Generalized Biconservative Surfaces In 3-dimensional Riemannian Space Forms

The theory of surface is an important part of classical differential geometry.Biconservative surfaces are closely related to biharmonic surfaces,which are a class of important surfaces in classical differential geometry,and have attracted much attention from scholars in recent years.In this paper,we consider a class of surfaces satisfying the equation A?H=kH?H in three-dimensional space forms,where A is the shape operator,H is the mean curvature and k is an arbitrary constant.Such surfaces are called generalized biconservative surfaces.In particular,the generalized biconservative surface is a biconservative surface when k=-1.Obviously,the notion of generalized biconservative surfaces is a natural generalization of biconservative surfaces.Compared with biconservative surfaces,the research found that the classification results of generalized biconservative surfaces are much richer.In this paper,the relation between generalized biconservative surfaces and linear Weingarten surfaces in three-dimensional space forms and the classifications of generalized Biconservative surfaces in three-dimensional sphere and hyperbolic space are deeply studied.The profile curve of generalized biconservative surface in three-dimensional space forms is characterized.By introducing the definition of generalized biconservative surfaces,the second chapter proved that generalized biconservative surfaces with nonconstant mean curvature in three-dimensional space forms must be linear Weingarten surfaces,and a rotational surface whose principal curvatures ?1,?2 satisfying(k-2)?1+k?2=0 must be a generalized biconservative surface.The third and fourth chapters respectively prove that generalized biconservative surfaces in the three-dimensional sphere and hyperbolic space are only surfaces with constant mean curvature or surfaces of revolution.According to Gauss formula,Weingarten formula and the integrability condition,the differential equation which generalized biconservative surfaces satisfied is obtained,the explicit parameter form of the generalized biconservative surface is further determined by solving the differential equation.The fifth charpter introduces a curvature energy function ? in three-dimensional space forms,the first variational formula and its corresponding Euler-Lagrange equation are obtained.And the profile curve of the generalized biconservative surface is characterized as the critical curve of ?,that is,for the curvature energy function ?,the profile curve of the generalized biconservative surface satisfies Euler-Lagrange equation.Furthermore,by proving that the binormal evolution surface generated by the critical curve with nonconstant curvature is a generalized biconservative surface,a method to construct generalized biconservative surfaces in three-dimensional space forms is given.