Research and Reports on Mathematics.

All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.

Research Article, Res Rep Math Vol: 1 Issue: 1

A Simpler Proof of the Characterization of Quadric CMC Hypersurfaces in Sn+1

Aquino CP1* and De Lima HF2

1Department of Mathematics, Universidade Federal do Piaui, Teresina, Brazil

2Department of Mathematics, Universidade Federal de Campina Grande, Campina Grande, Paraiba, Brazil

*Corresponding Author : Aquino CP
Department of Mathematics, Universidade Federal do Piaui, Teresina, Brazil
Tel: (86) 3215-5525
E-mail: [email protected]

Received: August 10, 2017 Accepted: September 01, 2017 Published: September 07, 2017

Citation: Aquino CP, De Lima HF (2017) A Simpler Proof of the Characterization of Quadric CMC Hypersurfaces in Sn+1. Res Rep Math 1:1.

Abstract

In this short article, we present a new and simpler proof of a characterization of the quadric constant mean curvature hypersurfaces of the Euclidean sphere Sn+1, originally due to Alias, Brasil and Perdomo.

Keywords: Euclidean sphere; Constant mean curvature hypersurfaces; Support functions; Totally umbilical hypersurfaces; Clifford torus

Introduction

In 2008, Alias, Brasil and Perdomo studied complete hypersurfaces immersed in the unit Euclidean sphere equation, whose height and angle functions with respect to a fixed nonzero vector of the Euclidean space equation are linearly related. Let us recall that, for a fixed arbitrary vector equation the height and the angle functions naturally attached to a hypersurface equation endowed with an orientation ν are defined, respectively, by equation and equationIn this setting, they showed the following characterization result concerning the quadric constant mean curvature hypersurfaces of Sn+1 [1,2]:

Theorem 1

Let equation be a complete hypersurface immersed in Sn+1 with constant mean curvature. la=λ fa for some non-zero vector equation and some real number λ, then Σn is either a totally umbilical hypersurface or a Clifford torus equation, for some k = 0; 1;..; n and some k=0,1,…,n and ρ>0.

Later on, working with a different approach of that used in [2], the first and second authors characterized the totally umbilical and the hyperbolic cylinders of the hyperbolic space Hn+1as the only complete hypersurfaces with constant mean curvature and whose support functions with respect to a fixed nonzero vector a of the Lorentz- Minkowski space are linearly related (see Theorem 4:1 of [3,4], for the case that a is either space like or time like, and Theorem 4:2 of [5], for the case that a is a nonzero null vector). In this short article, our purpose is just to use a similar approach of that in [4,5] in order to present a new and more simple proof of Theorem 1 (cf. Section 3). For this, in Section 2 we recall some preliminaries facts concerning hypersurfaces immersed in Sn+1.

Preliminaries

Let equation be an orientable hypersurface immersed in the Euclidean sphere. We will denote by A the Weingarten operator of Σn with respect to a globally defined unit normal vector ν.

In order to set up the notation, let us represent by ∇0 , ∇and ∇ the Levi-Civita connections of equation, Sn+1 and Σn respectively. Then the Gauss and Weingarten formulas for Σn in Sn+1 are given, respectively, by

equation

and

equation

for all tangent vector fields equation

In what follows, we will work with the first three symmetric elementary functions of the principal curvatures λ1,… λn of ψ, namely:

equation

where i, j, k ∈ {1,…, n}.

As before, for a fixed arbitrary vector a equation let us consider the height and the angle functions naturally attached to which are defined, respectively, by equation and equation. A direct computation allows us to conclude that the gradient of such functions are given by equation and equation, where aΤ is the orthogonal projection of a onto the tangent bundle Τ Σ , that is,

equation

Taking into account that equation and using Gauss and Weingarten formulas, we obtain equation for all equation. We use this previous identity jointly with Codazzi equation to deduce that

equation

For all that equation Thus according to [6] (see also [3]), it follows from the last two identities that

equation (2.1)

equation (2.2)

where H = (1/n) S1 is the mean curvature function of Σn

For what follows, it is convenient to consider the so-called Newton transformation

equation

P1= S1-A

where I is the identity operator. Naturally associated with the Newton transformation P1, we have the Cheng-Yau’s square operator [7], which is the second order linear di erential operator equation given by

equation (2.4)

Here equation stands for the self-adjoint linear operator metrically equivalent to the hessian of h, and it is given by

equation

For all X, Y∈ x (Σ) .

Based on Reilly’s seminal paper [8-10], Rosenberg [6] showed the following idenfitities related to the action of equation on the functions la and fa:

equation (2.5)

And

equation (2.6)

To close this section, we quote a suitable Simons-type formula which can be found in [1] or [11].

equation (2.7)

Proof of Theorem

Now, we are in position to proceed with our alternative proof of Theorem 1.1. If λ=0 then la= λfa=0, that is

equation

for all equation and, consequently, Σn is a totally umbilical sphere of Sn+1.

So, let us assume that λ≠0. We have Δla=λΔfa and using the fact that H is constant, from (2.1) and (2.2) we conclude that

equation

Or equivalently,

equation

Hence, we get that

equation (3.1)

By (3.1), we obtain

equation

equation

equation

equation

Thus,

equation (3.2)

We define a function

equation by

equation

Suppose that h(p0)≠0 for some p0 ∈ Σn Since h is smooth, there exists a neighbourhood u of p0 in Σn in which h(p)≠0 all p∈ u From (3.2) we conclude that la=0 in u and, hence fa=0 in u. since λ≠0. we arrive at a contradiction because in Σn we have

equation

Therefore, h = 0 on Σn, that is,

equation (3.3)

Consequently, S2 is constant on Σn . Repeating the previous argument for the operator L1 and using the fact that S2 is constant, we also obtain that

equation (3.4)

We observe that the above equation shows that S3 is also constant on Σn . We also note that this argument shows, in fact, that Sr is a constant function on Σn for all 2≤r≤n. From (2.7) we get

equation

More precisely,

equation (3.5)

We observe that if H=0, then S1=0 and, consequently, |A|2=−2s2 From (3.3), we have 2S2 =−n and |A|2 = n. Therefore, since

equation

We have that equation and, hence, from Theorem 4 of [9], we conclude that Σn must be a Clifford torus

equation, for some k=0,1,…,n and some ρ>0.

Now, suppose that H≠0 By equation (3.4) we get

equation

that is,

equation (3.7)

From equation (3.5) we have

equation (3.8)

Furthermore, from a straightforward computation we can verify that

equation(3.9)

Hence, if S2 = 0 we obtain of (3.9) that equation consequently, |∇A|2 =0 and, since Σn is complete, it follows once more from Theorem 4 of [9] that Σn must be a Clifford torus.

If S2≠0 then equation implies

equation (3.10)

We note that (3.10) and (3.9) imply equation and, hence, repeating the previous argument we also get that Σn is a Clifford torus. Therefore, we conclude the proof of Theorem 1.

References

Track Your Manuscript

Recommended Conferences

Share This Page