λ-Surfaces In 3-Dimensional Euclidean Space

In this paper,a class of linear Weingarten surfaces is studied,which orig-inates from the self-shrinkers in the mean curvature fow.The self-shrinkers in three-dimensional Euclidean space that satisfesX,ξ+f=0,where X is the im-mersion of M→Rⁿ⁺¹,ξis the normal vector,and f is the mean curvature.This type of surface describes the”blow-up”behavior of the mean curvature fow at the frst type of singularity,so autogenous contractions play an important role in the study of the mean curvature fow.From the perspective of variation,theλ-Sur-faces theory is a natural extension of self-shrinkers surfaces,Scholars have made some research progress on the classifcation ofλ-surfaces,however,the research on the classifcation and rigidity ofλ-surfaces under certain curvature assump-tions has not been completely resolved.This problem is also a widely concerned problem in diferential geometry,and has important signifcance of the study,and many research results have been obtained.This paper mainly gets a com-plete classifcation of two types linear Weingartenλ-surfaces in three-dimensional Euclidean space.In the frst chapter,we introduce the concept of mean curvature fow and the research progress at home and abroad,give the research background of self-shrinking surfaces andλ-surfaces and the related research status,and give the defnitions oftwo types of linear Weingarten surfaces and the main theorems of this paper.The second chapter introduces the relevant concepts of submanifold geometry,and gives Gauss equation and Codazzi equation,etc.,which prepare for the follow-up research.In chapter three,we study the classifcation of two types linear Weingartenλ-surfaces in R³.By reducing the structural equations,we transform the second-order nonlinear partial diferential equations into ordinary diferential equations,obtain several important lemmas,and then we transformλ-surfaces into second-order nonlinear partial diferential equations,further,ac-cording to the linear Weingarten assumption,the second-order nonlinear partial diferential equations are reduced to an interesting ordinary diferential equation.Through the discussion of the solution of the diferential equation,a complete characterization of two types linear Weingarten surfaces is given,and it is proved that:Theorem 0.1 Let X:M→R³be aλ-surfaces which satisfes the frst linear Weingarten assumption,then M must satisfy one of the following case:（1）M is plane R²;（2）M is cylinder （?）;（3）M is round sphere （?）;（4）M is a generalized cylinder γ×R.Theorem 0.2 Let X:M→R³be aλ-surfaces which satisfes the second linear Weingarten assumption,then M must satisfy one of the following case:（1）M is plane R²;（2）M is cylinder （?）;（3）M is round sphere （?）;（4）M is a generalized cylinder γ×R.These two results are generalizations of linear Weingarten self-shrinkers.