| We consider a complete non-compact Riemiannian manifold ( M, g) with the metrics g(t) evolving by Ricci flow, that is, 6gij6t = --2Rij. With the assumption that L2 Sobolev embedding hold on the initial manifold (M, g(o)) and the restriction on Ricci curvature tensor |Ric| and its gradient |∇Ric| over the space-time M x [0, T), we show that the L2 Sobolev embedding also holds on the manifold (M, g(t)), with the constants depending only on the initial metric g0 and the time parameter t. The organization of the dissertation is as follows: we start in Chapter one with a brief overview for Ricci flow. In chapter two, we introduce the necessary background of Riemannian geometry, preparing for the extensive computation of evolution equations for geometric quantities under Ricci flow. The main results of this dissertation, Sobolev embedding on complete non-compact manifolds under Ricci flow, is contained in Chapter three. Then we include some other results in Chapter four, and the appendix at the end. |
