Researches On Some Geometric And Analytic Problems Under Geometric Flows

We study a series of geometric and analytic problems under geometric flows and focus on the curvature estimates of complete noncompact gradient expanding Ricci solitons,together with rigidity and classification of gradient shrinking and expanding K(?)hler-Ricci solitons.Moreover,we prove priori estimates of several partial differen-tial equations under the Ricci flow,K(?)hler-Ricci flow and Yamabe flow,respectively.We also derive the monotonicity for the first eigenvalue of a geometric operator under the Yamabe flow.Besides,we show global Hessian estimates for an Allen-Chan equa-tion on a complete manifold with a fixed Riemannian metric.The specific problems that we solved are below.1.On gradient Ricci solitons.(1)On a four-dimensional complete noncompact gradient expanding Ricci soliton with nonnegative Ricci curvature,the Riemann curvature tensor and its covariant derivatives are bounded.(2)On a four-dimensional complete noncompact gradient expanding Ricci soliton with nonnegative Ricci curvature,the Ricci curvature Ric can be controlled by the scalar curvature R,that is,for each 0<a<1,there is a constant c>0 such that|Ric|~2?c R~a.(3)The Riemannian curvature tensor of an n-dimensional complete noncompact gradient expanding Ricci soliton with nonnegative Ricci curvature grows at most polynomially in the distance function.(4)A compact gradient shrinking K(?)hler-Ricci soliton with subharmonic scalar curvature is K(?)hler-Einstein.(5)An n-dimensional(n=5 or n?7)complete noncompact gradient shrinking K(?)hler-Ricci soliton with constant scalar curvature and nonnegative fourth order divergence Bochner tensor is rigid.2.On priori estimates of partial differential equations.(1)Gradient estimates for the forward conjugate heat equation under the Ricci flow.(2)Harnack estimates for a nonlinear diffusion equation on K(?)hler manifolds with the metric is fixed and evolves under the K(?)hler-Ricci flow,respectively.(3)A series of gradient estimates and differential Harnack inequalities for two parabolic partial differential equations under the Yamabe flow.(4)A type gradient estimate and global Hessian estimates for an Allen-Cahn equation on a complete Riemannian manifold.3.On eigenvalue problems of a geometric operator.A monotonicity formula for the first eigenvalue of a geometric operator under the Yamabe flow.