Rotational hypersurfaces of the sphere

Rotational hypersurfaces of the sphere

Guoxin Wei

Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea

e-mail: weigx03@mails.tsinghua.edu.cn

(2000 Mathematics Subject Classification : 53C42, 53C20. )

Abstract. In this paper, we give some integral inequalities related to m-minimal ro- tational hypersurfaces in a unit sphere Sⁿ⁺¹(1). As their applications, we prove that Sⁿ⁻¹

³qn−m n

´

×S¹¡p_m

n

¢and round geodesic spheres are the onlyn-dimensional com- pact embedded m-minimal rotational hypersurfaces (1 ≤ m ≤ n−1) in a unit sphere Sⁿ⁺¹(1). Whenm= 1, our result reduces to the result of T. Otsuki [14], [15], Brito and Leite [5].

1 Introduction

Do Carmo and Dajczer [7] defined the rotational hypersurfaces in space forms and studied the rotational hypersurfaces with constant mean curvature in space forms. Some years later, Leite [10] classified the complete rotational hypersurfaces with constant scalar curvature in space forms. In [16], Palmas studied the rota- tional hypersurfaces with constantHmin space forms, whereHmis them-th mean curvature.

The embeddability of rotational minimal hypersurfaces of Sⁿ⁺¹(1) has been treated by Otsuki in a long series of papers that started with [14]. From a different point of view, Brito and Leite [5] also considered the embeddability of compact minimal rotational hypersurfaces ofSⁿ⁺¹(1). They proved the following important result:

Theorem 1.1. ([14], [15], [5]) There are no compact minimal embedded rotational Key words and phrases : Rotational hypersurfaces, embedded hypersurfaces, Rieman- nian product.

Some of this work was done while the author was visiting postdoctoral fellow at the De- partment of Mathematics, Kyungpook National University, Korea, in the year of 2006. He would like to express his thanks to Professor Y. J. Suh for his hospitality and for the grant from Korea Research Foundation, Proj. No. R14-2002-003-01001-0, Korea 2006.

225

hypersurfaces of Sⁿ⁺¹ other than Clifford tori and round geodesic spheres.

From [16], we know that there also exist many compact immersed m-minimal ro- tational hypersurfaces of Sⁿ⁺¹. From Theorem 1.1, we know that the following problem is interesting:

Problem. Does there exist any n-dimensional compact embedded m-minimal ro- tational hypersurface (1 ≤ m ≤ n−1) in Sⁿ⁺¹ other than Riemannian product Sⁿ⁻¹

³qn−m n

´

×S¹¡p_m

n

¢and round geodesic spheres?

Li and the author [13] solved that problem. In fact, that problem was proved to be equivalent to the determination of the periods of the solutions of a certain ordinary differential equation. Li and the author [13] gave estimates on those periods deciding the original problem of embeddability of non-standardm-minimal compact rotational hypersurfaces. Recently, Suh and the author [18] gave another estimation better than that of Li and the author [13].

2 Preliminaries

LetM be a rotational hypersurface ofSⁿ⁺¹, that is, invariant by the orthogonal groupO(n) considered as a subgroup of isometries ofSⁿ⁺¹(1). Let us parametrize the profile curveαinS²(1) byy1=y1(s)≥0,yn+1=yn+1(s) andyn+2=yn+2(s).

We take ϕ(t1,· · · , tn−1) = (ϕ1,· · ·, ϕn) as an orthogonal parametrization of the unit sphereSⁿ⁻¹(1). It follows that the rotational hypersurface (see [7])

x:Mⁿ ,→Sⁿ⁺¹(1)⊂Rⁿ⁺²,

(s, t1,· · · , tn−1)7→(y1(s)ϕ1,· · ·, y1(s)ϕn, yn+1(s), yn+2(s)). (2.1) ϕi=ϕi(t1,· · · , tn−1), ϕ²₁+· · ·+ϕ²_n= 1 (2.2) is a parametrization of a rotational hypersurface generated by a curve y1 =y1(s), yn+1 = yn+1(s) and yn+2 = yn+2(s). Since the curve {y1(s), yn+1(s), yn+2(s)}

belongs toS²(1) and the parameterscan be chosen as its arc length, we have y₁²(s) +y_n+1² (s) +y²_n+2(s) = 1, y˙₁²(s) + ˙y²_n+1(s) + ˙y_n+2² (s) = 1 (2.3) where the dot denotes the derivative with respect tosand from (2.3) we can obtain yn+1(s) andyn+2(s) as functions ofy1(s). In fact, we can write

y1(s) = cosr(s), yn+1(s) = sinr(s) cosθ(s), yn+2(s) = sinr(s) sinθ(s). (2.4) We can deduce from (2.3) that

˙

r²+ ˙θ²sin²r= 1. (2.5)

It follows from equation (2.5) that ˙r² ≤ 1. Combining these with ˙r² = _1−y^y˙122 1, we have

˙

y²₁+y²₁≤1. (2.6)

Writingf(s) =y1(s), do Carmo and Dajczer proved the following

Lemma 2.1. ([7]) Let Mⁿ be a rotational hypersurface of Sⁿ⁺¹(1). Then the principal curvatures λi ofMⁿ are

λi=λ=− q

1−f²−f˙²

f (2.7)

fori= 1,· · ·, n−1, and

λn=µ= f¨+f q

1−f²−f˙²

. (2.8)

3 m-minimal rotational hypersurfaces in Sⁿ⁺¹(1)

The m-th mean curvatureHm of the hypersurface can be given in such a way that

C_n^mHm= X

1≤i1<i2<···<im≤n

λi1λi2· · ·λim (3.1)

whereC_n^m= n(n−1)···(n−m+1)

m(m−1)···1 ,λis⁰ are the principal curvatures ofM. WhenHm= 0, a hypersurfaceM is said to be m-minimal.

ExampleMm,n−m=Sⁿ⁻¹

³qn−m n

´

×S¹¡p_m

n

¢, 1≤m≤n−1.

ThenMm,n−mhas two distinct constant principal curvatures λ1=· · ·=λn−1=p

m/(n−m), λn=−p

(n−m)/m.

Hence,Hm≡0 and the square norm of the second fundamental form ofMm,n−m

satisfies

S= n(m²−2m+n) m(n−m) .

If M is am-minimal rotational hypersurface (m < n) in Sⁿ⁺¹(1), then we can deduce that

0 =C_n^mHm=C_n−1^m−1λ^m−1µ+C_n−1^m λ^m. That is,

λ^m−1{(n−m)λ+mµ}= 0. (3.2)

By putting (2.7) and (2.8) into (3.2), we get the following result of Oscar Palmas [16]:

Lemma 3.1. ([16]) The rotational hypersurface Mⁿ in Sⁿ⁺¹(1) is m-minimal (m < n)if and only if f satisfies the following differential equation:

(n−m)(1−f²−f˙²)^m2 −m(1−f²−f˙²)^m−22 ( ¨f +f)f = 0. (3.3)

Equation (3.3) is equivalent to its first order integral

f^n−m(1−f²−f˙²)^m2 =K, (3.4)

whereK is a constant.

For a constant solutionf =f0in (3.3), one has that

f₀²=n−m

n , K0=³m n

´^m

2

µn−m n

¶^m(n−m)

2n

. (3.5)

Moreover, the constant solutions of equation (3.3) correspond to the Riemannian productSⁿ⁻¹³q

n−m n

´

×S¹¡p_m

n

¢.

Now we follow the techniques in the paper of Leite [10] and Palmas [16] to study (3.4). Equation (3.4) tells us that a local solution f of (3.3) paired with its first derivative is a subset, denoted by (f,f˙), of a level curve for the functionGmdefined by

G_m(u, v) =u^n−m(1−u²−v²)^m2, (3.6)

withu >0 andu²+v²≤1.

Let us map the open half plane {(u, v) | u >0} by level curveG_m =K (see Figure 1). Each curve is a smooth union of two graphs

v=± s

1−u²− µ K

u^n−m

¶_2/m

, (3.7)

except for the level K0 given by (3.5). The level curve Gm = K0 consists of the unique critical point ofGm, which is on the horizontal axis, as it can be seen from

∇Gm(u, v) =u^n−m−1¡

1−u²−v²¢^m−2

2 ¡

(n−m)(1−v²)−nu²,−muv¢

. (3.8)

v

1 u 1

−1 0

K=0 K>0

K=K0

.

Figure 1: Level curves forK≥0.

ForK = 0, the level curveu²+v²= 1 is a semi-circle. ForK6= 0, we can get easily that the level curve is closed in the open half plane (in fact, in the semicircular region, see Figure 1).

We consider the foliation of the open half plane by level curvesGm=K. Since Gmhas a maximum atK0,K∈[0, K0]. Clearly any curve at an intermediate level K is compact and the associated solutionsr(s) attains a unique minimumr1>0.

Now we have to consider three cases.

Case 1: K=K0.

The value K = K0 implies λ1 = · · · = λn−1 = −q

m

n−m, λn =

qn−m m by (2.7), (2.8) and (3.5), corresponding to the Riemannian productSⁿ⁻¹

³qn−m n

´

× S¹¡p_m

n

¢.

Case 2: K= 0.

K = 0 gives us a totally geodesic n-sphere. In fact, fromK = 0 and equation (3.4), we get f²+ ˙f² = 1. Integration of f²+ ˙f² = 1 with f(0) = 0, we obtain f = sinsand θ =constant, so the profile curve is a great circle which generates a totally geodesic n-sphere.

Case 3: K∈(0, K0).

IfK∈(0, K0), then we have

f²+ ˙f²<1, 0< f <1, f 6= constant. (3.9) It follows fromf(s) =y1(s) = cosr(s) that

0<cosr <1, 0<sinr <1. (3.10) Usingf(s) = cosr(s), (2.5) can be written as

θ˙²= 1−r˙²

sin²r = 1−f²−f˙²

(1−f²)² , (3.11)

we can deduce from (3.9) and (3.11) that

θ˙6= 0, r <˙ 1. (3.12)

We see from (2.7) and (3.9) thatλ=−

√1−f²−f˙²

f 6= 0, then it follows from (3.2) that

(n−m)λ+mµ= 0, λ6=µ, (3.13)

that is,M has two distinct principal curvatures such that the multiplicity of one of the principal curvatures is 1 inSⁿ⁺¹(1).

In this case, we know that any solution f(s) of (3.3) is periodic and period P(K) is given by the following integral:

P(K) = 2 Z _f1

f0

s 1

K^−m2f^2(n−m)m (1−f²)−1× 1

1−f²df, (3.14) wheref0andf1are the two solutions of f^n−m(1−f²)^m2 =K, 0< K < K0.

4 Some integral inequalities and their applications

In this section, we will give some integral inequalities related to rotational hy- persurfaces inSⁿ⁺¹(1) and their applications.

Theorem 4.1. ([13])For any integern, mand1≤m≤n−1, we have

π < P(K)<2π. (4.1)

Theorem 4.2. ([18])For any integern, mand1≤m≤n−1, we have π < P(K)<

µ 1

√2 + n

√n²+ 4n−4

¶

π. (4.2)

As the application of Theorem 4.1 and Theorem 4.2, we get:

Theorem 4.3. ([13]) There are no compact embedded m-minimal rotational hy- persurfaces with (1 ≤ m ≤ n−1) of Sⁿ⁺¹ other than the Riemannian product Sⁿ⁻¹³q

n−m n

´

×S¹¡p_m

n

¢and round geodesic spheres,where Sⁿ⁻¹(a) denotes the (n−1)-dimensional sphere of radiusa.

Remark 4.1. Whenm= 1, our Theorem 4.3 reduces to Theorem 1.1.

Theorem 4.4. ([17]) If M is an n-dimensional compact connected m-minimal rotational hypersurface(1≤m≤n−1, n≥3) inSⁿ⁺¹(1). Then the square norm S of the second fundamental form ofM satisfies

Z

M

½

S−n(m²−2m+n) m(n−m)

¾

dV ≤0 (4.3)

with equality holding if and only if M is isometric to the Riemannian product Sⁿ⁻¹

³qn−m n

´

×S¹¡p_m

n

¢.

Acknowledgements Some of this work was done while the author was visiting postdoctoral fellow at the Department of Mathematics, Kyungpook National Uni- versity, Korea, in the year of 2006. He would like to express his thanks to Professor Y. J. Suh for his hospitality and for the grant from Korea Research Foundation, Proj. No. R14-2002-003-01001-0, Korea 2006.

References

[1] H.Alencar, M.do Carmo and M.F.Elbert, Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces, J. Reine Angew. Math.

554(2003), 201-216.

[2] H.Alencar, M.do Carmo and W.Santos, A gap theorem for hypersurfaces of the sphere with constant scalar curvature one, Comment. Math. Helv. 77(2002), 549-562.

[3] J.L.M.Barbosa and A.G.Colares, Stability of hypersurfaces with constant r- mean curvature,Ann. Global Anal. Geom.15(1997), 277-297.

[4] J.L.M.Barbosa and M.do Carmo, On stability of cones inRⁿ⁺¹with zero scalar curvature,Ann. Global Anal. Geom.28(2005), 107-127.

[5] F.Brito and M.L.Leite, A remark on rotational hypersurfaces ofSⁿ,Bull. Soc.

Math. Belg.–Tijdschr. Belg. Wisk. Gen.42(1990), 3,ser. B 303-318.

[6] S.Y.Cheng and S.T.Yau, Hypersurfaces with constant scalar curvature,Math.

Ann. 225(1977), 195-204.

[7] M.do Carmo and M.Dajczer, Rotational hypersurfaces in spaces of constant curvature,Trans. Amer. Math. Soc.277(1983), 685-709.

[8] M.do Carmo and M.F.Elbert, On stable complete hypersurfaces with vanishing r-mean curvature,Tohoku Math. J.(2)56(2004), no.2, 155-162.

[9] J.Hounie and M.L.Leite, Uniqueness and nonexistence theorems for hypersur- faces with Hk = 0,Ann. Global Anal. Geom.17(1999), no.5, 397-407.

[10] M.L.Leite, Rotational hypersurfaces of space forms with constant scalar cur- vature, Manuscripta Math.67(1990), 285-304.

[11] H.Li, Hypersurfaces with constant scalar curvature in space forms,Math. Ann.

305(1996), 665-672.

[12] H.Li, Global rigidity theorems of hypersurface,Ark.Math.35(1997), 327-351.

[13] H.Li and G.Wei, Compact embedded rotation hypersurfaces ofSⁿ⁺¹, to appear in Bull. Braz. Math. Soc.

[14] T.Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curva- ture,Amer. J. Math.92(1970), 145-173.

[15] T.Otsuki, On integral inequalities related with a certain non-linear differential equation, Proc. Japan Acad.48(1972), 9-12.

[16] O.Palmas, Complete rotation hypersurfaces withHk constant in space forms, Bol. Soc. Bras. Mat. 30(1999), 139-161.

[17] G.Wei, Rigidity theorem for hypersurfaces in a unit sphere, Monatsh. Math.

149(2006), 343-350.

[18] G.Wei and Y.J.Suh, Integral inequalities related to rotational hypersurfaces in a unit sphere, preprint.