Studies On Mean Curvature Flow And Related Topics

In this thesis,we mainly study the mean curvature flow and some related prob-lems.We investigate the convergence of the mean curvature flow of closed symplectic surfaces in CP2,and prove that the symplectic mean curvature flow in CP2 converges to CP1 if the initial surface satisfies suitable pointwise curvature pinching condition or integral curvature pinching condition.We investigate the pinching phenomena of the tracefree second fundamental form A of complete self-shrinkers of higher codi-mension,and we prove some integral curvature pinching theorems under the condition that the mean curvature is nowhere vanishing or suitably bounded,and we also ob-tain some rigidity theorems for self-shrinkers satisfying pointwise curvature pinching conditions on |A|2.We study the Myers type theorem for submanifolds with pinched extrinsic curvatures,and prove the compactness theorem for hypersurfaces with finite total curvature by using the mean curvature flow.We also obtain a compactness theo-rem for locally conformal flat complete Riemannian manifolds with finite Lq-norm of the tracefree Ricci curvature tensor.In Chapter 3,we study the convergence of the mean curvature flow of sympletic surfaces in CP2.The mean curvature flow may be used to deform a symplectic surface in a Kahler surface to a holomorphic curve.However,there are not many theorems about the longtime existence and the convergence of symplectic mean curvature flow.Chen-Li-Tian[25]proved that if the symplectic surface is a graph,then the mean curva-ture flow has a solution on[0,?)and converges to a holomorphic curve.Han-Li[51]proved that if the initial symplectic is close to a holomorphic curvature in a Kahler-Einstein surface with positive scalar curvature,then the mean curvature flow has a solution on[0,?)and converges to this holomorphic curve.In this chapter,we prove that for the symplectic mean curvature flow in CP2,if the initial surface satisfies a pointwise curvature pinching condition,then the evolving surface becomes more and more umbilical and the limit surface is in fact CP1.This is a refinement of a theorem in Han-Li-Yang[52].We also prove a convergence of mean curvature flow of symplectic surfaces in CP2 under an integral curvature pinching condition.In Chapter 4,we investigate the pinching phenomena of the tracefree second fun-damental form of complete self-shrinkers of higher codimension.The self-shrinker plays an important role in the study of the mean curvature flow,which is a model of singularities of type I and arises naturally as the tangent flow of mean curvature flow at singular times.Abresch-Langer[2],Huisken[46,47],Colding-Minicozzi[32]inves-tigated the geometric classification of self-shrinkers of codimension 1.Smoczyk[75]studied the relation of self-shrinkers of arbitrary codimension with nowhere vanishing mean curvature and parallel normalized mean curvature vector and minimal subman-ifolds in spheres.Le-Sesum[58]and Cao-Li[17]obtained a classification theorem for self-shrinkers with polynomial volume growth and second fundamental form satisfying|A|2?1/2.Ding-Xin[34]proved a gap theorem for complete self-shrinkers under the in-tegral pinching of the second fundamental form.Without the assumption of polynomial volume growth,Cheng-Peng[28]proved that a complete self-shrinker is isometric to the Euclidian space if the second fundamental form satisfies sup|A|2<1/2.In this chapter,we first assume the mean curvature is nonzero everywhere and the self-shrinker is of polynomial volume growth,and prove that if the tracefree second fundamental form A satisfies ?A?n<C(n)for a positive constant C(n)depending only on the dimension n of the self-shrinker,then it is isometric to the sphere Sn((?)).Secondly,we show if the mean curvature vector H of the self-shrinker satisfies sup |H|<(?)and A sat-isfies ?A?n<D(n,sup |H|)for a positive constant D(n,sup |H|)depending n and sup |H|,then it is isometric to the Euclidean space Rn.We also obtain some rigidity theorems for self-shrinkers satisfying pointwise curvature pinching conditions on |A|2.In Chapter 5,we study the Myers type theorem for submanifolds with pinched extrinsic curvatures.Given a complete hypersurface in the Euclidean space,we assume that the second fundamental form is bounded and the mean curvature is bounded from below by a positive constant.We prove that if the L9-norm of the tracefree second fundamental form for some q ?2 is finite,then the hypersurface is compact.We prove this theorem by contradiction and use the mean curvature flow to deform the hypersurface.In Chapter 6,we study the compactness of complete and locally conformally flat Riemannian manifolds.We prove that for a complete and locally conformally flat Rie-mannian manifold with positive Yamabe constant,if the Ricci curvature is bounded from above,the scalar curvature is bounded from below by a positive constant,and the Lq-norm(q?2)of the tracefree Ricci curvature tensor is finite,then the manifold is compact.As in chapter 5,we will use the Yamabe flow to prove our theorem.