the short time existence of solutions to the Dirichletand Neumann boundary problems of the cross curvature flow.Secondly, we consider the prescribed Gaussian curvature problem on compactmanifolds with negative Euler characteristic. We prove via the heat flow method thefollowing result obtained by Berger[1], Kazdan and Warner[2] via variational method.For a given smooth negative function f on a closed surface M and in any conformalclass [g0] of metrics, there exists a metric g such that it has the Gaussian curvaturef. Using the heat flow method, we obtain a new result on compact surfaces withsmooth boundary. Namely, for a compact surface M with boundary and a given smoothnegative function f on the boundary flM, we find that in any conformal class [g0] withvanishing Gaussian curvature, there is a metric g such that (M, g) is flat and the geodesiccurvature of flM is f.The last part of the thesis is devoted to the eigenvalue problem of the driftingLaplacian operator, which is also called the weighted Laplacian operator. This operatoris closely related to the Ricci soliton, which plays an important role in the Ricci flowtheory. We generalize some results of Korevaar[3] and S. T. Yau[4] to gain a Hessianestimate of the first eigenfunction. Then, we use this Hessian estimate to get a lowerbound of the diflerence of the first and second eigenvalues of the drifting Laplacian. Atthe end, we also find a lower bound when the Hessian estimate does not hold.