Josai Mathematical Monographs

vol 3 (2001), pp. 71-80

MAXIMUM PRINCIPLES AND ITS APPLICATIONS TO SUBMANIFOLDS

KAZUHIRO NONAKA

Abstract. In this paper, we study n-dimensional complete and connected submanifolds in an (n + p)-dimensional Euclidean space E^{n+p}. The classification of n-dimensional complete and connected submanifolds in an (n + p)-dimensional Euclidean space E^{n+p} is given under some conditions on submanifolds.

1. Introduction

The purpose of this paper is to study n-dimensional connected submanifolds in an (n + p)-dimensional Euclidean space E^{n+p}. In 1900, Liebmann proved that compact surfaces with constant Gaussian curvature in E^{3} and a compact surfaces with constant mean curvature and with nonnegative Gaussian curvature in E^{3} are the standard spheres. In 1951, Hopf [8] proved the following theorem.

Theorem 1.1 (H. Hopf). Let M be a compact surface with constant mean curvature |H| and with genus zero in E^{3}, then M is a standard sphere.

The theorem of Hopf was extended to complete surfaces in E^{3} by Klotz-Osserman [9] as following.

Theorem 1.2 (T. Klotz and R. Osserman). Let M be a complete and connected surface with constant mean curvature |H| in E^{3}. If the Gaussian curvature G of M is nonnegative, then M is a plane E^{2} in E^{3}, a sphere S^{2}(c) in E^{3} or a cylinder S^{1}(c) × E^{1} in E^{3}.

From the equation of Gauss, we know that the Gaussian curvature G of a surfaces in E^{3} is nonnegative if and only if ⟨h⟩^{2} ≤ n^{2}|H|^{2}/(n − 1), where n = 2 and ⟨h⟩ is the length of the second fundamental form of a surfaces in E^{3} and |H| is the mean curvature of a sufaces in E^{3}. The result due to Klotz-Osserman is extended to higher codimensions by Shen in [14] as following.

Theorem 1.3 (B. Y. Shen). Let M be a complete and connected surface with parallel mean curvature vector H in an (2 + p)-dimensional Euclidean space E^{2+p}. If the second fundamental form h of M satisfies

⟨h⟩^{2} ≤ \frac{n^{2}|H|^{2}}{n − 1} (n = 2) ,

Date: February, 2001.

then M is a plane E^{2} in E^{2+p}, a sphere S^{2}(c) in E^{2+p}, a cylinder S^{1}(c) × E^{1} in E^{2+p} or a product surface with circles S^{1}(c_{1}) × S^{1}(c_{2}) in E^{2+p}.

For connected hypersurfaces in an (n + 1)-dimensional Euclidean space E^{n+1}, in [12], Nomizu and Smyth proved the following :

Theorem 1.4 (K. Nomizu and B. Smyth).

(1) Let M be a compact hypersurface with constant mean curvature |H| in an (n + 1)- dimensional Euclidean space E^{n+1}. If the sectional curvatures of M are nonnegative, then M is a hypersphere S^{n}(c) in E^{n+1}.

(2) Let M be a complete and connected hypersurface with constant mean curvature |H| and with constant scalar curvature r in an (n + 1)-dimensional Euclidean space E^{n+1}. If the sectional curvatures of M are nonnegative, then M is a hyperplane E^{n} in E^{n+1}, a hypersphere S^{n}(c) in E^{n+1} or the generalized cylinder S^{n−k}(c) × E^{k} (1 ≤ k ≤ n − 1) in E^{n+1}.

In the case of complete and connected hypersurfaces in E^{n+1}, Nomizu and Smyth assumed three conditions, that is, constant scalar curvature and constant mean curvature and nonnegative sectional curvature, but the conditions of the theorem of Nomizu and Smyth are too strong. In [5] and [6], Cheng and Yau improved the theorem of Nomizu and Smyth as following.

Theorem 1.5 (S. Y. Cheng and S. T. Yau).

(1) Let M be a complete and connected hypersuface with constant scalar curvature r in an (n + 1)-dimensional Euclidean space E^{n+1}. If the sectional curvatures of M are nonnegative, then M is a hyperplane E^{n} in E^{n+1}, a hypersphere S^{n}(c) in E^{n+1} or the generalized cylinder S^{n−k}(c) × E^{k} (1 ≤ k ≤ n − 1) in E^{n+1}.

(2) Let M be a complete and connected hypersurface with constant mean curvature |H| in an (n + 1)-dimensional Euclidean space E^{n+1}. If the sectional curvatures of M are nonnegative, then M is a hyperplane E^{n} in E^{n+1}, a hypersphere S^{n}(c) in E^{n+1} or the generalized cylinder S^{n−k}(c) × E^{k} (1 ≤ k ≤ n − 1) in E^{n+1}.

We can expect that complete and connected hypersufaces with constant mean curvature |H| and with constant scalar curvature r in an (n + 1)-dimensional Euclidean space E^{n+1} are the hyperplane E^{n} in E^{n+1}, the hypersphere S^{n}(c) in E^{n+1} or the generalized cylinder S^{n−k}(c) × E^{k} (1 ≤ k ≤ n − 1). But it is known that the above conjecture holds when the dimension of hypersurfaces in E^{n+1} is 3.

Remark 1.1. It is known that compact hypersurfaces with constant mean curvature and with constant scalar curvature in an (n + 1)-dimensional Euclidean space E^{n+1} are the hypersphere S^{n}(c) in E^{n+1}.

On the other hand, Cheng and Nonaka [3] studied submanifolds with higher dimensions and higher codimensions in a Euclidean space. They extended the theorem of Klotz and Osserman to higher dimensions and higher codimensions.

Theorem 1.6 (Q. M. Cheng and K. Nonaka). Let M be an n-dimensional complete and connected submanifold with parallel mean curvature vector H in an (n + p)-dimensional Euclidean space E^{n+p} (n ≥ 3). If the second fundamental form h of M satisfies

⟨h⟩^{2} ≤ \frac{n^{2}|H|^{2}}{n − 1} , (1.1)

then M is a totally geodesic Euclidean space E^{n} in E^{n+p}, a totally umbilical sphere S^{n}(c) in E^{n+p} or the generalized cylinder S^{n−1}(c) × E^{1} in E^{n+p}.

Remark 1.2. In [14], Shen intended to prove the theorem 1.6 by making use of the result of Motomiya [11]. But, since the result of Motomiya is wrong (see the section 2), the proof of Shen about the theorem 1.6 is not valid. A proof of the theorem 1.6 was given by Cheng and Nonaka [3] (see the section 3).

2. The Maximum Principle

In this section, we shall mention the maximum principles which play on an important role in the study of differential geometry on Riemannian manifolds. First of all, we state the well known theorem which is called Hopf’s maximum principle as the following.

Hopf’s maximum principle :

Let M be an n-dimensional connected Riemannian manifold. If a C^{2}-function f satisfies Δf ≥ 0 (resp. Δf ≤ 0) on M and has a maximum (resp. a minimum) on M, then f is a constant function, where Δ denotes the Laplacian on M.

When the Riemannian manifolds are compact, Hopf’s maximum principle is often used in the following form.

Hopf’s maximum principle :

Let M be an n-dimensional compact Riemannian manifold. If a C^{2}-function f on M satisfies Δf ≥ 0 or Δf ≤ 0 on M, then M is a constant function.

When the Riemannian manifolds are complete and connected, Omori and Yau proved a very important theorem which is called the genaralized maximum principle. In [13], Omori proved the following.

Omori’s maximum principle :

Let M be an n-dimensional complete and connected Riemannian manifold with the sectional curvatures bounded from below. If a C^{2}-function f is bounded from above, then for all ε > 0, there exists a point x ∈ M such that

sup f − ε < f(x) ,
||grad f(x)|| < ε ,
max{∑_{i,j=1}^{n} X^{i}X^{j}∇_{i}∇_{j}f(x) | X ∈ T_{x}(M), |X| = 1} < ε ,

where ∇ denotes the Riemannian connection on M and X = ∑_{i=1}^{n} X^{i}∂/∂x^{i} for the natural frame ∂/∂x^{i} (i = 1, 2, ⋯, n).

In this paper, we shall make use of the following convention on the ranges of indices :

1 ≤ i, j, k, ⋯ ≤ n, n + 1 ≤ α, β, γ, ⋯ ≤ n + p.

In [15], Yau generalized Omori’s maximum principle.

The generalized maximum principle :

Let M be an n-dimensional complete and connected Riemannian manifold with Ricci curvature bounded from below. If a C^{2}-function f is bounded from above, then for all ε > 0, there exists a point x ∈ M such that

sup f − ε < f(x) , (2.1) ||grad f(x)|| < ε , (2.2) Δf(x) < ε . (2.3)

In [11], Motomiya intended to improve the generalized maximum principle due to Omori and Yau as the following.

Motomiya’s wrong result :

Let M be an n-dimensional complete and connected Riemannian manifold with Ricci curvature bounded from below. If a C^{2}-function f is bounded from above and has no maximum, then for all ε > 0, there exists a point x ∈ M such that

sup f − ε < f(x) < sup f − \frac{ε}{2} , (2.4)

||grad f(x)|| < ε , (2.5) Δf(x) < ε . (2.6)

But this result is wrong. In fact, Cheng and Wu gave the following counter example in [4].

Counter example of result of Motomiya : Let M = E^{2}, f(x, y) = − exp(cx) (c ≥ 2). Obviously, M is a 2-dimensional complete and connected Riemannian manifold with Ricci curvature (=0) bounded from below and a smooth function f(< 0) is bounded from above and f has no maximum on M. So, if for ε > 0, there exists a point q = (x, y) ∈ M such that (2.4)-(2.6) hold, then from (2.5), we see that

||grad f(q)|| = c exp(cx) < ε .

Hence, we have

−\frac{ε}{c} < − exp(cx) = f(q) . (2.7)

On the other hand, since sup f = 0, from (2.4), we have

−ε < f(q) < −\frac{ε}{2} . (2.8)

Taking c ≥ 2, we find that (2.7) and (2.8) are contradictory.

The Motomiya’s wrong result was used by Shen in [14] in order to prove the theorem 1.6 in the section 1. Hence, Shen’s proof is not valid. A proof of the theorem 1.6 was given by Cheng and Nonaka [3] by making use of the generalized maximum principle due to Omori and Yau (see the section 3).

3. Proofs of The Theorems

In this section, we shall prove the theorem 1.4, 1.5 and 1.6. We denote by h the second fundamental form of M and choose an orthonormal frame field e_{1}, e_{2}, ⋯, e_{n+p} on E^{n+p}, restricted to M, so that e_{1}, e_{2}, ⋯, e_{n} are tangent to M and h_{ji}^{n+1} = h_{ji} = ρ_{j}δ_{ji}. Let ω_{1}, ω_{2}, ⋯, ω_{n+p} be dual coframe field on E^{n+p}, that is, ω_{i}(e_{j}) = δ_{ij} and (ω_{ji}) denotes the Riemannian connection form of M and (ω_{αβ}) is the connection form in the normal bundle T^{⊥}(M) of M. We can prove that the second fundamental form h of M can be given by

h = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n} h_{ij}^{α}ω_{i} ⊗ ω_{j}e_{α} .

The mean curvature vector H of M is defined by

H = \frac{1}{n} ∑_{α=n+1}^{n+p} h_{ii}^{α}e_{α} .

We denote by K_{ijkℓ} the components of the curvature tensor of M and by K_{αβkℓ} the components of the normal curvature tensor of M, then the equation of Gauss (3.1) and the equation of Ricci (3.2) are given by

K_{ijkℓ} = c(δ_{ik}δ_{jℓ} − δ_{iℓ}δ_{jk}) + ∑_{α=n+1}^{n+p}(h_{ik}^{α}h_{jℓ}^{α} − h_{iℓ}^{α}h_{jk}^{α}) , (3.1)

K_{αβkℓ} = ∑_{i=1}^{n}(h_{ik}^{α}h_{iℓ}^{β} − h_{iℓ}^{α}h_{ik}^{β}) . (3.2)

From the equation of Gauss (3.1), we have

r = n(n − 1)c + n^{2}|H|^{2} − ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} ,

where r denotes the scalar curvature of M.

Defining h_{ijk}^{α} by

∑_{k=1}^{n} h_{ijk}^{α}ω_{k} = dh_{ij}^{α} + ∑_{k=1}^{n} h_{jk}^{α}ω_{ik} + ∑_{k=1}^{n} h_{ik}^{α}ω_{jk} , (3.3)

then we have the equation of Codazzi as the following.

h_{ijk}^{α} − h_{ikj}^{α} = 0 . (3.4)

Morover, by taking exterior differentiation of (3.3) and defining h_{ijkℓ}^{α} by

∑_{ℓ=1}^{n} h_{ijkℓ}^{α}ω_{ℓ} = dh_{ijk}^{α} + ∑_{ℓ=1}^{n} h_{ℓjk}^{α}ω_{iℓ} + ∑_{ℓ=1}^{n} h_{iℓk}^{α}ω_{jℓ} + ∑_{ℓ=1}^{n} h_{ijℓ}^{α}ω_{kℓ} , (3.5)

then we have

h_{ijkℓ}^{α} − h_{ijℓk}^{α} = ∑_{t=1}^{n} h_{it}^{α}K_{tjkℓ} + ∑_{t=1}^{n} h_{tj}^{α}K_{tikℓ} − ∑_{β=n+1}^{n+p} h_{ij}^{β}K_{αβkℓ} . (3.6)

The Laplacian Δh_{ij}^{α} of the second fundamental form h of M is defined by

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{ijkk}^{α} , (3.7)

then, from the equation of Gauss (3.1) and the equation of Codazzi (3.4) and (3.6), we have

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{kkij}^{α} + ∑_{k=1}^{n} (∑_{t=1}^{n} h_{kt}^{α}K_{tijk} + ∑_{t=1}^{n} h_{ti}^{α}K_{tkjk} − ∑_{β=n+1}^{n+p} h_{ki}^{β}K_{αβjk}) . (3.8)

Proof of Theorem 1.4. (1). From (3.8), by a direct computation, we have

\frac{1}{2}Δ ⟨h⟩^{2} = ∑_{i,j,k=1}^{n}(h_{ijk})^{2} + ∑_{i<j}(ρ_{i} − ρ_{j})^{2}K_{ij} , (3.9)

where ⟨h⟩ denotes the length of the second fundamental form h of M and K_{ij} = ρ_{i}ρ_{j} (i ≠ j) is the sectional curvatures of M for the plane section spanned by e_{i} and e_{j}. Since the sectional curvatures of M are nonnegative and M is compact, by Hopf’s maximum principle, we have

∑_{i,j,k=1}^{n}(h_{ijk})^{2} = 0 , ∑_{i,j=1}^{n}(ρ_{i} − ρ_{j})^{2}K_{ij} = 0 . (3.10)

Hence, M is isoparametric and the number of distinct principal curvatures are at most two. Therefore, by a theorem due to Cartan [1], M is a totally umbilical sphere.

(2). From the equation of Gauss, we see that ⟨h⟩^{2} is constant. Then we have Δ ⟨h⟩^{2} = 0. Hence, (3.10) holds. We find that M is isoparametric and the number of distinct principal curvatures are at most two. Therefore, (2) holds.

Proof of Theorem 1.5. (1). In [6], S. Y. Cheng and S. T. Yau introduced a differential operater □ which is defined by

□f = ∑_{i,j=1}^{n}(n|H|δ_{ij} − h_{ij})∇_{i}∇_{j}f (3.11)

for any C^{2}-function f on M, where |H| denotes the mean curvature of M. We can prove that it is self-adjoint.

When M is compact, we consider □(n|H|). Since the scalar curvature r of M is constant, by a direct computation,

□(n|H|) = ∑_{i,j,k=1}^{n}(h_{ijk})^{2} − n^{2}||grad H||^{2} + ∑_{i<j}(ρ_{i} − ρ_{j})^{2}K_{ij} , (3.12)

where K_{ij} = ρ_{i}ρ_{j} (i ≠ j) is the sectional curveture of M for the plane section spanned by e_{i} and e_{j}. From the equation of Gauss, we can prove

∑_{i,j,k=1}^{n}(h_{ijk})^{2} − n^{2}||grad H||^{2} ≥ 0 . (3.13)

Hence, from Stokes formula, we have

∫_{M} □(n|H|) = 0 . (3.14)

Then we have □(n|H|) = 0. Now, Since M is compact, there exists a point x ∈ M such that the scalar curvature r > 0 at x. Then, the scalar curvature r > 0 on M since r is constant. Hence, we see that □ is elliptic. We infer that |H| is constant. Therefore, from the theorem 1.4, we see that M is a totally umbilical sphere S^{n}(c).

In the following, we consider a complete and connected hypersurface M in E^{n+1}. Since the scalar curvature r = ∑_{i≠j} K_{ij} is nonnegative constant, r = 0 on M if and only if K_{ij} = 0 for all distinct i and j. In this case that the scalar curvature r of M is zero, (1) holds from the theorem due to Hartman and Nirenberg [7]. Hence, we may asuume that the scalar curvature r of M is positive constant.

Let ξ be a unit normal vector field on M. Now, M is convex because the sectional curvatures of M are nonnegative. Then, since the Gauss image of a complete convex hypersurface M in E^{n+1} lies in a closed hemisphere, there exists a unit vector X̃ on E^{n+1} such that g̃( ξ, X̃) ≥ 0 on M(for see [17]), where g̃ is a Euclidean metric of E^{n+1}. By a direct computation, we have

□g̃( ξ, X̃) = − ∑_{k,ℓ=1}^{n}(n|H|δ_{kℓ} − h_{kℓ}) ∑_{i=1}^{n}(h_{ki}h_{iℓ})g̃( ξ, X̃) . (3.15)

Since we see that (n|H|δ_{kℓ} − h_{kℓ}) ∑_{i=1}^{n}(h_{ki}h_{iℓ}) > 0 as above, we find that □g̃( ξ, X̃) ≤ 0 on M and g̃( ξ, X̃) ≥ 0 on M. Since the differential operater □ is elliptic, we inter that g̃( ξ, X̃) is constant. Hence, g̃(ξ, X̃) = 0 on M or g̃(ξ, X̃) > 0 on M. Therefore we conclude that (1) holds from this assertion (see [6] for details).

(2). Let ξ be a unit normal vector field on M. Then we have the well known formula

Δξ = − ⟨h⟩^{2} ξ , (3.16)

where ⟨h⟩ denotes the length of the second fundamental form h of M. Since the sectional curvatures of M are nonnegative, there exists a unit vector X̃ on E^{n+1} such that g̃(ξ, X̃) ≥ 0 on M as the proof of (1). Then, from (3.16), we can prove that g̃(ξ, X̃) has to be identically zero unless ⟨h⟩^{2} tends to zero. If the mean curvature |H| of M is nonzero constant, ⟨h⟩^{2} is bounded from below by a positive constant. Hence, either |H| = 0 on M or g̃(ξ, X̃) = 0 on M. Then (2) holds from this assertion (see [5] for details).

Proof of Theorem 1.6. Since the mean curvature vector H of M is parallel, that is, ω_{αβ} = 0, the mean curvature |H| of M is constant. we now consider the case |H| = 0 and case |H| ≠ 0 separately.

Case(i) : |H| = 0 on M From the assumption of theorem 1.6, we have

⟨h⟩^{2} ≤ \frac{n^{2}|H|^{2}}{n − 1} = 0 ,

that is, M is an n-dimensional totally geodesic Euclidean space E^{n} in E^{n+1}. so the theorem holds.

Case(ii) : |H| ≠ 0 on M

Let e_{1}, e_{2}, ⋯, e_{n+p} be an orthonormal frame field on E^{n+p} such that H = |H|e_{n+1}. We set

|T |^{2} = ∑_{α=n+2}^{n+p} trace A_{α}^{2} , (3.17)

where trace A_{α} = ∑_{i=1}^{n} h_{ii}^{α}. Then we find that the nonnegative smooth function |T |^{2} is defined globally on M. From the following two algebraic lemmas and the fundamental equations about submanifolds, we have

\frac{1}{2}Δ|T |^{2} ≥ \frac{n − 3}{2}|T |^{4} , (3.18)

where Δ denotes the Laplacian on M (for see [3]).

Lemma 3.1 (B. Y. Chen and M. Okumura [2]). Let a_{1}, a_{2}, ⋯, a_{n}, b be n + 1 real number (n > 1) satisfies the following inequality,

(∑_{i=1}^{n} a_{i})^{2} ≥ (n − 1) ∑_{i=1}^{n} a_{i}^{2} + b (resp. >) .

Then we have

2a_{i}a_{j} ≥ \frac{b}{n − 1} (resp. >)

for all distinct i and j.

Lemma 3.2 (A. M. Li and J. M. Li [10]). Let A_{1}, A_{2}, ⋯, A_{p} be symmetric (n × n) - matrices. Then we have

∑_{α,β=1}^{p}{N(A_{α}A_{β} − A_{β}A_{α}) + S_{αβ}^{2}} ≤ (∑_{α=1}^{p} N(A_{α}))^{2}

and equality holds if and only if one of the following conditions holds :

(i) A_{1} = A_{2} = ⋯ = A_{p} = 0

(ii) only two of A_{1}, A_{2}, ⋯, A_{p} are different from zero. Moreover assuming A_{1} ≠ 0, A_{2} ≠ 0, then S_{1} = S_{2} and there exists an (n × n)-matrix T such that

tT A_{1}T = \sqrt{\frac{S_{1}}{2}} (\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & −1 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 \end{matrix}) , ^{t}T A_{2}T = \sqrt{\frac{S_{1}}{2}} (\begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 1 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 \end{matrix})

where S_{α} = N(A_{α}) = trace (^{t}A_{α}A_{α}), S_{αβ} = trace (A_{α}A_{β}).

Condition (1.1) implies that |T |^{2} is bounded from above by n^{2}|H|^{2}/(n − 1) and by the lemma 3.1, we can prove that the sectional curvatures of M are nonnegative (for see [2]). Hence, we can apply the generalized maximum principle due to Omori and Yau to function |T |^{2}. Then there exists a sequence {x_{k}} ⊂ M such that

lim_{k→∞} |T |^{2}(x_{k}) = sup |T |^{2} , (3.19)

lim sup_{k→∞} Δ|T |^{2}(x_{k}) ≤ 0 . (3.20)

From (3.18), (3.19) and (3.20), we see that

(3.21)  lim sup_{k→∞} Δ|T |^{2}(x_{k}) = (n − 3)(sup |T |^{2})^{2} = 0 .

Hence, if n ≥ 4, then we have |T |^{2} = 0 on M. In case of n = 3, from (3.21), we can prove that |T |^{2} = 0 on M by the lemma 3.1 and the lemma 3.2 (for see [3]). From which, we denote N_{1} the normal subbundle spanned by e_{n+2}, e_{n+3}, ⋯, e_{n+p} of the normal bundle T^{⊥}(M) of M, then M is geodesic with respect to N_{1}. Since the mean curvature vector H of M in E^{n+p} is parallel, we see that N_{1} is parallel. Hence, from the theorem 1 in [16], we conclude that M lies (n + 1)-dimensional totally geodesic Euclidean space E^{n+1}. Then we find that the mean curvature |H′|(= |H|) of M in E^{n+1} is constant and the sectional curvatures of M in E^{n+1} are nonnegative. The theorem 1.6 can be obtain from the theorem 1.5 in section 1.

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Graduate School of Science, josai University, Sakado, Saitama 350- 0295, Japan