Differential Harnack Estimates Under Geometric Flows

In this paper, we consider the following two problems:Differential Harnack estimates under geometric flows; Vanishing theorems for killing vector fields.In chapter one, our purpose is to study equations with potentials on a complete Riemannian manifold ?M,g?t?? evolving by the general geometric flow where, p is a constant, S_ij?t? is a symmetric two-tensor and S=g^ijS_ij is the trace of Sij We obtain Aronson-Bernilan and Li-Yau-Hamilton type differential estimates for positive solutions.In chapter two. we study vanishing theorems for Killing vector fields on complete hypersurfaces in an ?n+1?-dimensional hyperbolic space Hⁿ⁺¹? -1? with constant sec-tional curvature-1. We derive vanishing theorems for Killing vector fields with bounded L²-norm in terms of the bottom of the spectrum of the Laplace operator.