HYPERSURFACES IN EUCLIDEAN SPACES

Qing-Ming Cheng*

Abstract. In this paper, we shall discuss complete hypersurfaces in Euclidean spaces with constant scalar curvature or constant mean curvature. In particular, Hopf’s conjecture and Yau’s conjecture will be mentioned.

1. Introduction and Preliminaries.

The study of hypersurfaces in Euclidean spaces has a long and interesting history. It is well-known that Liebmann showed that standard round spheres are the only oriented compact surfaces in the Euclidean three-space R^{3} with constant scalar curvature. He also proved that a strictly convex, compact surface of constant mean curvature in R^{3} is necessarily a standard round sphere. It is obvious that the mean curvature and the scalar curvature of a given hypersurface in Euclidean spaces are the basic invariants with important geometric meaning.

In this article, we shall study hypersurfaces in Euclidean spaces with constant mean curvature or constant scalar curvature.

Let M be an n-dimensional hypersurface in a Euclidean (n+1)-space R^{n+1}. We choose a local orthonormal frame field {e_{1}, …, e_{n+1}} in R^{n+1} so that, restricted to M, e_{1}, …, e_{n} are tangent to M. Let ω_{1}, ⋯, ω_{n+1} denote the dual coframe field in R^{n+1}. We restrict these forms to M. Then, in M,

(1.1)  ω_{n+1} = 0.

Let ω_{ij} denote the connection forms of R^{n+1}. From Cartan’s Lemma we may write

(1.2)  ω_{n+1i} = ∑_{j=1}^{n} h_{ij}ω_{j}, h_{ij} = h_{ji}.

Hence, the structure equations of M are given by

(1.3)  \{ \begin{matrix}dω_{i} + ∑_{j} ω_{ij} ∧ ω_{j} = 0, ω_{ij} + ω_{ji} = 0, \\ dω_{ij} + ∑_{k} ω_{ik} ∧ ω_{kj} = Ω_{ij}, \\ Ω_{ij} = \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l}, \end{matrix}

2000 Mathematics Subject Classification. Primary 53C42.

Key words and phrases: Constant scalar curvature, principal curvature, geodesic and locally conformally flat hypersurface.

* Research partially Supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Curvature, Japan.

where Ω_{ij} (resp. R_{ijkl}) denotes the curvature form (resp. the components of the curvature tensor) of M. The Gauss equation is given by

(1.4)  R_{ijkl} = h_{ik}h_{jl} − h_{il}h_{jk}.

We define the second fundamental form α⃗ by

(1.5)  α⃗ = ∑_{i,j=1}^{n} h_{ij}ω_{i} ⊗ ω_{j}e_{n+1}.

The mean curvature H of M is defined by

(1.6)  nH = ∑_{i} h_{ii}.

The covariant derivative ∇α⃗ of the second fundamental form α⃗ of M with components h_{ijk} is given by

(1.7)  ∑_{k} h_{ijk}ω_{k} = dh_{ij} + ∑_{k} h_{jk}ω_{ik} + ∑_{k} h_{ik}ω_{jk}.

Then, we have Codazzi equation

(1.8)  h_{ijk} = h_{ikj}.

From (1.4), we have

(1.9)  R = n^{2}H^{2} − ∑_{i,j=1}^{n} h_{ij}^{2},

where R denotes the scalar curvature of M.

2. Hypersurfaces in Euclidean spaces with constant mean curvature

In 1951, Hopf [12] proved that the standard round spheres are the only compact surfaces with constant mean curvature, which are homeomorphic to the standard round 2-dimensional sphere S^{2}(c), in R^{3}. Also, he proposed the following

Hopf’s conjecture. Let M be an oriented compact hypersurface with constant mean curvature in R^{n+1}. Must M be the standard round sphere?

To this conjecture, Alexandrov [2] obtained an important result. Namely, he showed that Hopf’s conjecture is true if M is an embedded hypersurface in R^{n+1}. That is, Theorem 2.1 ([2]). If M is an embedded compact hypersurface in R^{n+1} with constant mean curvature, then M is isometric to a standard round sphere.

The basic problem of whether there exist any other oriented compact immersed hypersurfaces in R^{n+1} with constant mean curvature was open until Hsiang [13] and Wente [22] discovered the examples which are not standard round sphere. Wente constructed an immersed torus in R^{3} with constant mean curvature. More surprisingly, Abresch [1] has shown that some very old surfaces with constant mean curvature studied in the nineteenth century (see Darboux [8, p. 217]) are compact surfaces with genus one. Also, Kapouleas [14] has constructed new examples of compact surfaces immersed into R^{3} with constant mean curvature and higher genus.

On the other hand, in 1841, Delaunay discovered a beautiful method of constructing rotational symmetric hypersurfaces with constant mean curvature in R^{3}. Namely, by rolling a given conic section on a line in a plane and rotating about that line the trace of a focus, one obtains a surface with constant mean curvature in R^{3}. Conversely, except standard round spheres, all rotational symmetric surfaces with constant mean curvature in R^{3} can be constructed in this way. From the viewpoint of equivariant differential geometry, a natural generalization of the SO(2) rotation on R^{3} is simply an orthogonal transformation group (G, R^{n}) with codimension-two principal orbit type. By exploiting the study of generalized rotational hypersurfaces of O(k) × O(l)-type, Hsiang [13] constructed a variety of important examples of compact hypersurfaces with constant mean curvature in R^{n+1}, which are diffeomorphic to n-dimensional sphere S^{n}(c), n > 2. Thus, the result of Hopf is a special phenomenon valid only in R^{3}. Hence, the assumption of embedding in the theorem 2.1 of Alexandrov is a necessity which can not be replaced by immersion.

As a complete version of the result of Hopf, Klotz and Osserman [15] studied complete surfaces with constant mean curvature in R^{3}. They proved the following

Theorem 2.2 ([15]). Let M be a complete surface with constant mean curvature in R^{3}. If the Gaussian curvature is nonnegative, then M is isometric to a plane, a sphere or a cylinder.

As a generalization of theorem 2.2, Cheng and Yau [7] prove the following

Theorem 2.3 ([7]). Let M be a complete hypersurface with constant mean curvature in R^{n+1}. If the sectional curvature of M is nonnegative, then M is isometric to a standard round sphere, a hyperplane R^{n} or a Riemannian product S^{k}(c) × R^{n−k}, 1 ≤ k ≤ n − 1.

Theorems in this section can be proved by using methods similar to those in the next section.

3. Hypersurfaces in Euclidean spaces with constant scalar curvature

In this section, we shall consider complete hypersurfaces in Euclidean spaces with constant scalar curvature. In [6], in order to investigate these hypersurfaces, Cheng and S. T. Yau introduced a new differential operator □ defined by

(3.1)  □f = ∑_{i,j=1}^{n}(nHδ_{ij} − h_{ij})∇_{i}∇_{j}f

for any C^{2}-function f defined on M, which is formally self-adjoint. By making use of this differential operator, they proved the following

Theorem 3.1 ([6]). Let M be a complete oriented hypersurface in R^{n+1} with constant scalar curvature. If the sectional curvature is nonnegative, then M is isometric to a standard round sphere, a Euclidean space R^{n} or a generalized cylinder S^{n−k}(c) × R^{k} for 1 ≤ k ≤ n − 1.

Proof. (1) When M is compact, we consider □(nH). We then obtain, by a simple computation,

(3.2)  □(nH) = ∑_{i,j,k=1}^{n} h_{ijk}^{2} − n^{2}|gradH|^{2} + ∑_{i<j}(λ_{i} − λ_{j})^{2}K_{ij}

since the scalar curvature is constant. From the Gauss equation, we can prove

(3.3)  ∑_{i,j,k=1}^{n} h_{ijk}^{2} − n^{2}|gradH|^{2} ≥ 0.

Hence, from Stokes formula, we infer that M is isoparametric, that is, M is isometric to a standard round sphere because M is compact.

(2) When M is complete and noncompact, we consider □⟨N, a⃗⟩, where N is the unit normal vector field and a⃗ is a fixed unit vector. Note that, since M is of nonnegative sectional curvature, M is convex. Hence there exsits a unit vector a⃗ in the Euclidean space so that ⟨N, a⃗⟩ ≥ 0 on M (see Wu [23]). By the same curvature assumption, we also know that the scalar curvature R = ∑_{i≠j} K_{ij} ≥ 0. Since the scalar curvature is constant, R = 0 at some point if and only if the sectional curvature is zero. Hence, in the case that the scalar curvature is zero, Theorem 3.1 is true from a theorem due to Hartman and Nirenberg [10]. If the scalar curvature is positive, we can prove that the operator □ is elliptic. By a direct computation, we infer that

(3.4)  □⟨N, a⃗⟩ = − ∑_{k,l=1}^{n}(nHδ_{kl} − h_{kl}) ∑_{i=1}^{n}(h_{ki}h_{il})⟨N, a⃗⟩.

From Maximum Principle, we obtain that ⟨N, a⃗⟩ is constant. Therefore, ⟨N, a⃗⟩ > 0 or ⟨N, a⃗⟩ ≡ 0. From this assertion, we infer that the Theorem 3.1 holds.

By using a similar method, we can prove that Theorem 2.3 in Section 2 is true. In 1982, Yau [24] proposed to study oriented compact hypersurfaces with con-

stant scalar curvature in R^{n+1} and conjectured the following

Yau’s conjecture. The standard round spheres are the only possible oriented compact hypersurfaces in R^{n+1} with constant scalar curvature.

By making use of the so-called Reilly’s formula, Ros [20] showed that Yau’s conjecture is true if M is an embedded hypersurface. Namely, Theorem 3.2 ([20]). If M is an embedded compact hypersurface with constant scalar curvature in R^{n+1}, then M is isometric to a standard round sphere.

On the other hand, the topological types of compact locally conformally flat hypersurfaces in R^{n+1}, n > 3, were determined by do Carmo, Dajczer and Mercuri [9]. They also described the extrinsic geometry of compact locally conformally flat hypersurfaces in R^{n+1}. Finally, Pinkall [18] determined their intrinsic conformal geometry. He proved that every compact locally conformally flat hypersurface in R^{n+1}, n > 3, is conformally equivalent to a classical Schottky manifold. That is,

Theorem 3.3 ([8]). Every compact locally conformally flat hypersurface in R^{n+1} with n ≥ 4 is conformally equivalent to the standard round sphere or a classical Schottky manifold.

Since Yamabe’s problem was affirmatively solved, that is, any metric on a compact manifold can be conformally deformed to a metric of constant scalar curvature, it is natural to ask the following

Problem 1. Whether do there exist classical Schottky manifolds with constant scalar curvature such that they can be immersed into R^{n+1} as locally conformally flat hypersurfaces? Namely, is Yau’s conjecture true for the class of locally conformally flat hypersurfaces?

In [3], the author studied oriented complete hypersurfaces with constant scalar curvature in R^{n+1} and proved that Yau’s conjecture is true for the class of locally conformally flat hypersurfaces (see Corollary 1).

Theorem 3.4. If M is an oriented complete hypersurface in R^{n+1}, n > 2, with constant scalar curvature n(n − 1)r and with two distinct principal curvatures, then M is isometric to one of the following hypersurfaces:

(1) R^{n−k} × S^{k}(c), for 1 ≤ k ≤ n − 1,

(2) a complete non-compact hypersurface of revolution S^{n−1}(c(s)) × M^{1}, where S^{n−1}(c(s)) is of constant curvature [\frac{d}{ds}\{log |λ^{2} − r|^{1/n}\}]^{2} + λ^{2} and M^{1} is a plane curve. Moreover, w = {λ^{2} − r}^{−1/n} satisfies the following ordinary differential equation of order 2:

\frac{d^{2}w}{ds^{2}} − w(\frac{n − 2}{2} \frac{1}{w^{n}} − r) = 0,

where λ is the principal curvature with multiplicity n − 1.

Theorem 3.5. If M is an oriented complete locally conformally flat hypersurface in R^{n+1}, n > 3, with constant scalar curvature n(n − 1)r, then M is isometric to one of the following hypersurfaces:

(1) a standard round sphere,

(2) R^{n},

(3) R^{n−1} × S^{1}(c),

(4) R^{1} × S^{n−1}(c),

(5) a complete non-compact hypersurface of revolution S^{n−1}(c(s)) × M^{1}, where S^{n−1}(c(s)) is of constant curvature [\frac{d}{ds}\{log |λ^{2} − r|^{1/n}\}]^{2} + λ^{2} and M^{1} is a plane curve. Moreover, w = {λ^{2} − r}^{−1/n} satisfies the ordinary differential equation of order 2 in Theorem 3.4.

Corollary 1. If M is an oriented compact locally conformally flat hypersurface in R^{n+1}, n > 3, with constant scalar curvature n(n − 1)r, then M is isometric to a standard round sphere S^{n}(r).

In order to prove Theorem 2.1 and Theorem 3.2, we prove several lemmas, which are used in the later.

Let M be a compact hypersurface in R^{n+1}. Let ϕ denote the position vector field of R^{n+1} relative to the origin. With a straightforward calculation we then have on M

(3.5)  Δ⟨ϕ, ϕ⟩ = 2n(1 + H⟨ϕ, N⟩),

where N is a unit normal field on M. By applying the differential operator □ defined by

□f = ∑_{i,j=1}^{n}(nHδ_{ij} − h_{ij})∇_{i}∇_{j}f,

where f is a C^{2}-function on M, we also obtain on M

(3.6)  □⟨ϕ, ϕ⟩ = n(n − 1)H + R⟨ϕ, N⟩.

Lemma 1 (Minkowski formulas). Let M be a compact hypersurface in R^{n+1}. Then we have

(3.7)  ∫_{M}(1 + H⟨ϕ, N⟩)dM = 0

and

(3.8)  ∫_{M}(n(n − 1)H + R⟨ϕ, N⟩)dM = 0,

where dM denotes the volume element on M.

Now we assume that M is a compact embedded hypersurface in R^{n+1}. Then, there exists a compact domain Ω ⊂ R^{n+1} such that the boundary ∂Ω = M holds. Let Δ̄ denote the Euclidean Laplacian of R^{n+1}. We then have

(3.9)  Δ̄⟨ϕ, ϕ⟩ = 2(n + 1)

on R^{n+1}. Hence, from the divergence theorem, we obtain

(3.10)  − ∫_{M}⟨ϕ, N⟩dM = (n + 1)volΩ,

where N is the inner unit normal field of Ω.

Lemma 2. Let M be a compact embedded hypersurface in R^{n+1}. If the mean curvature H of M with respect to the inner normal N is positive everywhere on M, then we have

(3.11)  ∫_{M} \frac{1}{H} dM ≥ (n + 1)volΩ,

and the equality holds if and only if M is isometric to a standard round sphere.

Remark 2. The Lemma 2 can be proved by the so-called Reilly’s formula (see Ros [20]) and it can also be proved by using the idea of Heintze and Karcher [11] (see Montiel and Ros [17]).

Proof of Theorem 2.1. Since M is compact, one can find a point where all the principal curvatures with respect to the inner normal vector field N are positive. Since H is constant, we have H > 0. According to the formula (3.10), we have

− ∫_{M}⟨ϕ, N⟩dM = (n + 1)volΩ.

From the Lemma 1, we then have

∫_{M} \frac{1}{H} dM = − ∫_{M}⟨ϕ, N⟩dM = (n + 1)volΩ

since H is contant. From Lemma 2, we conclude that M is isometric to a standard sphere.

Proof of Theorem 3.2. From the Gauss equation, we have

R = n^{2}H^{2} − S ≤ n(n − 1)H^{2},

since S = ∑_{i=1}^{2} λ_{i}^{2} ≥ nH^{2} holds, where λ_{i} are the principal curvatures of M. By using the Minkowski formula (3.8), and the argument similar to that in the proof of Theorem 2.1, we see that Theorem 3.2 is true.

When we do not assume that M is embedded into R^{n+1}, according to the Minkowski formulas, we can prove the following

Theorem 3.6. Let M be an oriented compact hypersurface in R^{n+1} with constant mean curvature. If the scalar curvature is constant, then M is isometric to a standard round sphere.

Proof. From the Minkowski formulas, we have

∫_{M}(1 + H⟨ϕ, N⟩)dM = 0,

∫_{M}{n(n − 1)H + R⟨ϕ, N⟩}dM = 0.

Since the mean curvature H and the scalar curvature R of M are constant, we then have

∫_{M}{n(n − 1)H^{2} − R}dM = 0.

Because of the inequality

R = n^{2}H^{2} − S ≤ n(n − 1)H^{2},

we therefore have R = n(n − 1)H^{2}. Hence, M is totally umbilical, and thus M is isometric to a standard round sphere.

It is well-known that the Riemannian product S^{k}(c)×R^{n−k} for k = 1, 2, ⋯, n−1 are oriented complete hypersurfaces with constant mean curvature and constant scalar curvature in R^{n+1}, which are called generalized cylinders. From Theorem 3.6, it is natural to ask the following

Problem 2. Let M be an oriented complete hypersurface in R^{n+1} with constant mean curvature. If the scalar curvature is constant, then is M isometric to a standard round sphere, a Euclidean space R^{n} or a Riemannian product S^{k}(c) × R^{n−k} for k = 1, 2, ⋯, n − 1?

When n = 2, it is obvious that Problem 2 holds. When n = 3, Wan and the author [5] gave an affirmative answer for it, that is, we proved the following

Theorem 3.7 ([5]). Let M be an oriented complete hypersurface in R^{4} with constant mean curvature. If the scalar curvature is constant, then M is isometric to a standard round sphere, the Euclidean three-space R^{3} or the Riemannian product S^{k}(c) × R^{3−k} for k = 1, 2.

From Theorem 3.4, we know that there are many complete hypersurfaces, which have two distinct principal curvatures, one of which is simple, but are not isometric to S^{n−1}(c) × R and S^{1}(c) × R^{n−1}. The following theorem gives a characterization of the Riemannian products S^{n−1}(c) × R and S^{1}(c) × R^{n−1}.

Theorem 3.8. Let M be an oriented complete hypersurface in R^{n+1} with constant scalar curvature n(n − 1)r and with two distinct principal curvatures, one of which is simple. If S ≥ n(n−1)r/(n−2), then M is isometric to the Riemannian product S^{n−1}(c) × R or S^{1}(c) × R^{n−1}.

Corollary 2. Let M be an oriented complete locally conformally flat hypersurface with constant scalar curvature n(n − 1)r in R^{n+1}, n > 3. If S ≥ n(n − 1)r/(n − 2), then M is isometric to one of the following hypersurfaces:

(1) R^{n},

(2) S^{1}(c) × R^{n−1},

(3) S^{n−1}(c) × R.

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Department of Mathematics Faculty of Science Josai University Sakado, Saitama 350-0295 Japan E-mail address: cheng@math.josai.ac.jp