DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 51–60

SOME APPLICATIONS OF THE EXTRINSIC DISTANCE

TO IMMERSED AND EMBEDDED SUBMANIFOLDS

Vicente Miquel

§ 0. Introduction

We survey here some recent results about embedded or immersed submanifolds of complex or real space forms obtained by F.J. Carreras, F. Giménez and the author (cfr [CGM1,2], [GM2] and [Mi]). The common thread of these results is in their proofs: A basic tool used in all of them is a computation of some laplacians (in the submanifold) of some well chosen functions of the extrinsic distance to a point of the ambient space.

§ 1. Notation

Let ψ : M ⟶ \overline{M} be an isometric immersion of a compact Riemannian manifold M, of dimension m, into an n-dimensional Riemannian manifold \overline{M}.

Let o ∈ \overline{M} and let r : \overline{M} ⟶ IR be the function defined by

r(p) = d(o, p) for every p ∈ \overline{M}, d = distance in \overline{M}.

1991 Mathematics Subject Classification. 53C42, 58G25.

Key words and phrases. Isometric immersion, bounded mean curvature, constant mean curvature, geodesic ball, complex space form, estimate of first Dirichlet eigenvalue, immersion with bounded Ricci and scalar curvatures, Hopf hypersurface.

This paper is in final form and no version of it will be submitted for publication elsewhere.

*Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 “G.A.D.G.E.T. II”

We shall also denote by r the composition r ≡ r ∘ ψ : M ⟶ IR. It is called the extrinsic distance to o in M.

∂_{r} and ∂_{r}^{⊤} will denote the gradients of r in \overline{M} and in M respectively. Let us observe that ∂_{r}^{⊤}(m) is the image by ψ_{∗}^{−1} of the component of ∂_{r}(ψ(m)) tangent to ψ_{∗}(T_{m}M), for every m ∈ M.

If f : IR ⟶ IR is any function, f(r) will denote the composition f∘r : M ⟶ IR. Then the notation f(r) reminds us that f(r) is an extrinsic radial function on M, i.e., that it depends only on the extrinsic distance to o.

By cut(o) we shall denote the set of cut points of o in \overline{M}.

If R denotes the curvature tensor of \overline{M}, R(r) will be the (1, 1)-tensor on \overline{M} − cut(o) defined by

R(r)A = R(∂_{r}, A)∂_{r} for every vector A on \overline{M} − cut(o).

It is called radial curvature operator ([GW]). Observe that R(r)∂_{r} = 0.

By R_{λ}(r) we shall denote the radial curvature operator of a (real or complex) space form IK^{n}(λ).

We denote by S(r) the (1, 1)-tensor on \overline{M} defined by

S(r)(A) = −\overline{∇}_{A}∂_{r} for every A tangent to \overline{M},

Let us observe that S(r)∂_{r} = 0, and S(r) restricted to the vectors tangent to the geodesic sphere ∂B_{r}(o) of \overline{M} of centre o and radius r is the Weingarten map of this sphere, and tr S(r) is (n − 1) times the mean curvature of this sphere.

When \overline{M} = IK^{n}(λ), then S(r) will be denoted by S_{λ}(r).

We shall define the “transplanted” operator \overline{R}_{λ}(r) of R_{λ}(r) in \overline{M} in the following way: let ϕ : T_{p′}IK^{n}(λ) ⟶ T_{p}\overline{M} be an (holomorphic if \overline{M} and IK^{n}(λ) are Kähler) isometry, then

\overline{R}_{λ}(r)A(q) = \overline{τ}_{t} ∘ ϕ ∘ τ_{t}^{−1} ∘ R_{λ}(r) ∘ τ_{t} ∘ ϕ^{−1}(\overline{τ}_{t}^{−1}A), t = distance(o, q) in \overline{M},

where \overline{τ}_{t} is the parallel transport along the minimizing geodesic γ of \overline{M} from p to q and τ_{t} is the parallel transport along the minimizing geodesic exp_{p′}sϕ^{−1}(γ′(0)) of IK^{n}(λ) from p′ to exp_{p′}tϕ^{−1}(γ′(0)).

The “transplanted” operator \overline{S}_{λ}(r) is defined analogously.

We shall denote also by R(r), \overline{R}_{λ}(r), S(r) and \overline{S}_{λ}(r) the quadratic forms associated to these operators and the metric of \overline{M}.

Given two quadratic forms, A and B, we shall say that A ≥ B if A − B is positive semidefinite.

Given a real number λ, we shall denote by s_{λ}(t) the solution of

s′′ +λ s = 0,  s(0) = 0,  s′(0) = 1,

and let

c_{λ} = s′_{λ}, ta_{λ} = λ\begin{matrix}s & λ \\ c & λ \end{matrix} , co_{λ} = \begin{matrix}c & λ \\ s & λ \end{matrix} .

§ 2. Basic formula

Let \overline{∇} and \overline{Δ} denote the covariant derivative and the Laplacian respectively on \overline{M}. Let us denote by ∇ and Δ the corresponding operators on M.

From the definition of S(r) it follows

(2.1)  \overline{∇}^{2}r(A, B) = −⟨S(r)A, B⟩ for every A, B on \overline{M} − cut(o).

In the local computations below, we shall denote by the same letter a local vector field X on M and its image ψ_{∗}X.

If α is the second fundamental form of the immersion ψ : M ⟶ \overline{M}, then an easy computation shows that

(2.2)  ∇^{2}r(X, Y) = \overline{∇}^{2}r(X, Y) + ⟨α(X, Y), ∂_{r}⟩

for every X, Y tangent to M.

If {e_{i}}_{i=1}^{m} is a local orthonormal frame of vector fields tangent to M and H denotes the mean curvature vector of the immersion ψ : M ⟶ \overline{M}, from (2.1) and (2.2), it follows that

(2.3)  Δr = ∑_{i=1}^{m}⟨S(r)(e_{i}), e_{i}⟩ − m⟨H, ∂_{r}⟩.

If f : IR ⟶ IR is any C^{2} function, it follows from (2.3) that

(2.4)  Δf(r) = −f′′(r)|∂_{r}^{⊤}|^{2} + f′{∑_{i=1}^{m}⟨S(r)(e_{i}), e_{i}⟩ − m⟨H, ∂_{r}⟩}.

In the next sections we shall give applications of the formula (2.4) to some problems. Our emphasis will be more on computations than on arguments. For more details, see the references.

§ 3. Immersibility into a geodesic ball

Riemannian case.

Here IK^{n}(λ) will denote the simply connected real space form of dimension n and constant sectional curvature λ, and B_{ρ}^{λ,n} will be a geodesic ball of radius ρ in this space. One has

R_{λ}(r)A = λA, S_{λ}(r)A = − co_{λ}(r)A for every A orthogonal to ∂_{r}.

Moreover 3.1. Lemma. If R(r) ≤ \overline{R}_{λ}(r) in B_{t}(o) for some t ≤ c(o) if λ ≤ 0, and t ≤ min\{c(o), \frac{π}{\sqrt{λ}}\} if λ > 0, then

(3.1.1)  S(r) ≤ \overline{S}_{λ}(r) for every q ∈ B_{t}(o).

If the equality in (3.1.1) holds in ∂B_{ρ}(o), for some ρ < t, then there exists an isometry between B_{ρ}(o) and B_{ρ}^{λ,n} which takes ∂B_{ρ}(p) onto ∂B_{ρ}^{λ,n}.

Then, under the general hypothesis R(r) ≤ \overline{R}_{λ}(r), Lemma 3.1 says that

∑_{i=1}^{m}⟨S(r)e_{i}, e_{i}⟩ ≤ ∑_{i=1}^{m}⟨\overline{S}_{λ}(r)(e_{i} − ⟨e_{i}, ∂_{r}⟩∂_{r}), e_{i}⟩ = −m co_{λ}(r) + co_{λ}(r)|∂_{r}^{⊤}|^{2}.

By substitution of this inequality in (2.4), we get

Δf(r) ≤ (−f′′(r) + f′(r) co_{λ}(r)) |∂_{r}^{⊤}|^{2} − mf′(r) (co_{λ}(r) + ⟨H, ∂_{r}⟩).

Now we have to choose a function f which is the best suited for each situation. In this case, it is that which makes the coefficient of |∂_{r}^{⊤}|^{2} vanish, for which f must satisfy f′(r) = s_{λ}(r). With this choice we get

Δf(r) ≤ −m(c_{λ}(r) + ⟨s_{λ}(r) H, ∂_{r}⟩).

If |H| ≤ co_{λ}(ρ) (with ρ ≤ \frac{π}{\sqrt{λ}} if λ > 0) and ψ(M) ⊂ B_{ρ}(o),

then

Δf(r) ≤ −m c_{λ}(r) + m s_{λ} /r)|H|

≤ −m c_{λ}(r) + m\frac{s_{λ}(r)}{s_{λ}(ρ)} c_{λ}(ρ) ≤ −m c_{λ}(r) + m\frac{s_{λ}(r)}{s_{λ}(r)} c_{λ}(r) = 0.

From Hopf principle, this inequality implies that Δf(r) = 0. Then the inequalities in the above formula must all be equalities and we have:

r|_{M} = ρ, S(r) = s_{λ}(r)Id on the vectors tangent to M, ∂_{r} = N on M, |H| = co_{λ}(ρ) and −⟨H, ∂_{r}⟩ = co_{λ}(r). Then H = − co_{λ}(ρ)∂_{r}, and ψ : M ⟶ ∂B_{ρ}(o) is minimal.

If m = n − 1, ψ(M) = ∂B_{ρ}(o), ψ is an embedding and B_{ρ}(o) is isometric to B_{ρ}^{λ,n}.

This is essentially the union of Markvorsen Theorem ([Ma]) and a compact version of a Theorem by Jorge and Xavier ([JX]).

Kähler case.

Here IK^{n}(λ) will denote the complex space form of real dimension 2n and constant holomorphic sectional curvature 4λ, and B_{ρ}^{λ,n} will be a geodesic ball of radius ρ in this space. Now

R_{λ}(r)J∂_{r} = 4λJ∂_{r}, R_{λ}(r)X = λX,
S_{λ}(r)J∂_{r} = − co_{4λ}(r)J∂_{r} and S_{λ}(r)X = −co_{λ}(r)X

for every X orthogonal to ∂_{r} and J∂_{r}.

We have 3.2. Lemma. If R(r) ≤ \overline{R}_{λ}(r) in B_{t}(o) for some t ≤ c(o) if λ ≤ 0, and t ≤ min\{c(o), \frac{π}{2\sqrt{λ}}\} if λ > 0, then

(2.1.1)  S(r) ≤ \overline{S}_{λ}(r) for every q ∈ B_{t}(o).

Moreover, if the equality in (2.1.1) holds in ∂B_{ρ}(o), for some ρ < t, then there exists a holomorphic isometry between B_{ρ}(o) and B_{ρ}^{λ,n} which takes ∂B_{ρ}(o) onto ∂B_{ρ}^{λ,n}.

Computing as above, we get

Δf(r) ≤(−f′′(r) + f′(r) co_{λ}(r)) |∂_{r}^{⊤}|^{2}
+ f′(r) ta_{λ}(r) |(J∂_{r})^{⊤}|^{2} − mf′(r) (co_{λ}(r) + ⟨H, ∂_{r}⟩).

The result analogous to that in the riemannian case will be

3.3. Theorem ([CGM2]). Let us suppose that R(r) ≤ \overline{R}_{λ}(r) in B_{ρ}(p), ψ(M) ⊂ B_{ρ}(p), and

m|H| ≤ (m − 1) co_{λ}(ρ) + co_{4λ}(ρ).

When λ ≤ 0, let us assume, also, that JN M ⊂ T M. Then ψ(M) ⊂ ∂B_{ρ}(p), m H = −{(m − 1) co_{λ}(ρ) + co_{4λ}(ρ)}∂_{r}, and J∂_{r} is tangent to ψ(M) at each point.

If m = 2n − 1, then B_{ρ}(p) is isometric to a geodesic ball of radius ρ in IK^{n}(λ), M is isometric to ∂B_{ρ}^{λ,n} and ψ is an embedding.

Now, the proof requires to distinguish between the cases λ > 0 and λ < 0. The reason of that is the second term in the above expression of Δf(r), which does not appear in he Riemannian case.

For λ > 0 we are, as above, interested in the vanishing of the term containing |∂_{r}^{⊤}|^{2} and in bounding |(J∂_{r})^{⊤}| by 1. Arguments similar to those in the previous case give us the conclusion of Theorem 3.3.

For λ < 0 we add the condition JN M ⊂ T M (which is always satisfied if m = 2n − 1). Now it is convenient to work with |∂_{r}^{⊤}|^{2} and |(J∂_{r})^{⊤}|^{2} together. For it we need that f satisfies the equation

−f′′ + f′ co_{λ} = f′ ta_{λ},

and it is enough to take f with f′ = s_{4λ}. Now, computing, and bounding |Π_{N M⊕JN M}∂_{r}| by 1, one gets the same result that with λ > 0.

Let us denote by Π : ∂B_{ρ}^{λ,n} ⟶ I CP^{n−1}(1/ s_{λ}^{2}(ρ)) the Hopf fibration on the geodesic sphere ∂B_{ρ}^{λ,n}. Two interesting corollaries (which have no Riemannian analogs) are

3.4. Corollary ([CGM2]). Let M be of dimension 2n − 2. Let us suppose that \overline{M} = IK^{n}(λ), ψ(M) ⊂ B_{ρ}^{λ,n} and (2n − 2)|H| ≤ (2n − 3) co_{λ}(ρ) + co_{4λ}(ρ). When λ ≤ 0, we assume also that JN M ⊂ T M. Then there is a compact riemannian manifold G of dimension 2n − 3, a riemannian submersion π : M ⟶ G, and a minimal immersion φ : G ⟶ I CP^{n−1}(1/ s_{λ}^{2}(ρ)) such that Π ∘ ψ = φ ∘ π. Moreover, ψ(M) = Π^{−1}(φ(G)).

3.5. Corollary ([CGM2]). Let M and ψ be as in Corollary 3.3, but with n = 2. Then ψ : M ⟶ IK^{2}(λ) is an isometric immersion of a flat torus in IK^{2}(λ), such that ψ(M) is an embedded torus and also a tubular hypersurface of radius (π/4) s_{λ}(ρ) around an integral curve of J∂_{r} in ∂B_{ρ}^{λ,2}.

A natural question to be posed: Apart from the technical reason given by the formula for Δf(r), why are different (for m < 2n − 1)) the hypotheses when λ < 0 and when λ > 0?.

Let us observe that

m co_{λ}(ρ) < (m − 1) co_{λ}(ρ) + co_{4λ}(ρ) if λ < 0

and

m co_{λ}(ρ) > (m − 1) co_{λ}(ρ) + co_{4λ}(ρ) if λ > 0.

That is, the bound for m|H| is the lowest of the possible sums of principal curvatures of ∂B_{ρ}^{λ,n} if λ > 0, but not if λ < 0. This explains why, to get the same strong conclusion, we have been obliged to add conditions.

The lowest of the possible sums of principal curvatures of ∂B_{ρ}^{λ,n} for λ < 0 is m co_{λ}(ρ). If we take it as an upper bound of |H| and choose f as in the Riemannian case, we get

Δ(−\frac{1}{λ} c_{λ}) ≤ λ\begin{matrix}s & 2 λ \\ c & λ \end{matrix} |(J∂_{r})^{⊤}|^{2} − m c_{λ} −m⟨H, ∂_{r}⟩ s_{λ} ≤ −m c_{λ} +m |H| s_{λ} ≤ 0.

Applying again Hopf principle and using known facts about the Hopf fibration on ∂B_{ρ}^{λ,n} one gets

3.6. Theorem ([CGM2]). If ψ : M ⟶ I CH^{n}(λ), λ < 0, is an isometric immersion such that ψ(M) ⊂ B_{ρ}(p) and |H| ≤ co_{λ}(ρ), then 1 ≤ m ≤ n − 1, |H| = co_{λ}(ρ) and there exists a minimal totally real isometric immersion φ : M ⟶ I CP^{n−1}(1/ s_{λ}^{2}(ρ)) such that ψ is a lifting of φ by Π.

§ 4. Bounds for the first Dirichlet eigenvalue of an extrinsic ball of a complex submanifold.

In this section ψ : M ⟶ \overline{M} will be a Kähler embedding of a Kähler manifold M of real dimension 2q into a Kähler manifold \overline{M} of real dimension 2n satisfying R(r) ≤ R_{λ}(r). Then H = 0, |J∂_{r}| = |∂_{r}|, and the formula for the laplacian becomes

Δf(r) ≤ (−f′′(r) + f′(r)(co_{λ}(r) + λ\begin{matrix}s & λ \\ c & λ \end{matrix}))|∂_{r}^{⊤}|^{2} − 2qf′(r) co_{λ}(r).

We take f(r) positive and such that the coefficient of |∂_{r}^{⊤}| vanishes.

Now, we follow a technique of [CLY]. We consider the eigenfunction ϕ associated to the first eigenvalue μ_{1}(B_{ρ}^{λ,q}) of the Dirichlet eigenvalue problem on B_{ρ}^{λ,q}, which is a radial function. We define ϕ over M by ϕ(m) = ϕ(r(m)), and we change the variable r by f and consider ϕ as a function of f. Denoting by ^{.} and ′ the derivatives respect to f and r respectively, one gets

Δϕ = −ϕ^{. .}f′^{2}|∂_{r}^{⊤}|^{2} + ϕ^{.}Δf.

If ϕ^{. .} ≥ 0 and ϕ^{.} ≤ 0, then one concludes

Δϕ ≥ −ϕ^{. .}(f(r))f′^{2}(r) + 2qf′(r) co_{λ}(r)ϕ^{.} = μ_{1}(B_{ρ}^{λ,q})ϕ.

Now, by using a classical Lemma of Barta for lower bounds of the first eigenvalue one gets

4.1. Theorem ([CGM1]). Let us assume that R(t) ≤ R_{λ}(t). If cut(o) ∩ B_{ρ}(o) = (and ρ < \frac{π}{2\sqrt{λ}} if λ > 0), then

μ_{1}(B_{ρ}(o) ∩ M) ≥ λ_{1}(B_{ρ}^{λ,q}).  (4.1.1)

Moreover, if the equality holds in (4.1.1), then B_{ρ}(o) ∩ M is holomorphically isometric to a geodesic ball of radius ρ in IK^{q}(λ).

Two main facts used in the proof of the above Theorem are the inequalities ϕ^{.} ≤ 0 and ϕ^{. .} ≥ 0. The first is an obvious consequence of the properties of the functions ϕ and f. The study of the second inequality requires a more elaborated argument.

§ 5. Looking for a Kähler version of an Alexandrov Theorem.

A well known theorem by A. D. Alexandrov says that a connected compact embedded hypersurface of constant mean curvature in the Euclidean space or in the real hyperbolic space must be a geodesic sphere of the ambient space. It seems natural to conjecture a Kähler version. Using the notation of section 3, Kähler case, this conjecture will say that a connected compact embedded real hypersurface of constant mean curvature in IK^{n}(λ) (or in some geodesic ball if λ > 0) must be a geodesic sphere. We have proved a weaker result. To state it we need to introduce some notation:

If L is the Weingarten map of M associated to a unit normal vector field N, then the JN- normal curvature k_{JN} of M is defined by k_{JN} = ⟨L JN , JN⟩ and the JN-mean curvature of M is defined by H_{JN} = (1/(2n − 2))(tr L − k_{JN}).

We say that a real hypersurface M is a Hopf hypersurface if JN is an eigenvector of the Weingarten map L.

5.1. Theorem ([Mi]). Let M be a connected compact embedded Hopf hypersurface of IK^{n}(λ) with constant mean curvature and k_{JN} ≥ k_{λ}, where k_{λ} is the value of the JN-normal curvature of a geodesic sphere of IK^{n}(λ) having a JN-mean curvature equal to that of M. If λ > 0, we suppose also that it is contained in a geodesic ball of radius R < π/(2\sqrt{λ}). Then M is a geodesic sphere of IK^{n}(λ).

To prove Theorem 5.1 we have used a special Laplacian that we introduce now:

For a (1, 1)-tensor φ on M, the φ-divergence div_{φ} of a vector field X on M is defined as the function div_{φ} X given by

(div_{φ} X)(p) = − tr(Y ↦ φ(∇_{Y} X))(p) = − ∑_{i=1}^{2n−1} < φ(∇_{e_{i}}X), e_{i} > (p).

One gets a nice div_{φ}-theorem similar to the well known divergence theorem when div_{φ} = 0. In fact, let us denote by div the usual divergence operator in M, and by ω the volume form of M, then

5.2. Proposition. Given a vector field X on M,

−(div_{φ} X)ω = dι_{φ(X)}ω

if and only if (div φ)X = 0.

The φ-Laplacian Δ_{φ} is defined by

Δ_{φ}f = div_{φ} grad f

for every C^{2} function f on M.

Now, let us denote by {JN}^{⊥} the orthogonal complement in T M of the space < {JN} > generated by JN. Let η and ζ be, respectively, the (1, 1)-tensors on M defined by the projections from T M onto {JN}^{⊥} and < {JN} > respectively, that is

ζ(X) = ⟨X, JN⟩JN and η = Id −ζ, for every X ∈ T P.

The following elementary proposition is crucial.

Proposition ([Mi]). M is a Hopf hypersurface of IK^{n}(λ) if and only if div η = 0 or, equivalently, div ζ = 0.

Then for a Hopf hypersurface M, computations like those in preceeding sections give rise to the formula

Δ_{η}f(r) = {−f′′(r) + f′(2 co_{λ}(r) − co_{4λ}(r))}|∂_{r}^{J}|^{2}

− (2n − 2)f′(r)(co_{λ}(r) + H_{JN}⟨N, ∂_{r}⟩),

where ∂_{r}^{J} = η(∂_{r}^{⊤}), and the convenient choice of f(r) is f′(r) = ta_{λ}(r), which makes the coefficient of |∂_{r}^{J}|^{2} to vanish.

The combination of this expression with one for \overline{Δ}f(r) and a formula for the volume in [GM1], in a similar way as S. Montiel and A. Ros do in [MR] complete the proof of Theorem 5.1.

§ 6. More about immersibility into a geodesic ball.

There are many theorems about nonimmersibility of compact riemannian manifolds with sectional curvature bounded from above inside a metric ball of a simply connected space form. Recently, S. Deshmukh and M.A. Al-Gwaiz ([DA]) have proved a theorem, which deals with Ricci curvatures instead of sectional curvatures. This theorem works for immersions into a euclidean space of even dimension. They used the following well known integral formula

∫_{M}\{ρ(X, X) + \frac{1}{2}|L_{X}g|^{2} − |∇X|^{2} − | div X|^{2}\}dM = 0.

We saw that a good choice of the vector field X in the above formula and a combination of it with formula (2.4) (now taking f′ = s_{4λ}) allow to extend their theorem (with slightly weaker hypothesis) to immersions in every real space form IK^{n}(λ) of any dimension. In particular, we have

6.1. Theorem ([GM2]). Let M be a compact (n − 1)-dimensional Riemannian manifold. Let us define the tangent vector field u on M by u(m) = s_{λ}(r(m))∂_{r}^{⊤}(m). Let us suppose that its Ricci curvature ρ and scalar curvature τ satisfy

∫_{M}{ρ(u, u) + τ|u|^{2}}dM ≥ (n + 1)(n − 2)λ ∫_{M} |u|^{2}dM

(and τ ≥ 0 if λ < 0) and

∫_{M} τdM ≤ (n − 1)(n − 2) vol(M) s_{λ}^{−2}(R).

If there is an isometric immersion ψ of M into IK^{n}(λ) contained in B_{R}(o), then ψ(M) is the boundary of this geodesic ball.

References

[CGM1] F. J. Carreras, F. Giménez and V. Miquel, A Comparison Theorem for the First Dirichlet Eigenvalue of a Domain in a Kaehler Submanifold, J. Australian Math. Soc. A . 56 (1994), 267–277.

[CGM2] F. J. Carreras, F. Giménez and V. Miquel, Immersions of compact riemannian manifolds into a ball of a complex space form, Mathematische Zeit.. To appear.

[CLY] S.Y.Cheng, P.Li and S.T. Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984), 1033–1065.

[DA] S. Deshmukh and M. A. Al-Gwaiz, Compact hypersurfaces in even dimensional euclidean space and in the sphere, Quart. J. Math. Oxford 45 (1994), 151–157.

[GM1] F. Giménez and V. Miquel, Volume estimates for real hypersurfaces of a Kähler manifold with strictly positive holomorphic sectional and antiholomorphic Ricci curvatures, Pacific J. Math. 142 (1990), 23–39.

[GM2] F. Giménez and V. Miquel, On nonimmersibility of compact hypersurfaces into a ball of a simply connected space form, Israel J. Math.. To appear.

[GW] R.E. Greene and H. Wu, Lecture in Mathematics 699 (1979), Function Theory on Manifolds Which Possess a Pole, Springer-Verlag.

[JX] L. P. Jorge and F.V. Xavier, An inequality between the exterior diameter and the mean curvature of bounded immersions, Math. Z. 178 (1981), 77–82.

[Ma] S. Markvorsen, A Sufficient Condition for a Compact Immersion to be Spherical, Math. Zeit. 183 (1983), 407–411.

[Mi] V. Miquel, Compact Hopf Hypersurfaces of Constant Mean Curvature in Complex Space Forms, Annals of Global Analysis and Geometry 12 (1994), 211–218.

[MR] S. Montiel and A. Ros, Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, in “Differential Geometry”, edited by B.L. Lawson and K. Tenenblat, Longman Sc. and Tech., 1991, pp. 279–296.

Departamento de Geometría y Topología Universidad de Valencia Burjasot (Valencia) Spain