Research On The Curvature Flows In Space Forms

In this thesis,we mainly study the evolution of closed hypersurfaces in space forms by curvature flows,for which the speed is a homogeneous function of degree greater than one or a non-homogeneous function of principal curvatures,and the evolution of complete non-compact hypersurfaces in Euclidean space by curvature flows with degree of homogeneity not less than one.This thesis is divided into seven chapters.Chapter 1 is introduction,which mainly introduces the background and the devel-opments of curvature flows and the main results of this thesis.In Chapter 2,we mainly introduce the preliminaries of this thesis.It contains the fundamental formulas of hypersurfaces,the properties of inverse-concave curvature function and the regularity results for parabolic equations.In Chapter 3,we give the short-time existence for the contracting curvature flows of closed hypersurfaces in space forms,and we also give the proofs of evolution equations for some important geometric quantities.Chapter 4 is devoted to the study of contracting curvature flow with degree of homogeneity greater than one(F~?-flow for short)of pinched hypersurfaces in space forms,that is,the normal speed is a power?>1 of curvature function F which is monotone,symmetric,homogeneous of degree one.Firstly,by maximum principle we prove the pinching estimate is preserved under the F~?-flow,and the hypersurfaces shrink to a single point in finite time.Secondly,after a appropriate rescaling,we get the uniform bounds for the normalized principal curvatures by the pinching estimate and prove the normalized flow converges to the unit sphere smoothly and exponentially.In Chapter 5,we consider the non-homogeneous contracting curvature flow(?(F)-flow for short)of closed hypersurfaces in space forms,that is,the normal speed is a general non-homogeneous function?of curvature function F which is monotone,symmetric,inverse-concave,homogeneous of degree one.Firstly,we prove the convexity is preserved under the?(F)-flow by the tensor maximum principle and the properties of inverse-concave function.Secondly,by applying the Gauss map parametrization of flows,we get the estimates for all higher derivatives of principal curvatures.Finally,we prove hypersurfaces shrink to a single point in finite time.Chapter 6 mainly contains the evolution of complete non-compact hypersurfaces in Euclidean space by curvature flow with degree of homogeneity not less than one(F~?-flow for short),that is,the normal speed is a power??1 of curvature function F which is monotone,symmetric,inverse-concave,homogeneous of degree one.Firstly,interior a prior estimate for the gradient function,and local estimates for the principal curvatures and all the derivatives of the second fundamental form are shown by the properties of inverse-concave curvature function.Secondly,based on these local estimates,we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence.Finally,for special inverse-concave curvature function,by constructing an appropriate barrier,we prove that the complete non-compact smooth strictly convex solution exists and remains a graph for all times.In Chapter 7,we summarize the main contents of this thesis and put forward some questions that will be further studied in the future.