Hypersurfaces with constant scalar curvature in a hyperbolic space form^{∗}

Liu Ximin

Abstract

Let M^{n} be a complete hypersurface with constant normalized scalar curvature R in a hyperbolic space form H_{1}^{n+1}. We prove that if R̄ = R+1 ≥ 0 and the norm square |h|^{2} of the second fundamental form of M^{n} satisfies nR̄ ≤ sup |h|^{2} ≤ \frac{n}{(n−2)(nR̄−2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n], then either sup |h|^{2} = nR̄ and M^{n} is a totally umbilical hypersurface; or sup |h|^{2} = \frac{n}{(n−2)(nR̄−2)}[n(n−1)R̄^{2} −4(n−1)R̄+n], and M^{n} is isometric to S^{n−1}(r)× H^{1}(−1/(r^{2} + 1)), for some r > 0. Key words and phrases: hypersurface, hyperbolic space form, scalar curvature 1991 Mathmatics Subject Classification: 53C42, 53A10.

1 Introduction

Let R^{n+1}(c) be an (n+1)-dimensional Riemannian manifold with constant sectional curvature c. We also call it a space form. When c > 0, R^{n+1}(c) = S^{n+1}(c) (i.e. (n + 1)-dimensioanl sphere space); when c = 0, R^{n+1}(c) = R^{n+1} (i.e. (n + 1)- dimensioanl Euclidean space); when c < 0, R^{n+1}(c) = H^{n+1}(c) (i.e. (n + 1)- dimensioanl hyperbolic space). We simply denote H^{n+1}(−1) by H^{n+1}. Let M^{n} be an n-dimensional hypersurface in R^{n+1}(c), and e_{1}, …, e_{n} a local orthonormal frame field on M^{n}, ω_{1}, …, ω_{n} its dual coframe field. Then the second fundamental form of M^{n} is

h = ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j}.  (1)

Further, near any given point p ∈ M^{n}, we can choose a local frame field e_{1}, …, e_{n} so that at p, ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j} = ∑_{i} k_{i}ω_{i} ⊗ ω_{j}, then the Gauss equation says

R_{ijij} = c + k_{i}k_{j}, i ≠ j.  (2)

∗This work is supported in part by the National Natural Science Foundation of China

n(n − 1)(R − c) = n^{2}H^{2} − |h|^{2},  (3)

where R is the normalized scalar curvature, H = \frac{1}{n} ∑_{i} k_{i} the mean curvature and |h|^{2} = ∑_{i} k_{i}^{2} the norm square of the second fundamental form of M^{n}.

As it is well known, there are many rigidity results for minimal hypersurfaces or hypersurfaces with constant mean curvature H in R^{n+1}(c) (c ≥ 0) by use of J. Simons’ method, for example, see [1], [4], [5], [8], [12] etc., but less of that were obtained for hypersurfaces immersed into a hyperbolic space from. Walter [13] gave a classification for non-negatively curved compact hypersurfaces in a space form under the assumption that the rth mean curvature is constant. Morvan-Wu [7], Wu [14] also proved some rigidity theorems for complete hypersurfaces M^{n} in a hyperbolic space form H^{n+1}(c) under the assumption that the mean curvature is constant and the Ricci curvature is non-negative. Moreover, they proved that M^{n} is a geodesic distance sphere in H^{n+1}(c) provided that it is compact.

On the other hand, Cheng-Yau [3] introduced a new self-adjoint differential operator □ to study the hypersurfaces with constant scalar curvature. Later, Li [6] obtained interesting rigidity results for compact hypersurfaces with constant scalar curvature in space-forms using the Cheng-Yau’s self-adjoint operator □.

In the present paper, we would like to use Cheng-Yau’s self-adjoint operator □ to study the complete hypersurfaces in a hyperbolic space form with constant scalar curvature, and prove the following theorem:

Theorem. Let M^{n} be an n-dimensional (n ≥ 3) complete hypersurface with constant normalized scalar curvature R in H^{n+1}. If

(1) R̄ = R + 1 ≥ 0,

(2) the norm square |h|^{2} of the second fundamental form of M^{n} satisfies

nR̄ ≤ sup |h|^{2} ≤ \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n],

then either

sup |h|^{2} = nR̄

and M^{n} is a totally umbilical hypersurface; or

sup |h|^{2} = \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n],

and M^{n} is isometric to S^{n−1}(r) × H^{1}(−1/(r^{2} + 1)), for some r > 0.

2 Preliminaries

Let M^{n} be an n-dimensional hypersurface in H^{n+1}. We choose a local orthonormal frame e_{1}, …, e_{n+1} in H^{n+1} such that at each point of M^{n}, e_{1}, …, e_{n} span the tangent space of M^{n} and form an orthonormal frame there. Let ω_{1}, …, ω_{n+1} be its dual coframe. In this paper, we use the following convention on the range of indices:

1 ≤ A, B, C, … ≤ n + 1; 1 ≤ i, j, k, … ≤ n.

Then the structure equations of H^{n+1} are given by

dω_{A} = ∑_{B} ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0,  (4)

dω_{AB} = ∑_{C} ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑_{C,D} K_{ABCD}ω_{C} ∧ ω_{D},  (5)
K_{ABCD} = −(δ_{AC}δ_{BD} − δ_{AD}δ_{BC}).  (6)

Restrict these form to M^{n}, we have

ω_{n+1} = 0.  (7)

From Cartan’s lemma we can write

ω_{n+1i} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.  (8)

From these formulas, we obtain the structure equations of M^{n}:

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0,  (9)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},  (10)
R_{ijkl} = −(δ_{ik}δ_{jl} − δ_{il}δ_{jk}) + (h_{ik}h_{jl} − h_{il}h_{jk}),  (11)

where R_{ijkl} are the components of the curvature tensor of M^{n} and

h = ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j}  (12)

is the second fundamental form of M^{n}. We also have

R_{ij} = −(n − 1)δ_{ij} + nHh_{ij} − ∑_{k} h_{ik}h_{kj},  (13)

n(n − 1)(R + 1) = n^{2}H^{2} − |h|^{2},  (14)

where R is the normalized scalar curvature, and H the mean curvature. Define the first and the second covariant derivatives of h_{ij}, say h_{ijk} and h_{ijkl} by

∑_{k} h_{ijk}ω_{k} = dh_{ij} + ∑_{k} h_{kj}ω_{ki} + ∑_{k} h_{ik}ω_{kj},  (15)

∑_{l} h_{ijkl}ω_{l} = dh_{ijk} + ∑_{m} h_{mjk}ω_{mi} + ∑_{m} h_{imk}ω_{mj} + ∑_{m} h_{ijm}ω_{mk}.  (16)

Then we have the Codazzi equation

h_{ijk} = h_{ikj},  (17)

and the Ricci’s identity

h_{ijkl} − h_{ijlk} = ∑_{m} h_{mj}R_{mikl} + ∑_{m} h_{im}R_{mjkl}.  (18)

For a C^{2}-function f defined on M^{n}, we defined its gradient and Hessian (f_{ij}) by the following formulas

df = ∑_{i} f_{i}ω_{i}, ∑_{j} f_{ij}ω_{j} = df_{i} + ∑_{j} f_{j}ω_{ji}.  (19)

The Laplacian of f is defined by Δf = ∑_{i} f_{ii}.

Let ϕ = ∑_{ij} ϕ_{ij}ω_{i} ⊗ ω_{j} be a symmetric tensor defined on M^{n}, where

ϕ_{ij} = nHδ_{ij} − h_{ij}.  (20)

Following Cheng-Yau [3], we introduce an opertator □ associated to ϕ acting on any C^{2}-function f by

□f = ∑_{i,j} ϕ_{ij}f_{ij} = ∑_{i,j}(nHδ_{ij} − h_{ij})f_{ij}.  (21)

Since ϕ_{ij} is divergence-free, it follows [3] that the operator □ is self-adjoint relative to the L^{2} inner product of M^{n}, i.e.

∫_{M^{n}} f□g = ∫_{M^{n}} g□f.  (22)

We can choose a local frame field e_{1}, ….e_{n} at any point p ∈ M^{n}, such that h_{ij} = k_{i}δ_{ij} at p, by use of (21) and (14), we have

□(nH) = nHΔ(nH) − ∑_{i} k_{i}(nH)_{ii}
= \frac{1}{2}Δ(nH)^{2} − ∑_{i}(nH)_{i}^{2} − ∑_{i} k_{i}(nH)_{ii}
= \frac{1}{2}n(n − 1)ΔR + \frac{1}{2}Δ|h|^{2} − n^{2}|∇H|^{2} − ∑_{i} k_{i}(nH)_{ii}.  (23)

On the other hand, through a standard calculation by use of (17) and (18), we get

\frac{1}{2}Δ|h|^{2} = ∑_{i,j,k} h_{ijk}^{2} + ∑_{i} k_{i}(nH)_{ii} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}.  (24)

Putting (24) into (23), we have

□(nH) = \frac{1}{2}n(n − 1)ΔR + |∇h|^{2} − n^{2}|∇H|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}. (25)

From (11), we have R_{ijij} = −1 + k_{i}k_{j}, i ≠ j, and by putting this into (25), we obtain

□(nH) = \frac{1}{2}n(n − 1)ΔR + |∇h|^{2} − n^{2}|∇H|^{2} − n|h|^{2} + n^{2}H^{2} − |h|^{4} + nH ∑_{i} k_{i}^{3}. (26)

Let μ_{i} = k_{i} − H and |Z|^{2} = ∑_{i} μ_{i}^{2}, we have

∑_{i} μ_{i} = 0, |Z|^{2} = |h|^{2} − nH^{2},  (27)

∑_{i} k_{i}^{3} = ∑_{i} μ_{i}^{3} + 3H|Z|^{2} + nH^{3},  (28)

From (26)-(28), we get

□(nH) = \frac{1}{2}n(n−1)ΔR+|∇h|^{2}−n^{2}|∇H|^{2}+|Z|^{2}(−n+nH^{2}−|Z|^{2})+nH ∑_{i} μ_{i}^{3}. (29)

We need the following algebraic lemma due to M. Okumura [9] (see also [1]). Lemma 2.1. Let μ_{i}, i = 1, …, n, be real numbers such that ∑_{i} μ_{i} = 0 and

∑_{i} μ_{i}^{2} = β^{2}, where β = constant ≥ 0. then

−\frac{n − 2}{\sqrt{n(n − 1)}}β^{3} ≤ ∑_{i} μ_{i}^{3} ≤ \frac{n − 2}{\sqrt{n(n − 1)}}β^{3},  (30)

and the equality holds in (30) if and only if at least (n − 1) of the μ_{i} are equal. By use of Lemma 2.1, we have

□(nH) ≥ \frac{1}{2}n(n − 1)ΔR + |∇h|^{2} − n^{2}|∇H|^{2} +

(|h|^{2} − nH^{2})(−n + 2nH^{2} − |h|^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{|h|^{2} − nH^{2}}).  (31)

3 Umbilical hypersurface in a hyperbolic space form

In this section, we consider some special hypersurfaces in a hyperbolic space form which we will need in our latter discussions.

First we want to give a description of the real hyperbolic space-form H^{n+1}(c) of constant curvature c (< 0). For any two vectors x and y in R^{n+2}, we set

g(x, y) = ∑_{i=1}^{n+1} x_{i}y_{i} − x_{n+2}y_{n+2}.

(R^{n+2}, g) is the so-called Minkowski space-time. Denote ρ = \sqrt{−1/c}. We define

H^{n+1}(c) = {x ∈ R^{n+2} | x_{n+2} > 0, g(x, x) = −ρ^{2}}.

Then H^{n+1}(c) is a connected simply-connected hypersurface of R^{n+2}. It is not hard to check that the restriction of g to the tangent space of H^{n+1}(c) yields a complete Riemannian metric of constant curvature c. Here we obtain a model of a real hyperbolic space form.

We are interested in those complete hypersurfaces with at most two distinct constant principal curvatures in H^{n+1}(c). this kind of hypersurfaces was described by Lawson [5] and completely classified by Ryan [11].

Lemma 3.1 [11]. Let M^{n} be a complete hypersurface in H^{n+1}(c). Suppose that, under a suitable choice of a local orthonormal tangent frame field of T M^{n}, the shape operator over T M^{n} is expressed as a matrix A. If M^{n} has at most two distinct constant principal curvatures, then it is congruent to one of the following:

(1) M_{1} = {x ∈ H^{n+1}(c) | x_{1} = 0}. In this case, A = 0, and M_{1} is totally geodesic. Hence M_{1} is isometric to H^{n}(c);

(2) M_{2} = \{x ∈ H^{n+1}(c) | x_{1} = r > 0\}. In this case, A = \frac{1/ρ^{2}}{\sqrt{1/ρ^{2}+1/r^{2}}} I_{n}, where I_{n}

denotes the identity matrix of degree n, and M_{2} is isometric to H^{n}(−1/(r^{2} + ρ^{2}));

(3) M_{3} = \{x ∈ H^{n+1}(c) | x_{n+2} = x_{n+1} + ρ\}. In this case, A = \frac{1}{ρ}I_{n}, and M_{3} is isometric to a Euclidean space E^{n};

(4) M_{4} = \{x ∈ H^{n+1}(c) | ∑_{i=1}^{n+1} x_{i}^{2} = r^{2} > 0\}. In this case, A = \sqrt{1/r^{2} + 1/ρ^{2}}I_{n}, and M_{4} is isometric to a round sphere S^{n}(r) of radius r;

(5) M_{5} = {x ∈ H^{n+1}(c) | ∑_{i=1}^{k+1} x_{i}^{2} = r^{2} > 0, ∑_{j=k+2}^{n+1} x_{j}^{2} − x_{n+2}^{2} = −ρ^{2} − r^{2}}. In this case, A = λI_{k} ⊕ νI_{n−k}, where λ = \sqrt{1/ρ^{2} + 1/r^{2}}, and ν = \frac{1/ρ^{2}}{\sqrt{1/r^{2}+1ρ^{2}}} , M_{5} is

isometric to S^{k}(r) × H^{n−k}(−1/(r^{2} + ρ^{2})).

Remark 3.1. M_{1}, …, M_{5} are often called the standard examples of complete hypersurfaces in H^{n+1}(c) with at most two distinct constant principal curvatures. It is obvious that M_{1}, …, M_{4} are totally umbilical. In the sence of Chen [2], they are called the hyperspheres of H^{n+1}(c). M_{3} is called the horosphere and M_{4} the geodesic distance sphere of H^{n+1}(c).

Remark 3.2. Ryan [11] stated that the shape operator of M_{2} is A = \sqrt{1/r^{2} − 1/ρ^{2}}I_{n}, and M_{2} is isometric to H^{n}(−1/r^{2}), where r ≤ ρ. This is incorrect and we have corrected it here.

4 The proof of Theorem

The following lemma essentially due to Cheng-Yau [3].

Lemma 4.1. Let M^{n} be an n-dimensional hypersurface in H^{n+1}. Suppose that the normalized scalar curvature R = constant and R ≥ −1. Then |∇h|^{2} ≥ n^{2}|∇H|^{2}.

Proof. From (14),

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

Taking the covariant derivative of the above expression, and using the fact R = constant, we get

n^{2}HH_{k} = ∑_{i,j} h_{ij}h_{ijk}.

By Cauchy-Schwarz inequality, we have

∑_{k} n^{4}H^{2}(H_{k})^{2} = ∑_{k}(∑_{i,j} h_{ij}h_{ijk})^{2} ≤ (∑_{i,j} h_{ij}^{2}) ∑_{i,j,k} h_{ijk}^{2},

that is

n^{4}H^{2}∥∇H∥^{2} ≤ |h|^{2}|∇h|^{2}.

On the other hand, from R + 1 ≥ 0, we have n^{2}H^{2} − |h|^{2} ≥ 0. Thus

H^{2}|∇h|^{2} ≥ n^{2}H^{2}|∇H|^{2}

and Lemma 4.1 follows.

From the assumption of the Theorem that R is constant and R̄ = R + 1 ≥ 0 and Lemma 4.1 we have

□(nH) ≥ (|h|^{2} − nH^{2})(−n + 2nH^{2} − |h|^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{|h|^{2} − nH^{2}}). (32)

By Gauss equation (14) we know that

|Z|^{2} = |h|^{2} − nH^{2} = \frac{n − 1}{n}(|h|^{2} − nR̄).  (33)

From (32) and (33) we have

□(nH) ≥ \frac{n − 1}{n}(|h|^{2} − nR̄)ϕ_{H}(|h|),  (34)

where

ϕ_{H}(|h|) = −n + 2nH^{2} − |h|^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{|h|^{2} − nH^{2}}.

By (33) we can write ϕ_{H}(|h|) as

ϕR̄(|h|) = −n+2(n−1)R̄− \frac{n − 2}{n}|h|^{2}− \frac{n − 2}{n}\sqrt{(n(n − 1)R̄ + |h|^{2})(|h|^{2} − nR̄)}. (35)

Therefore (34) becomes

□(nH) ≥ \frac{n − 1}{n}(|h|^{2} − nR̄)ϕR̄(|h|),  (36)

It is a direct check that our assumption

sup |h|^{2} ≤ \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n]  (37)

is equivalent to

(−n + 2(n − 1)R̄ − \frac{n − 2}{n}sup|h|^{2})^{2} ≥ \frac{(n − 2)^{2}}{n^{2}} (n(n − 1)R̄ + sup|h|^{2})(sup|h|^{2} − nR̄).

(38)

But it is clear from (37) that (38) is equivalent to

−n + 2(n − 1)R̄ − \frac{n − 2}{n}sup|h|^{2} ≥ \frac{n − 2}{n}\sqrt{(n(n − 1)R̄ + sup|h|^{2})(sup|h|^{2} − nR̄)}.

(39)

So under the hyperthesis that

sup |h|^{2} ≤ \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n],

we have

ϕR̄(\sqrt{sup|h|^{2}}) ≥ 0.  (40)

On the other hand,

□(nH) = ∑_{i,j}(nHδ_{ij} − nh_{ij})(nH)_{ij} = ∑_{i}(nH − nh_{ii})(nH)_{ii}
= n ∑_{i} H(nH)_{ii} − n ∑_{i} k_{i}(nH)_{ii} ≤ (|H|_{max} − C)Δ(nH), (41)

where |H|_{max} is the maximum of the mean curvature H and C = min k_{i} is the minimum of the principal curvatures of M^{n}.

Now we need the following maximum principle at infinity for complete manifolds due to Omori [10] and Yau [15]:

Lemma 4.2. Let M^{n} be an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and f : M^{n} → R a smooth function bounded from below. Then for each ε > 0 there exists a point p_{ε} ∈ M^{n} such that

(i) |∇f|(p_{ε}) < ε, (ii) Δf(p_{ε}) > −ε, (iii) inf f ≤ f(p_{ε}) ≤ inf f + ε. From the hypothesis of the Theorem and Gauss equation, we know that the Ricci

curvature is bounded below. So we may apply Lemma 4.2 to the following smooth function f on M^{n} defined by

f = \frac{1}{\sqrt{1 + (nH)^{2}}} .

It is immediate to check that

|∇f|^{2} = \frac{1}{4} \frac{|∇(nH)^{2}|^{2}}{(1 + (nH)^{2})^{3}}  (42)

and that

Δf = −\frac{1}{2} \frac{Δ(nH)^{2}}{(1 + (nH)^{2})^{3/2}} + \frac{3}{4} \frac{|∇(nH)^{2}|^{2}}{(1 + (nH)^{2})^{5/2}} .  (43)

By Lemma 4.2 we can find a sequence of points p_{k}, k ∈ N in M^{n}, such that

lim_{k→∞}f(p_{k}) = inf f, Δf(p_{k}) > −\frac{1}{k}, |∇f|^{2}(p_{k}) < \frac{1}{k^{2}} .  (44)

Using (44) in the equations (42) and (43) and the fact that

lim_{k→∞}(nH)(p_{k}) = sup_{p∈M^{n}}(nH)(p),  (45)

we get

−\frac{1}{k} ≤ −\frac{1}{2} \frac{Δ(nH)^{2}}{(1 + (nH)^{2})^{3/2}} (p_{k}) + \frac{3}{k^{2}} (1 + (nH)^{2}(p_{k}))^{1/2}.  (46)

Hence we obtain

\frac{Δ(nH)^{2}}{(1 + (nH)^{2})^{2}} (p_{k}) < \frac{2}{k}(\frac{1}{\sqrt{1 + (nH)^{2}(p_{k})}} + \frac{3}{k}).  (47)

On the other hand, by (36) and (41), we have

\frac{n − 1}{n}(|h|^{2} − nR̄)ϕR̄(|h|) ≤ □(nH) ≤ n(|H|_{max} − C)Δ(nH).  (48)

At points p_{k} of the sequence given in (44), this becomes

\frac{n − 1}{n}(|h|^{2}(p_{k}) − nR̄)ϕR̄(|h|(p_{k})) ≤ □(nH(p_{k}))

≤ n(|H|_{max} − C)Δ(nH)(p_{k}).  (49)

Let k → ∞ and use (47) we have that the right hand side of (49) goes to zero, so we have either \frac{n−1}{n}(sup|h|^{2} − nR̄) = 0, i.e. sup |h|^{2} = nR̄ or ϕR̄(\sqrt{sup|h|^{2}}) = 0.

If sup|h|^{2} = nR̄, by (33) |Z|^{2} = \frac{n−1}{n}(|h|^{2} − nR̄) we have sup|Z|^{2} = \frac{n−1}{n}(sup|h|^{2} − nR̄) = 0, then |Z|^{2} = 0 and M^{n} is totally umbilical.

If ϕR̄(\sqrt{sup|h|^{2}}) = 0, it is easy to prove that

sup H^{2} = \frac{1}{n^{2}} [(n − 1)^{2} \frac{nR̄ + 2}{n − 2} − 2(n − 1) + \frac{n − 2}{nR̄ + 2}],

then equalities hold in (30) and Lemma 4.1, we follow that k_{i} = constant for all i and (n − 1) of the k_{i} are equal. After renumberation if necessary, we can assume

k_{1} = k_{2} = ⋯ = k_{n−1} k_{1} ≠ k_{n}.

Therefore, from Lemma 3.1, we know that M^{n} is a hypersurface in H^{n+1} with two distinct principal curvatures, and M^{n} is isometric to S^{n−1}(r) × H^{1}(−1/(r^{2} + 1)), for some r > 0. This completes the proof of Theorem.

When M^{n} is compact, we can prove

Corollary 1. Let M^{n} be an n-dimensional (n ≥ 3) compact hypersurface with constant normalized scalar curvature R in H^{n+1}. If

(1) R̄ = R + 1 ≥ 0,

(2) the norm square |h|^{2} of the second fundamental form of M^{n} satisfies

nR̄ ≤ |h|^{2} ≤ \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n],  (50)

then M^{n} is a totally umbilical hypersurface.

Proof. From (36) we have

□(nH) ≥ \frac{n − 1}{n}(|h|^{2} − nR̄)[−n + 2(n − 1)R̄ − \frac{n − 2}{n}|h|^{2}

−\frac{n − 2}{n}\sqrt{(n(n − 1)R̄ + |h|^{2})(|h|^{2} − nR̄)}],  (51)

It is a direct check that our assumption condition (50) is equivalent to

(−n + 2(n − 1)R̄ − \frac{n − 2}{n}|h|^{2})^{2} ≥ \frac{(n − 2)^{2}}{n^{2}} (n(n − 1)R̄ + |h|^{2})(|h|^{2} − nR̄). (52)

But it is clear from (50) that (52) is equivalent to

−n + 2(n − 1)R̄ − \frac{n − 2}{n}|h|^{2} ≥ \frac{n − 2}{n}\sqrt{(n(n − 1)R̄ + |h|^{2})(|h|^{2} − nR̄)},  (53)

therefore the right hand side of (51) is non-negative. Because M^{n} is compact and the operator □ is self-adjoint, we have ∫_{M^{n}} □(nH)dv = 0. Thus either

|h|^{2} = nR̄,  (54)

that is, |h|^{2} = nH^{2}, M^{n} is a totally umbilical hypersurface; or

|h|^{2} = \frac{n}{(n − 2)(nR̄ − 2)}[n(n − 1)R̄^{2} − 4(n − 1)R̄ + n].  (55)

In the latter case, equalities hold in (30) and Lemma 4.1, and it follows that M^{n} has at most two distinct constant principal curvatures. We conclude that M^{n} is totally umbilical from the compactness of M^{n}. This completes the proof of Corollary 1.

Corollary 2. Let M^{n} be an n-dimensional compact hypersurface with constant normalized scalar curvature R and R + 1 ≥ 0 in H^{n+1}. If M has non-negative sectional curvature, then M is a totally umbilical hypersurface.

Proof. Because M^{n} is compact and the operator □ is self-adjoint, form (25), we have

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}] = 0.  (56)

If M^{n} has constant normalized scalar curvature R and R ≥ −1, from Lemma 4.1, we have |∇h|^{2} ≥ n^{2}|∇H|^{2}. So if M has non-negative sectional curvature, form (56) we have |∇h|^{2} = n^{2}|∇H|^{2} and R_{ijij} = 0, when k_{i} ≠ k_{j} on M^{n}. Since R_{ijij} = −1 + k_{i}k_{j}, then either M^{n} is totally umbilical, or M^{n} has two different principal curvatures, in the latter case, M^{n} is still totally umbilical from the compactness of M^{n}. This completes the proof of Corollary 2.

Acknowledgements

This paper was written during the author’s stay at the Max-Planck-Institut für Mathematik in Bonn. The author would like to exprss his thanks to the institute for its hospitality and support.

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Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail: xmliu@dlut.edu.cn xliu@mpim-bonn.mpg.de