New Results In Ricci Flow And Yamabe Flow

In this thesis, we study two kinds of geometric flows: Ricci flow and Yamabeflow. We first consider the controlling problem of the full Riemannian curvature tensorunder Ricci flow. We then study the extension problem of Yamabe flow. Finally, westudy the Yamabe flow with the pinching condition Rc≥Rg > 0 on complete locallyconformally flat manifolds.An important problem in the study of Ricci flow is that to find the weakest conditionsthat provide control of the norm of the full Riemannian curvature tensor. Supposethat (M~n;g(t)) is a solution to the Ricci flow on a complete manifold on time interval[0;T) with bounded sectional curvature at t = 0. In Chapter 3, we show that Ln+2/2 norm bound of the scalar curvature and the Weyl tensor can control the norm of the full Riemanniancurvature tensor under the Ricci flow if M~n is closed and T <∞We thenget a similar result in K╝hler Ricci flow. Next we prove that, without condition T <∞C0 bound of the scalar curvature and the Weyl tensor can control the norm of the fullRiemannian curvature tensor under the Ricci flow on complete manifolds. Finally, weprove that if Ricci curvature is uniformly bounded on M~n0;T), where T <+∞thenthe full Riemannian curvature tensor stays uniformly bounded under the Ricci flow onM~n0;T).In Chapter 4, we study the extension problem of the Yamabe flowon closed manifolds. Suppose that (M~n;g(t)) is a solution to the Yamabe flow on aclosed manifold on time interval [0;T), where T <∞We first prove that the Yamabeflow can be extended over T provided the scalar curvature stays uniformly bounded on[0;T). Next, we show that the Yamabe flow with positive Yamabe invariant can beextended beyond T provided Ln+2/2 norm of the scalar curvature is bounded on [0;T).This latter result is obtained by the Nash-Moser iteration method.In Chapter 5, we study the Yamabe flow with the pinching condition Rc≥Rg>0on complete locally conformally flat manifolds, where e > 0 is an uniformly constant. We first prove that if (M~n,g(t)), n≥3, is an ancient solution to Yamabe ?ow on ann-dimensional complete locally conformally ?at Riemannian manifold with boundedcurvature satisfying Rc≥εRg > 0, then (M~n,g(t)) must be compact with constantsectional curvature. We then prove that if (M~n,g), n≥3, is an n-dimensional completelocally conformally ?at Riemannian manifold with bounded Ricci curvature satisfyingRc≥εRg > 0, then either M~n is compact or the Yamabe exists all thetime on M~n and sup Rc(x,t)→0 as t→∞.The blow up analysis is an important technique in the study of all problems inthis thesis. Especially, we deal with the situation of blow up analysis in noncompactmanifolds in Chapter 3 and Chapter 5. In this case, the injectivity radius can not becontrolled in general. Hence, the limits could be collapsing. Fortunately, we can usethe Fukaya-Glickenstein's technique overcome this problem.