Complete λ-Hypersurfaces in Euclidean Spaces*

Qingming CHENGGuoxin WEI

1Department of Applied Mathematics,Faculty of Science,Fukuoka University,814–0180,Fukuoka,Japan,E-mail:cheng@fukuoka-u.ac.jp

2School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China,E-mail:weiguoxin@tsinghua.org.cn

Abstract In this paper,the authors give a survey about λ-hypersurfaces in Euclidean spaces.Especially,they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.

Keywords Self-shrinker,λ-Hypersurface,Mean curvature flow,Weighted volume,Rigidity theorem

1 Introduction

2 Preliminaries

3 Examples of λ-Hypersurfaces

3.1 0-hypersurfaces

0-hypersurfaces are just self-shrinkers.In 1989,by using the shooting method for geodesics,Angenent[5]constructs compact embedded rotational 0-hypersurface,called“Angenent torus”,whose profile curve intersects symmetry axis perpendicularly.In 1994,Chopp[24]finds several new 0-hypersurfaces.Later,Drugan,Lee and Nguyen[31],Drugan and Kleene[32]construct an infinite number of complete,immersed and non-embedded rotational 0-hypersurfaces for each of the topological types:The sphere,the plane,the cylinder and the torus.These examples whose profile curves also intersect symmetry axis perpendicularly.Recently,Cheng and Wei[20]numerically compute and find many interesting compact immersed rotational 0-hypersurfaces whose profile curves do not intersect symmetry axis perpendicularly(see Figure 1).

Figure 1 The graph of profile curve of compact 0-hypersurface,0-hypersurface and half of 0-hypersurface,here n=2.

Figure 2 The profile curves of λ-hypersurfaces,here n=2,λ=0.1.

Figure 3 The graph of profile curve of λ-torus,λ-torus and half of λ-torus,here n=2,λ=0.1 and r0 ≈0.343.

In addition,Kapouleas,Kleene and M?ller[47](also see[48,56])and Nguyen[57–59]construct complete embedded 0-hypersurfaces with higher genus in R3.

3.2 λ-hypersurfaces with λ /= 0

Some of them are embedded,some of them are immersed.

3.2.1 λ-curves

There are no closed embedded 0-curves of mean curvature flow except circle with radius 1.But for λ-curves,their behaviors are different.For some λ ＜0,we can prove that there exist closed embedded λ-curves Γλin R2,which is not circle(also see[11]).Hence,for any positive integer n,there exist complete embedded λ-hypersurfaces,which are given by Γλ×Rn-1in Rn+1.

3.2.2 λ-torus

In the same paper,Cheng and Wei[20]also proved the following theorem.

Theorem 3.2For n ≥2 and small λ,there are many compact immersed λ-hypersurfaces in Rn+1.

Additional details on the behavior of the profile curves needed to be discussed and established.Here are some numerical approximation of profile curves and λ-hypersurfaces.The horizontal axis is the axis of rotation.For small λ,compact immersed λ-hypersurfaces can be given by rotating a closed curve in the upper half plane around the horizontal axis;see Figure 4.

Figure 4 The graph of profile curve of compact λ-hypersurface,λ-hypersurface and half of λ-hypersurface,here n=2,λ=0.1 and r0 ≈0.811.

Moreover,we also found many compact immersed rotational λ-hypersurfaces whose profile curves do not intersect r-axis perpendicularly;see Figure 5.

Figure 5 The graph of profile curve of compact λ-hypersurface,λ-hypersurface and half of λ-hypersurface,here n=2,λ=0.1 and r0 ≈0.811.

3.2.3 Some other λ-hypersurfaces

4 Rigidity Results of λ-Hypersurfaces

4.1 Rigidity results of 0-hypersurfaces

4.2 Rigidity results of λ-hypersurfaces with λ /= 0