Several Geometric Quantities Under Geometric Flows And Related Problems

In this thesis, we mainly study some problems for several geometric quanti-ties under geometric flows, including the first eigenvalue of the p-Laplace under the Ricci flow, upper bounds on the first eigenvalue for a diffusion operator, Har-nack estimates for heat-type equations, Ricci deformations of the metric on closed Riemannian orbifolds and geometric properties of the Yamabe flow on complete manifolds, etc.More precisely, in the second chapter, we investigate continuity, monotonic-ity and differentiability for the first eigenvalue of the p-Laplace operator along the Ricci flow on closed manifolds. We show that the first p-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first p-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a p-eigenvalue comparison-type theorem when its Euler charac-teristic is negative.In the third chapter, we give an upper bound on the first eigenvalue for a diffusion operator. Let L=△- (?)φ·(?) be a symmetric diffusion operator on a complete Riemannian manifold. We give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry-Emery Ricci curvature satisfying Ricm,n(L)≥- (n - 1), and therefore generalize a Cheng's result on the Laplacian to the case of the diffusion operator.In the fourth chapter, on one hand, we study Harnack estimates for a general nonlinear heat-type equation on a complete manifold. We first give a local Li-Yau type Harnack estimate for the positive solutions to the nonlinear heat-type equa-tion. Then we can also derive a elliptic type Harnack estimate for this equation with respect to evolving Riemannian metrics, and therefore give another answer to L. Ma's question. On the other hand, we establish an interesting interpolated Harnack inequality, connecting the constrained trace Li-Yau differential Harnack inequality for the heat equation to the constrained trace Chow-Hamilton Harnack inequality, and thereby generalize B. Chow's interpolated Harnack inequality.In the fifth chapter, we study the Ricci deformation of the metric on Rieman-nian orbifolds. We show that on any n-dimensional (n≥4) orbifold of positive scalar curvature the metric can be deformed into a metric of constant positive curvature, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not large compared to the scalar curvature at each point. Therefore we not only generalize the G. Huisken's result to Rie-mannian orbifolds, but also generalize 3-orbifolds result proved by R. Hamilton to n-orbifolds (n≥4).Finally, we study geometric properties of complete manifolds evolved by Yam-abe flow. We mainly prove local derivative estimates of Bernstein-Bando-Shi type under the Yamabe flow and use these to give a compactness theorem on complete locally conformally flat manifolds. We investigate some properties of two geo-metric invariants (asymptotic volume ratio and asymptotic scalar curvature ratio) under the Yamabe flow on complete noncompact locally conformally flat mani-folds. Meanwhile we find that any Type I ancient solution to the Yamabe flow on such manifolds with bounded positive curvature operator has infinte asymptotic scalar curvature ratio. Moreover, we discuss the connections between the Yam-abe soliton and topological property of Riemannian manifolds. We also prove a dimension reduction theorem for the Yamabe flow and give some applications.