Annales de l’institut Fourier (AIF)

Florit, Luis A.

Zheng, Fangyang

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L’INSTITUT FOURIER
Luis A. FLORIT & Fangyang ZHENG
Complete real Kähler Euclidean hypersurfaces are cylinders
Tome 57, no 1 (2007), p. 155-161.
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Ann. Inst. Fourier, Grenoble
57, 1 (2007) 155-161
COMPLETE REAL KÄHLER EUCLIDEAN
HYPERSURFACES ARE CYLINDERS
by Luis A. FLORIT & Fangyang ZHENG (*)
Abstract. — In this note we show that any complete Kähler (immersed) Eu-
clidean hypersurface M2n ⊂ R2n+1 must be the product of a surface in R3 with
an Euclidean factor Cn−1 ∼= R2n−2.
Résumé. — Dans cet article nous montrons que toute hypersurface Kählerienne
complète immergée dans un espace Euclidien M2n ⊂ R2n+1 est le produit d’une
surface de R3 et d’un facteur Euclidien Cn−1 ∼= R2n−2.
1. The statement
The purpose of this paper is to give a proof of the following result, that
was proved by Abe ([2]) under the additional assumption that either f is
real analytic, or the scalar curvature of M2n is everywhere nonnegative or
everywhere negative:
Theorem 1.1. — Let f : M2n → R2n+1 be an isometric immersion of
a complete Kähler manifold. Then M2n = Σ2 ×C n−1 and f = f1×ι split,
where f1 : Σ2 → R3 is an isometric immersion and ι is the identity map of
C n−1 ∼= R2n−2.
The problem of classifying all real Kähler hypersurfaces in real space
forms was raised in [12] and solved in [11], both locally and globally, when
the ambient space has non-zero constant curvature. In this case, the ex-
amples are essentially unique, even locally, being the product of two space
forms. One of the main ingredients used to show this is the well-known fact
Keywords: Kähler submanifolds, cylinders, splitting.
Math. classification: 53C40, 53C55.
(*) The first author is partially supported by CNPq.
The second author is partially supported by IHES and a NSF grant.
156 Luis A. Florit & Fangyand Zheng
that the index of relative nullity ν of the immersion satisfies that ν > 2n−2
([12]). Here ν(x) is the dimension of the nullity of the second fundamental
form at x ∈ M2n. The function r = n − ν is often called the rank of the
immersion.
The situation when the ambient space is flat is fairly more complicated.
The local classification for hypersurfaces without flat points was carried
out in [3]. It was shown that they are abundant and can be parametrized
by means of a pseudoholomorphic surface in S2n (the Gauss image) and
a smooth function over it. In particular, generically they are locally irre-
ducible.
Since any isometric immersion f : Nn → Rn+p with ν > n−1 is flat, the
classical Hartman’s cylinder theorem ([9]) implies that, when Nn is also
complete, Nn = Γ×Rn−1 and f = f1×ι, where Γ = R or S1, f1 : Γ → R1+p
is a smooth curve and ι is the identity map of Rn−1. In other words, f is
a (n− 1)-cylinder.
Later on, Abe ([1]) obtained the complex analogue, which states that
for any holomorphic isometric immersion f : M2n → C n+p of a complete
Kähler manifold into the complex Euclidean space, if ν > 2n− 2, it holds
that f = f1×ι, where f1 : Σ → C 1+p is a holomorphic curve and ι is the
identity map of C n−1. That is, any complete complex submanifold M2n
of the complex Euclidean space with (real) rank at most two is a complex
(n− 1)-cylinder.
After these two classical results, it is only natural to consider the cylinder
problem in the hybrid case, namely, for complete real Kähler submanifolds
with rank at most two. In view of the proof of Theorem 1.1, we will state
the question in the form of a conjecture:
Conjecture 1.2. — Let f : M2n → R2n+p be an isometric immersion
of a complete Kähler manifold into Euclidean space. If ν > 2n − 2, then
M2n = Σ2 ×R2n−2 and f = f1×ι split, where f1 : Σ2 → R2+p and ι is the
identity map of R2n−2.
Note that when M2n is flat (or more generally, when M2n has nonneg-
ative scalar curvature), the above is true by Hartman’s cylinder theorem.
The above conjecture was also proved by Dajczer and Gromoll in [4] under
the additional assumption that the set of non-flat points of M2n is dense
and connected. When M2n is not everywhere flat, it is easy to see that, if
the above conjecture is true, then the decomposition
M2n = Σ2×R2n−2 ∼= Σ2×C n−1 is a Kähler one. For related splitting results
of real Kähler Euclidean submanifolds in higher codimensions, see [6], [8]
and [7].
ANNALES DE L’INSTITUT FOURIER
KÄHLER HYPERSURFACES ARE CYLINDERS 157
2. The proof
Let f : M2n → R2n+1 be an isometric immersion of a Kähler manifold of
(real) dimension 2n, i.e., a real Kähler Euclidean hypersurface. Denote by
∆(x) the relative nullity of f at x ∈ M2n, that is, the kernel of the second
fundamental form at x. Its dimension ν(x) is the index of relative nullity
of f at x. It is well-known that, along any open subset where ν is constant,
∆ is a smooth distribution with totally geodesic leaves in both M2n and
R2n+1. Moreover, on the open subset where ν attains its minimum, the
leaves are complete if M2n is complete.
The next observation follows easily from the Gauss equation and the
fact that R(X, Y ) ◦ J = J ◦ R(X, Y ) for all X, Y ∈ TM , where R is the
curvature tensor of M2n.
Lemma 2.1. — ([12]) For any real Kähler hypersurface f : M2n →
R2n+1, the relative nullity index ν satisfies that ν > 2n − 2, and in the
open set of non-flat points
U = {x ∈ M2n : ν(x) = 2n− 2},
∆ is a complex distribution, that is, J∆ = ∆.
Let f : M2n → R2n+1 be as in Theorem 1.1. If M2n is flat, then by the
cylinder theorem of Hartman-Nirenberg ([10]), f is a (2n−1)-cylinder over
a plane curve f1 : Γ → R2. Let T be a unit tangent vector of Γ. Then JT
is constant in the Euclidean space since ∇̃T JT = ∇T JT = J∇T T = 0,
where ∇̃ and ∇ stand for the Levi-Civita connections of R2n+1 and M2n,
respectively. Therefore, JT defines a line R ⊂ R2n−1 and if we set Σ = Γ×R
as the product of Γ with that line, then we have the Kähler decomposition
M2n = Σ×C n−1 and Theorem 1.1 follows in this case. Hence, since an
arbitrary Euclidean hypersurface is flat if and only if its rank is at most
one, in view of Lemma 2.1 from now on we can assume that U is not empty.
By Lemma 2.1 each leaf of ∆ in U is a complete complex submanifold of
M2n that has to be C n−1 since f maps each leaf of ∆ in U onto a linear
subvariety of R2n+1, that is, an affine subspace of R2n+1. The aforemen-
tioned result in [2] is consequence of the following lemma together with the
Cheeger-Gromoll splitting theorem (see also [4]):
Lemma 2.2. — ([2]) In any connected component of U , the leaves of ∆
are mapped by f onto linear subvarieties in R2n+1 that are parallel to each
other.
TOME 57 (2007), FASCICULE 1
158 Luis A. Florit & Fangyand Zheng
Remark 2.3. — The same proof holds for any open subset of M2n with
a complex totally geodesic distribution ∆′ ⊂ ∆ with complete leaves of
complex codimension 1 in M2n.
Our goal is to extend this foliation in U to the entire M2n. First let us
fix some notations. Denote by
U ′ = {x ∈ M2n | ν(x) 6 2n− 1}, U1 = U ′ \ U.
Then U ′ is open, with M2n \ U ′ being the set of totally geodesic points.
U1 is the set of points where f has constant rank 1. For convenience, we
will denote by V the set of interior points of U1, and write F = U1 \ V .
Clearly, U ′ is the disjoint union of the open subsets U , V and the closed
set F , with F = U1 ∩ U .
Now let us construct a distribution L in the open set U ′. For any rank 2
point x ∈ U , we will simply set Lx = ∆x. For x ∈ V , define
Lx = ∆x ∩ J∆x,
where ∆x ∼= R2n−1 is the kernel of α and J is the almost complex structure.
Lx is again a J-invariant subspace of TxM of codimension 2. Since ∆ is a
totally geodesic foliation in V with flat leaves, L is also a totally geodesic
foliation in V and its leaves are flat complex submanifolds. Let us denote
by L(y) the leaf of L in U ∪ V passing through y.
For any x ∈ F , since F ⊆ U , there exists a sequence of points {xk} in
U approaching to x. Then by passing to a subsequence if necessary, we
may assume that Lxk converges, so lim L(xk) will be a totally geodesic flat
C n−1 passing through x that is mapped by f onto a linear subvariety. We
will call it a limit position at x.
We claim that the limit positions at any given x ∈ F must be unique.
To see this, let us assume there are two sequences {xk} and {yk} in U ,
both approaching x, such that P := lim L(xk) and Q := lim L(yk) both
exist but are different. Since P and Q are (closed) complex hypersurfaces
of M2n, both intersect at x transversally. So, nearby leaves will also have
to intersect. That is, for sufficiently large k, L(xk) and L(yk) will intersect
each other, which is only possible when they coincide. This contradicts
P 6= Q.
Now we have the uniqueness of limit position at any x ∈ F , let us just
denote it by F (x), and write Lx for the tangent space of F (x) at x. Since
for each k, Lxk is a J-invariant subspace where α vanishes, so Lx is a
J-invariant subspace of ∆x. Thus by the fact that ν(x) = 2n−1, we get
Lx = ∆x∩J∆x, consistent with our definition of L in V . In addition, since
each connected component of U is a cylinder over a surface in R3 by Lemma
ANNALES DE L’INSTITUT FOURIER
KÄHLER HYPERSURFACES ARE CYLINDERS 159
2.2, for any y ∈ F (x) we have that ν(y) = ν(x) = 2n− 1. Therefore, F (x)
must be contained in F , and then F is the disjoint union of these limit
positions.
Now let us fix any x ∈ V , and consider the leaf L(x) of L in V . We claim
that this totally geodesic submanifold is complete. Let γ : [0,∞) → M
be a geodesic such that γ(0) = x, and γ([0, a)) ⊆ L(x), where a > 0. It
is well-known (cf. [5]) that ν does not increase along such a geodesic. So
γ(a) ∈ U1. If y = γ(a) ∈ F , then since
γ′(t) ∈ Lγ(t) = ∆γ(t) ∩ J∆γ(t)
for any t < a, we would have γ′(a) ∈ Ly. But the limit position F (y)
through y is totally geodesic, so γ is contained in F (y) ⊂ F , contradicting
the assumption that x ∈ V . So we have proved that γ(a) ∈ V . This proves
the completeness of L(x).
In summary, we showed that the open subset U ′ is foliated by L, whose
leaves are totally geodesic flat C n−1 that are mapped under f onto linear
subvarieties in R2n+1. Also, in view of the flatness of the metric in V ,
Lemma 2.2 and the continuity of L, in each connected component of U ′,
these linear subvarieties are parallel to each other. (Another proof of the
parallelism of the leaves in V can be obtained with the standard splitting
tensor argument used in the proof of Lemma 2.2; see Remark 2.3).
Our next goal is to extend the foliation L across the set W := M \ U ′
of totally geodesic points. First, for any x ∈ U ′ \ U ′, the leaves of L give
limit positions at x, which are totally geodesic complex submanifolds (of
complex codimension 1) passing through x. Such limit positions at x must
be unique for the exact same reason as before (when we extended L from
U to F ). So U ′ is now foliated by flat, locally parallel C n−1.
Denote by W 0 the set of interior points of W . Suppose W 0 is not empty
and let Ω be a connected component of W 0. Fix any x ∈ Ω, and consider
H = Tf(x)M ∼= C n as a linear subvariety of R2n+1. For convenience, we
will identify a subset P ⊆ M with its image f(P ) if f |P is an embedding.
Denote by ∂Ω ⊂ U ′ the set of boundary points of Ω. Since f is totally
geodesic in W , we know that the restriction of f on Ω is an embedding,
and Ω ⊂ H. Note that ∂Ω 6= ∅ since W 6= M2n.
For any y ∈ ∂Ω, y is a boundary point of U ′. Hence, there exists a totally
geodesic flat L(y) ∼= C n−1 passing through y, and f embeds L(y) onto a
linear subvariety. Since Tf(y)M = H, we have L(y) ⊂ H as a complex
affine hyperplane. Thus, if z is another point in ∂Ω, unless L(y) is parallel
to L(z) they will intersect. Suppose w ∈ L(y) ∩ L(z). Since L(y) and L(z)
TOME 57 (2007), FASCICULE 1
160 Luis A. Florit & Fangyand Zheng
are both contained in U ′, w is a point in U ′, and then we have L(w), which
must agree with L(y) and L(z) by the uniqueness of the leaves of L on U ′.
Therefore, we get L(y) = L(z). That is, for any two points y and z in ∂Ω,
the complex linear subvarieties L(y) and L(z) in H are parallel to each
other.
We conclude that there is a unique complex linear subvariety of complex
codimension 1 in H passing through x, which is parallel to L(y) for all
y ∈ ∂Ω. Call it L(x). Clearly, L(x) ⊂ Ω since it cannot touch the boundary
part of Ω due to the fact that L(y) ⊂ U ′ for all y ∈ ∂Ω.
Now our entire manifold M2n is foliated by totally geodesic flat C n−1,
with each leaf mapped by f onto a linear subvariety, and nearby leaves are
parallel to each other. The proof of Theorem 1.1 now follows easily.
BIBLIOGRAPHY
[1] K. Abe, “A complex analogue of Hartman-Nirenberg cylinder theorem”, J. Differ-
ential Geom. 7 (1972), p. 453-460.
[2] ——— , “On a class of hypersurfaces of R2n+1”, Duke Math. J. 41 (1974), p. 865-
874.
[3] M. Dajczer & D. Gromoll, “Real Kähler submanifolds and uniqueness of the
Gauss map”, J. Differential Geom. 22 (1985), p. 13-28.
[4] ——— , “Rigidity of complete Euclidean hypersurfaces”, J. Differential Geom. 31
(1990), p. 401-416.
[5] M. Dajczer & L. Rodríguez, “Complete real Kähler minimal submanifolds”, J.
Reine Angew. Math. 419 (1991), p. 1-8.
[6] L. Florit, W. Hui & F. Zheng, “On real Kähler Euclidean submanifolds with
non-negative Ricci curvature”, J. Eur. Math. Soc. 7 (2005), p. 1-11.
[7] L. Florit & F. Zheng, “Complete real Kähler Euclidean submanifolds in codi-
mension two”, Preprint at http://www.preprint.impa.br/Shadows/SERIE_A/2004/
306.html.
[8] ——— , “A local and global splitting result for real Kähler Euclidean submanifolds”,
Arch. Math. (Basel) 84 (2005), p. 88-95.
[9] P. Hartman, “On isometric immersions in Euclidean space of manifolds with non–
negative sectional curvatures II”, Trans. Amer. Math. Soc. 147 (1970), p. 529-540.
[10] P. Hartman & L. Nirenberg, “On spherical image maps whose Jacobians do not
change sign”, Amer. J. Math. 81 (1959), p. 901-920.
[11] P. Ryan, “Kähler manifolds as real hypersurfaces”, Duke Math. J. 40 (1973), p. 207-
213.
[12] T. Takahashi, “A note on Kählerian hypersurfaces of spaces of constant curvature”,
Kumamoto J. Sci. (Math.) 9 (1972), p. 21-24.
Manuscrit reçu le 21 mai 2005,
accepté le 20 décembre 2005.
ANNALES DE L’INSTITUT FOURIER
KÄHLER HYPERSURFACES ARE CYLINDERS 161
Luis A. FLORIT
IMPA: Estrada Dona Castorina 110
22460–320, Rio de Janeiro (Brazil)
luis@impa.br
Fangyang ZHENG
Ohio State University
Columbus, OH 43210 (USA)
and
Zhejiang University
IMS
Hanzhou (China)
zheng@math.ohio-state.edu
TOME 57 (2007), FASCICULE 1


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