Acta Mathematica Sinica, English Series
1999, Oct., Vol.15, No.4, p. 555–560
On the Isometric Minimal Immersion into a Euclidean Space
Qing Chen
Department of Mathematics, The University of Science and Technology of China,
Hefei 230026, P. R. China
E-mail: [email protected]
Abstract We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean
space. A necessary condition for the isometric minimal immersion into a Euclidean space is obtained.
A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given.
Keywords Minimal immersion, Geodesic ball, Volume growth, Ruled submanifolds
1991MR Subject Classification 53A20, 53C42
1
Introduction
Let M be an n-dimensional Riemannian manifold and f : M → Rn+k an isometric immersion
into a Euclidean space Rn+k of dimension n+k. f is said to be a minimal immersion if the mean
curvature of f vanishes identically. It is well known that if a complete Riemannian manifold
admits an isometric minimal immersion into a Euclidean space, then M is not compact and the
Ricci curvature of M is non-positive definite. A natural question is:
Does every complete non-compact Riemannian manifold with non-positive definite Ricci
curvature admit an isometric minimal immersion into a Euclidean space?
In [3] S. S. Chern proposed to search for further necessary conditions on the isometric minimal immersion into Euclidean space. Recently, B. Y. Chen [1] gave a new necessary condition.
In this paper we will give another necessary condition for the isometric minimal immersions
into a Euclidean space. Namely,
Theorem 1
If there exists an isometric minimal immersion from an n-dimensional complete
Riemannian manifold M into a Euclidean space, then for any p ∈ M , the function
Vol (Bp (t))
tn
Received November 12, 1996, Revised December 12, 1997, Accepted August 10, 1998
This work is partially supported by the National Science Foundation of China
(1.1)
Qing Chen
556
is monotone non-decreasing with respect to t, where Bp (t) is the geodesic ball of M with center
p and radius t.
The well known monotonicity formula of the n-dimensional minimal submanifolds in Rn+k
says the function of the volume of extrinsic distance ball of radius t over tn is monotone nondecreasing (see [5]). Theorem 1 also generalizes the monotonicity formula to the case of intrinsic
distance ball.
A class of Riemannian manifolds is obviously satisfying Theorem 1, say the non-positively
curved manifolds (by the volume comparison theorem [4] we see that if the sectional curvature
of M is non-positive definite, function (1.1) is monotone non-decreasing when t is less than the
injective radius of a given point). It is not known wether a non-positively curved Riemanian
manifold can be isometrically immersed in a Euclidean space as a minimal submanifold. However, the structure of non-positively curved minimal hypersurfaces in a Euclidean space is so
clear that in Sec. 3 we will give a certain classification of such types of hypersurfaces. A simple
corollary of our classification is that there is no isometric minimal immersion of n-dimensional
Riemannian manifold M into Rn+1 , provided the sectional curvature of M is negatively definite.
2
Volume Growth Formula and Main Theorems
Let M be a complete n-dimensional submanifold of a Euclidean space Rn+k equipped with
the standard metric , . Denote by D and ∇, the covariant derivatives of Rn+k and M
respectively. For X, Y ∈ T M , we put B(X, Y ) = DX Y − ∇X Y , the second fundamental form
of M , and H =
1
n trB,
the mean curvature vector of M . M is minimal if H = 0. If we denote
the position vector of M in Rn+k by x, then
∆x = nH,
(2.1)
where ∆ is the Laplacian of M .
For any p ∈ M , let r(x) = |x − p|, the Euclidean distance related to p. From (2.1) we easily
obtain
1
1 2
∆r = ∆x − p, x − p = n + nH, x − p.
(2.2)
2
2
Let Bp (t) = {x ∈ M : ρp (x) = distM (x, p) < t}, the geodesic ball of radius t of M with center
p. When ∂Bp (t) is smooth, integrating (2.2) over Bp (t), we have (by Green’s formula)
r∇r, ∇ρp = nVol Bp (t) + n
H, x − p.
∂Bp (t)
(2.3)
Bp (t)
On the other hand, by co-area formula (see [5]),
d
Vol Bp (t) = Vol ∂Bp (t) =
dt
1.
(2.4)
∂Bp (t)
Thus we obtain
d
t Vol Bp (t) − nVol Bp (t) =
(t − r∇r, ∇ρ) +
nH, x − p,
dt
∂Bp (t)
Bp (t)
(2.5)
On the Isometric Minimal Immersion into a Euclidean Space
557
from which, and also from the fact that ρ(x) ≥ r(x) ∀x ∈ M , and |∇r| ≤ 1, |∇ρ| = 1, we get
d
t Vol Bp (t) − nVol Bp (t) ≥
nH, x − p.
(2.6)
dt
Bp (t)
If ∂Bp (t) is not smooth, for given > 0, we choose a smooth cut-off functions φ such that

if ρ(x) < t − (1 + θ),

 1,
φ(x) = 0,
if ρ(x) > t,


∈ [0, 1]
if t − (1 + θ) ≤ ρ(x) ≤ t,
and |∇φ| ≤
1
(where θ ∈ (0, 1) is a given constant). From (2.2) we have
1
φ∆r2
nφ(1 + H, x − p) =
Bp (t)
Bp (t) 2
=
−r∇r, ∇φ
Bp (t)
≤
≤
(2.7)
Bp (t)−Bp (t−(1+θ))
r|∇φ|
t
(Vol Bp (t) − Vol Bp (t − (1 + θ))) .
Letting → 0 in (2.7), we have
nVol Bp (t) +
Bp (t)
nH, x − p ≤ (1 + θ)t
d
Vol Bp (t).
dt
(2.8)
Now letting θ → 0 in (2.8), we find that (2.6) holds for every t > 0. Thus we have
Proposition
Let M be an immersed n-dimensional complete submanifold of Rn+k and p ∈
M . Then for every t > 0,
d Vol Bp (t)
−n−1
≥ nt
H, x − p.
dt
tn
Bp (t)
(2.9)
Now Theorem 1 follows readily from the proposition.
Theorem 2
Let M be an n-dimensional complete minimal submanifold in Rn+k . If
Vol Bp (t)
=1
t→+∞
ω n tn
lim
for a point p ∈ M , then M is a total geodesic submanifold, where ωn is the volume of a unit
ball in Rn .
Proof
Since
Vol Bp (t)
= 1,
t→0
ω n tn
lim
by the hypothesis and the monotonicity, we see that
Vol Bp (t) = ωn tn
for every
t > 0.
(2.10)
Qing Chen
558
Therefore (2.5) yields
∂Bp (t)
(t − r∇r, ∇ρ) = 0
(2.11)
when t is less then the injective radius of p. We can derive from (2.11) that ρ(x) = r(x) when
x is in a small neighborhood of p. This means every geodesic starting from p is a line. Then
locally M is flat, and hence M is flat. We complete the proof.
The following fact is well known; we present a simple proof here.
Corollary
Let M be an n-dimensional complete minimal submanifold in Rn+k . Let C(t) be
the Euclidean distance ball of Rn+k with radius t and center at origin o. If
lim
t→+∞
Vol (M ∩ C(t))
= 1,
ω n tn
then M is an affine n-plane.
Proof
Without loss of generality, suppose o ∈ M . It is obvious that Bo (t) ⊂ M ∩ C(t),
therefore
Vol Bo (t)
Vol (M ∩ C(t))
≤ lim
= 1.
t→+∞
ω n tn
ω n tn
Then the assertion follows by Theorem 2.
lim
t→+∞
From the proposition above we also have the following theorem, which generalizes the result
[5] concerning the volume growth estimate of extrinsic distance ball of submanifolds in Rn+k
to the intrinsic case.
Theorem 3
Let M be an immersed complete n-dimensional submanifold of a Euclidean space
Rn+k . Let p ∈ M and Bp (t) be the geodesic ball of M with center p and radius t. Suppose there
is a constant Λ such that
Bp (t)
holds for every t. Then function
|H| ≤
P (t)
eΛt Vol B
tn
Λ
Vol Bp (t)
n
is monotone non-decreasing of t.
From (2.6) and the fact r|∂Bp (t) ≤ t, we have
d Vol Bp (t)
−n−1
−n
≥ −t
nr(x)|H| ≥ −t
n|H| ≥ −Λt−n Vol Bp (t),
dt
tn
Bp (t)
Bp (t)
Proof
which deduces that
Vol Bp (t)
d Vol Bp (t)
d
Λt Vol Bp (t)
Λt
e
=e
+Λ
≥ 0.
dt
tn
dt
tn
tn
(2.12)
(2.13)
This proves the assertion.
3
Non-positive Curved Minimal Hypersurfaces
In this section we consider non-positively curved minimal submanifolds in a Euclidean space.
First we present the following two examples.
On the Isometric Minimal Immersion into a Euclidean Space
Example 1
559
Let N be an immersed minimal surface in Rk+2 . Then N × Rn−2 is a minimal
submanifold in Rn+k with non-positive definite sectional curvature.
Example 2
Let N be a minimal surface in a k + 2-dimensional unit sphere S k+2 , and CN
the 3-dimensional minimal cone in Rk+3 over N with vertex at origin. Then M = CN × Rn−3
is a non-positively curved minimal submanifold in Rn+k .
Example 1 is trivial, we refer to [1, Lemma 4.2] for the proof of Example 2.
Remark In [1] B. Y. Chen proved that a minimal submanifold in a Euclidean space must
satisfy inf K ≥ τ2 , where τ is the scalar curvature and inf K is the infimum of the sectional
curvatures. Examples 1 and 2 are two examples which satisfy inf K = τ2 , see Lemmas 4.1 and
4.2 of [1].
A submanifold M of Rn+k is m-ruled if there is an open dense subset M1 of M , that M1 is
foliated by m-dimensional totally geodesic submanifolds of Rn+k (see [1] and [6] Theorem 1). If
M is an n-dimensional m-ruled submanifold of Rn+k , then there exists an (n − m)-dimensional
submanifold N of M such that M1 is generated by a moving m-dimensional totally geodesic
submanifold Rm along N . In particular, if the moving Rm is normal to N at each point, then
the ruled submanifold M is called a normal m-ruled submanifold.
In the case of codimension one, we have
Theorem 4
Let M be a connected non-positively curved minimal hypersurface in Rn+1 . Then
M is a piece of the following minimal hypersurfaces in Rn+1 :
(1) a totally geodesic hypersurface,
(2) an (n − 2)-ruled hypersurface of Rn+1 . In particular, if M is normal (n − 2)-ruled, then
M is one of the following:
(2a) a product manifold N × Rn−2 , where N is a minimal surface in R3 ,
(2b) a product manifold CN × Rn−3 , where CN is a minimal cone in R4 .
Proof
We choose a suitable orthonormal frame e1 , e2 , . . . , en , en+1 at a given point p, such that
the second fundamental form of M takes the form
B(ei , ej ) = λi δij en+1 ,
i, j = 1, . . . , n,
(3.1)
where λi , i = 1, . . . , n are the principal curvatures of M . By the hypothesis, λi λj ≤ 0, for i = j.
This deduces that there are at most one positive λi and one negative λi , i.e., {λ1 , . . . , λn } =
{λ, −λ, 0, . . . , 0}. Set D(p) = {X ∈ Tp M | B(X, Y ) = 0, ∀Y ∈ Tp M }, then it is obvious that
dim D(p) = n − 2 or n.
If dim D(p) ≡ n, M is a piece of a totally geodesic hypersurface. If dim D(p) ≡ n, then
M1 = {p ∈ M | dim D(p) = n − 2} = {p ∈ M | λ(p) = 0} is an open subset of M . Supposing
M − M1 = φ, we notice that for any p0 ∈ M − M1 , and any neighborhood U of p0 in M ,
dim D(p)|U ≡ n (otherwise, U is totally geodesic and hence M is totally geodesic), i.e. M1 ∩U =
0. Hence M1 is an open dense subset of M .
Lemma
The distribution D is completely integrable on M1 and each leaf of the distribution
is a totally geodesic submanifold of Rn+1 .
Qing Chen
560
Proof of Lemma Take adapted local orthonormal frame fields {e1 , . . . , en+1 } of Rn+1 , such
that {e1 , . . . , en } are the principal directions of M , and D is locally spanned by {e3 , . . . , en }.
Let {ω1 , . . . , ωn+1 } be the dual frame fields, and {ωij , i, j = 1, . . . , n + 1} the connection
1-forms, and {ωin+1 , i = 1, . . . , n} the second fundamental form of M . Then
ω1n+1 = λω1 ,
ω2n+1 = −λω2 ,
ωjn+1 = 0
for j = 3, . . . , n.
(3.2)
Differentiating (3.2), we obtain from the structure equation [3]
0 = dωjn+1 = −
n
ωjk ∧ ωkn+1 = −λ(ωj1 ∧ ω1 − ωj2 ∧ ω2 )
k=1
for j ≥ 3. Since M1 = {p| λ(p) = 0}, we get from the above that
ωj1 ∧ ω1 − ωj2 ∧ ω2 = 0
for j ≥ 3,
(3.3)
which implies when j ≥ 3, 1-forms ωj1 and ωj2 are both the combinations of ω1 and ω2 .
Consequently, for i = 1, 2,
dωi = −
n
ωij ∧ ωj ≡ 0
mod{ω1 , ω2 }.
j=1
This means the distribution {ω1 = ω2 = 0}, i.e. the distribution D is completely integrable.
Furthermore, along each leaf of {ω1 = ω2 = 0}, ωj1 = ωj2 = 0 (j ≥ 3), this together with
ωjn+1 = 0 (j ≥ 3) deduces that each leaf is a totally geodesic submanifold in Rn+1 . This
completes the proof.
Continuation of the Proof of Theorem 3
The first assertion of (2) is a direct consequence of
the lemma. If M is normal ruled, in this case the dimension of the distribution D(p) on M1 is
n − 2. The scalar curvature of M is τ = −2λ2 , which yields inf K = −λ2 = τ2 . Hence from the
classification of [1] Theorem 5.1 we obtain that M1 is either (2a) or (2b). By the fact that the
two connected minimal hypersurfaces in Rn+1 coincide in a neighborhood and then coincide
identically, we see that M is either (2a) or (2b). This completes the proof.
Acknowledgement
The author would like to thank the referee for many useful suggestions,
and in particular for pointing out a gap in the proof of Theorem 4 in an earlier version of this
paper.
References
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568–578
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constant length. Functional Analysis and Related Fields. Berlin, New York: Springer-Verlag, 1970, 59–75
[3] S S Chern. Minimal submanifolds in a Riemannian manifold. Lawrence, Kansas, 1968
[4] S Gallot, D Hulun, J Lafontaine. Riemannian Geometry. Springer-Verlag, 1987
[5] L Simon. Lectures on Geometric Measure Theory. C M A Australian National University, 1983, 3
[6] F Zheng. Isometric embedding of Kahler manifolds with nonpositive sectional curvature. Math Ann, 1996,
304(4): 769–784