The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups

Abstract

In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the (2m+1)-dimensional Heisenberg group we also derive necessary and sufficient conditions for the Gauss map to be harmonic and prove that for m=1 all CMC-surfaces with the harmonic Gauss map are "cylinders".

Full text

arXiv:0803.2214v1 [math.DG] 14 Mar 2008
The Gauss Map of Hypersurfaes in
2
-Step Nilpotent
Lie Groups
E. V. Petrov
V. N. Karazin Kharkiv National University
4 Svobody sq., Kharkiv, 61077, Ukraine
E-mail: petrovuniver.kharkov.ua
In this paper we onsider smo oth oriented hypersurfaes in
2
-step nilp otent Lie
groups with a left invariant metri and derive an expression for the Laplaian
of the Gauss map for suh hypersurfaes in the general ase and in some
partiular ases. In the ase of CMC-hypersurfae in the
2m+ 1
-dimensional
Heisenberg group we also derive neessary and suient onditions for the
Gauss map to be harmoni and prove that for
m= 1
all CMC-surfaes with
the harmoni Gauss map are ylinders.
2000 Mathematis Subjet Classiation.
Primary 53C40. Seondary 53C42,
53C43, 22E25.
Keywords.
2-step nilpotent Lie group, Heisenberg group, left invariant metri,
Gauss map, harmoni map, minimal submanifold, onstant mean urvature.
It is proved in [12℄ that the Gauss map of a smooth
n
-dimensional oriented
hypersurfae in
Rn+1
is harmoni if and only if the hypersurfae is of a
onstant mean urvature (CMC). The same is proved for the ases of
S3
,
whih is a Lie group and thus has a natural denition of the Gauss map [9℄,
and, in dierent settings, of
H3
[10℄. A generalization of this proposition
to the ase of Lie groups with a bi-invariant metri (this lass of Lie groups
inludes, for example, abelian groups
Rn+1
and
S3∼
=SU (2)
) is proved in [6℄.
In this paper we use methods of [6℄ for an investigation of the Gauss map of a
hypersurfae in some
2
-step nilpotent Lie group with a left invariant metri.
The theory of suh groups is highly developed (see, for example, [3℄ and [4℄).
The paper is organized as follows. After some preliminary information
(setion 1), in setion 2 we obtain an expression for the Laplaian of the
Gauss map of a hypersurfae in a
2
-step nilpotent Lie group (Theorem 1).
1

Using this expression we prove some fats onerning relations between har-
moni properties of the Gauss map and the mean urvature of the hyper-
surfae (see setion 3), in partiular, a suient ondition for the stability
of CMC-hypersurfaes (Proposition 6). In setion 4 we onsider the ases of
Heisenberg type groups and Heisenberg groups. We show the harmoniity of
the Gauss map of a hypersurfae in suh groups is, in general, not equiva-
lent to the onstany of the mean urvature. Also we obtain neessary and
suient onditions for this equivalene in the partiular ase of Heisenberg
groups (Prop osition 7).
The author is grateful to prof. L. A. Masal'tsev for onstant attention to
this work. The author would also thank prof. Yu. A. Nikolayevsky and prof.
A. L. Yampolsky for many useful advies onerning language and style.
1 Preliminaries
Let us reall some basi denitions and fats about the stability of on-
stant mean urvature hypersurfaes in Riemannian manifolds. Suppose
M
is
a smooth
n
-dimensional manifold immersed in a smo oth
n+ 1
-dimensional
Riemannian manifold as a CMC-hypersurfae. Denote by
η
a unit normal
vetor eld of
M
. Let
D⊂M
be a ompat domain. The
index form
of
D
is a quadrati form
Q(·,·)
on
C∞(D)
dened by the equation
Q(w, w) = −Z
D
wLw dVM,
(1)
where
dVM
is the volume form of the indued metri on
M
,
L
is the
Jaobi
operator
∆M+Ric(η, η) + kBk2
,
Ric(·,·)
is the Rii tensor of the ambient
manifold,
kBk
is the norm of the seond fundamental form of the immersion,
and
∆M
is the Laplaian of the indued metri (see, for example, [2℄).
Let
M
be a minimal hypersurfae (a hypersurfae of a nonzero onstant
mean urvature, respetively). A ompat domain
D⊂M
is alled
stable
if
Q(w, w)>0
for every funtion
w∈C∞(D)
vanishing on
∂D
(for every
w∈C∞(D)
vanishing on
∂D
and with
R
D
w dVM= 0
, respetively). The
hypersurfae
M
is
stable
if every ompat domain
D⊂M
is stable, and is
unstable
otherwise (see, for example, [1℄). It is proved in [7, Theorem 1℄ that
if the
Jaobi equation
Lw = 0
admits a solution
w
stritly positive on
M
,
then
M
is stable.
2

Let
(M, g)
be a smo oth Riemannian manifold. Denote by
∆M
the Lapla-
ian of
g
. For eah
φ∈C∞(M, S n)
denote by
∆Mφ
the vetor
(∆Mφ1,...,
∆Mφn+1)
, where
(φ1,...,φn+1)
is the oordinate funtions of
φ
for the stan-
dard embedding of a unit sphere
Sn֒→Rn+1
. It is well known that the
harmoniity of
φ
is equivalent to the equation
∆Mφ= 2e(φ)φ
, where
e(φ)
is
the energy density funtion of
φ
(see [14, p. 140, Corollary (2.24)℄).
Suppose
M
is an oriented hypersurfae in a
n+ 1
-dimensional Lie group
N
with a left invariant Riemannian metri. Fix the unit normal vetor eld
η
of
M
with resp et to the orientation. Let
p
be a point of
M
. Denote
by
La
the left translation by
a∈N
, and let
dLa
be the dierential of this
map. We an onsider
p
as an element of
N
if we identify this point with
its image under the immersion. Let
G
be the map of
M
to
Sn⊂ N
suh
that
G(p) = (dLp)−1(η(p))
for all
p∈N
, where
N
is the Lie algebra of
N
.
We all
G
the
Gauss map
of
M
. It is proved in [6℄ that if a metri of
N
is
bi-invariant (see [11℄ on a struture of suh Lie groups), then the Gauss map
is harmoni if and only if the mean urvature of
M
is onstant.
Now we onsider the ase of nilpotent Lie groups. Let
N
be a nite
dimensional Lie algebra over
R
with a Lie braket
[·,·]
. The lower entral
series of
N
is dened indutively by
N1=N
,
Nk+1 =Nk,N
for all
positive integers
k
. The Lie algebra
N
is alled
k
-step nilpotent
if
Nk6= 0
and
Nk+1 = 0
. A Lie group
N
is alled
k
-step nilpotent
if its Lie algebra
N
is
k
-step nilp otent.
In the sequel, we onsider a
2
-step nilpotent onneted and simply on-
neted Lie group
N
and its Lie algebra
N
. Let
Z
be the enter of
N
. Sine
N
is
2
-step nilp otent,
06= [N,N]⊂ Z
. Supp ose that
N
is endowed with a
salar produt
h·,·i
. This salar produt indues a left invariant Riemannian
metri on
N
, whih we also denote by
h·,·i
. Let
V
be an orthogonal om-
plement to
Z
in
N
with respet to
h·,·i
. Then
[V,V] = [N,N]⊂ Z
. For
eah
Z∈ Z
a linear op erator
J(Z): V → V
is well dened by
hJ(Z)X, Y i=
h[X, Y ], Zi
, where
X, Y ∈ V
are arbitrary vetors.
An important lass of
2
-step nilp otent groups onsists of so-alled
2m+1
-
dimensional
Heisenberg groups
, whih appear in some problems of quantum
and Hamiltonian mehanis [8℄. The Lie algebra of a Heisenberg group has
a basis
K1,...,Km
,
L1,...,Lm
,
Z
and the struture relations
[Ki, Lj] = δijZ, [Ki, Kj] = [Li, Lj] = [Ki, Z] = [Li, Z] = 0,16i, j 6m,
where
δij
is the Kroneker symbol. We introdue a salar produt suh that
this basis is orthonormal. The three-dimensional Heisenberg group with a
3

left invariant Riemannian metri is often denoted by
Nil
and is a three-
dimensional Thurston geometry. A Lie algebra
N
is of
Heisenberg type
if
J(Z)2=−hZ, ZiId |V
, for every
Z∈ Z
[4℄. Its Lie group
N
is alled a
Lie group of
Heisenberg type
. This lass of groups ontains, for example,
Heisenberg groups and quaternioni Heisenberg groups [3, p. 617℄. A general
approah to the struture of
2
-step nilpotent Lie algebras was developed in
the pap er [5℄.
The Riemannian onnetion asso iated with
h·,·i
is dened on left invari-
ant elds by (see [3℄)
∇XY=1
2[X, Y ], X, Y ∈ V;
∇XZ=∇ZX=−1
2J(Z)X, X ∈ V, Z ∈ Z;
∇ZZ∗= 0, Z, Z∗∈ Z.
(2)
From this one an obtain for the urvature tensor
R(X, Y )X∗=1
2J([X, Y ])X∗
−1
4J([Y, X ∗])X
+1
4J([X, X∗])Y,
X, X∗, Y ∈ V;
R(X, Z)Y
R(X, Y )Z
=
=−1
4[X, J (Z)Y],
−1
4[X, J (Z)Y]
+1
4[Y, J (Z)X],
X, Y ∈ V, Z ∈ Z;
R(X, Z)Z∗
R(Z, Z∗)X
=
=−1
4J(Z)J(Z∗)X,
−1
4J(Z∗)J(Z)X
+1
4J(Z)J(Z∗)X,
X∈ V, Z, Z∗∈ Z;
R(Z, Z∗)Z∗∗ = 0, Z, Z∗, Z∗∗ ∈ Z.
(3)
And the Rii tensor is dened by
Ric(X, Y ) = 1
2
l
P
k=1hJ(Zk)2X, Y i, X, Y ∈ V;
Ric(X, Z) = 0, X ∈ V, Z ∈ Z;
Ric(Z, Z∗) = −1
4Tr(J(Z)J(Z∗)), Z, Z∗∈ Z.
(4)
Here
dim Z=l
, and
Z1,...,Zl
is an orthonormal basis for
Z
.
2 The Laplaian of the Gauss map
Suppose
dim N= dim N=n+ 1
,
dim Z=n−q+ 1
, where
n
and
q
are
positive integers,
q6n
.
4

Let
M
be a smooth oriented manifold,
dim M=n
. Suppose
M→
N
is an immersion of this manifold in
N
as a hypersurfae, and
η
is the
unit normal vetor eld of
M
in
N
. For eah point
p
of
M
, supp ose that
η(p) = Yn+1 =Xn+1 +Zn+1
, where
Xn+1 ∈ V
,
Zn+1 ∈ Z
. Throughout this
paper, we denote by
Xi, Yi, Zi
elements of
TpN
as well as the orresponding
left invariant vetor elds, whih are elements of
N
. Cho ose an orthonormal
frame
{Y1,...,Yn}
in the vetor spae
TpM⊂TpN
suh that for
16i6q−1
Yi=Xi
,
Yq=Xq−Zq
, and for
q+ 1 6i6n Yi=Zi
, where
X1,...,Xq
are elements of
V
,
Zq,...,Zn
belong to
Z
,
Xn+1 =λXq
,
Zn+1 =µZq
, where
λ>0
and
µ>0
,
|Xq|=|Zn+1|
,
|Zq|=|Xn+1|
. Let
E1,...En
be an
orthonormal frame dened on some neighborhoo d
U
of
p
suh that
Ei(p) = Yi
and
(∇EiEj)T(p) = 0
, for all
i, j = 1,...n
(suh a frame is alled geodesi
at
p
). Here we denote by
(·)T
the pro jetion to
TpM
.
We an rewrite (4) in the following form
Ric(X, Y ) = 1
2
n+1
P
k=qhJ(Zk)2X, Y i, X, Y ∈ V;
Ric(X, Z) = 0, X ∈ V, Z ∈ Z;
Ric(Z, Z∗) = −1
4P
16k6q, k=n+1hJ(Z)J(Z∗)Xk, Xki, Z, Z∗∈ Z.
(5)
In partiular, for all
X, Y ∈ V
P
16i6q, i=n+1hJ([X, Xi])Xi, Y i
=P
16i6q, i=n+1
n+1
P
j=qh[X, Xi], Zjih[Xi, Y ], Zji
=−
n+1
P
j=qP
16i6q, i=n+1hJ(Zj)X, XiihJ(Zj)Y , Xii
=
n+1
P
j=qhJ(Zj)2X, Y i= 2 Ric(X, Y ).
(6)
For
16i, j 6n
, denote by
bij =h∇EiEj, ηi
the oeients of the seond
fundamental form of the immersion, by
kBk
the norm of this form, and by
H
the mean urvature of the immersion on
U
. Sine the frame is orthonormal
over
U
,
H=1
n
n
P
i=1
bii,
kBk2=P
16i,j6n
(bij )2.
(7)
5

Suppose that on
U
η=
n+1
X
j=1
ajYj,
where
{aj}n+1
j=1
are some funtions on
U
. It is lear that
aj(p) = δj n+1
. Then
the Gauss map
G:U→Sn⊂Rn+1
takes the form
G=
n+1
X
j=1
ajYj(e).
In partiular,
G(p) = Yn+1(e)
. Denote by
∆
the Laplaian
∆M
of the indued
metri on
M
.
Theorem 1.
Let
M
be a smooth oriented manifold immersed in a 2-step
nilpotent Lie group
N
as a hypersurfae and
G
be the Gauss map of
M
.
Then, in the above notation
∆G(p) =
q
P
k=1 −Yk(nH) +
q−1
P
j=1hJ([Xk, Xj])Xj, Xn+1i
+ 4hR(Xk, Zn+1)Zn+1, Xn+1i − 2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Xki
+2
q
P
i=1
biq (p)hJ(Zq)Xi, Xki+nH(p)hJ(Zn+1)Xn+1, XkiYk(e)
+
n
P
k=q+1 −Yk(nH)Yk(e)
+ q−1
P
j=1hJ([Xn+1, Xj])Xj, Xn+1i+ 4hR(Xn+1, Zn+1)Zn+1, Xn+1i
−2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Xn+1i+ 2
q
P
i=1
biq (p)hJ(Zq)Xi, Xn+1i
−kBk2(p)−Ric(Yn+1, Yn+1 )Yn+1(e).
(8)
Here
Yk(nH)
denotes the derivative of the funtion
nH
with respet to
the vetor eld
Yk
.
6

P r o o f. Sine the frame
E1,...,En
is geo desi at
p
, the Laplaian at this
point has the form
∆G(p) =
n+1
X
j=1
n
X
i=1
EiEi(aj)Yj(e).
(9)
For
16i6n
we have on
U
∇Eiη=
n+1
X
j=1
Ei(aj)Yj+
n+1
X
j=1
aj∇EiYj,
(10)
∇Ei∇Eiη=
n+1
X
j=1
EiEi(aj)Yj+ 2
n+1
X
j=1
Ei(aj)∇EiYj+
n+1
X
j=1
aj∇Ei∇EiYj.
(11)
Considering this expression at
p
and taking its salar produt with
Yk
for
16k6n+ 1
, we get
h∇Ei∇Eiη, Yki=EiEi(ak) + 2
n+1
X
j=1
Ei(aj)h∇EiYj, Yki+h∇Ei∇EiYn+1, Yki.
Then for salar oeients in (9) we have
n
P
i=1
EiEi(ak) =
n
P
i=1h∇Ei∇Eiη, Yki
−2
n+1
P
j=1
n
P
i=1
Ei(aj)h∇EiYj, Yki −
n
P
i=1h∇Ei∇EiYn+1, Yki.
(12)
For
16k6n
, the rst expression in (7) and the denition of a seond
fundamental form imply at
p
Yk(nH) = Ek n
X
i=1 h∇EiEi, ηi!=
n
X
i=1 h∇Ek∇EiEi, ηi
=
n
X
i=1 hR(Ek, Ei)Ei, ηi+h∇Ei∇EkEi, ηi+h∇[Ek,Ei]Ei, ηi!
=
n
X
i=1 hR(Yk, Yi)Yi, Yn+1i+h∇Ei∇EkEi, ηi!.
7

The seond equality in the equation above follows from the fat that the
projetion
(∇EiEi)T= 0
at
p
and the vetor
∇Ekη
is tangent to
M
. The
fourth equality is a onsequene of
[Ek, Ei] = ([Ek, Ei])T= (∇EkEi− ∇EiEk)T= 0
at
p
. Sine
h[Ek, Ei], ηi= 0
on
U
, and
[Ek, Ei](p) = 0
, at
p
we have
0 = h∇Ei[Ek, Ei], ηi=h∇Ei∇EkEi− ∇Ek∇EiEi, ηi,
for
16i6n
, hene
Yk(nH) =
n
X
i=1 hR(Yk, Yi)Yi, Yn+1i+h∇Ei∇EiEk, ηi!.
(13)
Dierentiating
hEk, ηi= 0
two times with resp et to
Ei
(here we put
16k, i 6n
) and using
h∇EiEk,∇Eiηi(p) = 0
, we derive from (13)
n
P
i=1h∇Ei∇Eiη, Yki=−
n
P
i=1h∇Ei∇EiEk, ηi
=−Yk(nH) +
n
P
i=1hR(Yk, Yi)Yi, Yn+1i=−Yk(nH) + Ric(Yk, Yn+1).
(14)
For
16i6n
, dierentiating
hη, ηi= 1
two times with respet to
Ei
, we get
2h∇Ei∇Eiη, ηi+2h∇Eiη, ∇Eiηi= 0
. This equation and the seond expression
in (7) imply at
p
n
P
i=1h∇Ei∇Eiη, Yn+1i=−
n
P
i=1h∇Eiη, ∇Eiηi
=−
n
P
i=1
n
P
j=1h∇Eiη, Ejih∇Eiη, Eji
=−P
16i,j6nh∇EiEj, ηi2=−kBk2(p).
(15)
Consider the salar produts
h∇Eiη, Yki
at the p oint
p
, for
16i6n
,
16k6n+ 1
. As
|η|=|Yn+1|= 1
, we obtain from (10)
0 = h∇Eiη, ηi(p) = h∇Eiη, Yn+1 i=Ei(an+1) + h∇EiYn+1, Yn+1i=Ei(an+1).
For
16k6n
,
hEk, ηi= 0
imply
8

bik(p) = h∇EiEk, ηi(p) = −h∇Eiη, Eki(p) = −h∇Eiη, Yki
=−Ei(ak)− h∇EiYn+1, Yki.
Hene at
p
we have
−2
n+1
X
j=1
n
X
i=1
Ei(aj)h∇EiYj, Yki
= 2
n
X
j=1
n
X
i=1 bij (p) + h∇YiYn+1, Yji!h∇YiYj, Yki.
It follows from (2) that for
16i, j 6q
the expression
bij (p)h∇XiXj, Yki
is skew-symmetri with resp et to
i, j
; hene the sum of suh terms with
respet to
i
and
j
vanishes. Sum up other expressions using the symmetry
∇XZ=∇ZX
for all
X∈ V
,
Z∈ Z
and the symmetry of the seond
fundamental form. We obtain
−2
n+1
P
j=1
n
P
i=1
Ei(aj)h∇EiYj, Yki=−2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Yki
+2
q
P
i=1
biq(p)hJ(Zq)Xi, Yki+ 2 P
16i,j6nh∇YiYn+1, Yjih∇YiYj, Yki.
(16)
Now we an omplete the pro of of the theorem using the following tehnial
lemmas.
Lemma 2.
The last summand on the right hand side of
(16)
is equal to
2P
16i,j6nh∇YiYn+1, Yjih∇YiYj, Yki
=












2 Ric(Yk, Yn+1) + 4hR(Xk, Zn+1)Zn+1, Xn+1i,16k6q−1;
2 Ric(Xq, Xn+1) + 2 Ric(Zq, Zn+1)
+4hR(Xq, Zn+1)Zn+1, Xn+1i
−4hR(Xn+1, Zq)Zn+1, Xn+1i,
k=q;
−2 Ric(Yk, Yn+1)
+4hR(Xn+1, Zk)Zn+1, Xn+1i,q+ 1 6k6n;
2 Ric(Xn+1, Xn+1)−2 Ric(Zn+1, Zn+1)
+8hR(Xn+1, Zn+1)Zn+1, Xn+1i,k=n+ 1.
(17)
9

Lemma 3.
The last summand on the right hand side of
(12)
an be redued
to the form
−
n
P
i=1h∇Ei∇EiYn+1, Yki
=








nH(p)hJ(Zn+1)Xn+1, Xki − Ric(Yk, Yn+1),16k6q−1;
−Ric(Xq, Xn+1)−Ric(Zq, Zn+1 )
+4hR(Xn+1, Zq)Zn+1, Xn+1i,k=q;
Ric(Yk, Yn+1)−4hR(Xn+1, Zk)Zn+1, Xn+1i, q + 1 6k6n;
−Ric(Xn+1, Xn+1) + Ric(Zn+1, Zn+1)
−4hR(Xn+1, Zn+1)Zn+1, Xn+1i,k=n+ 1.
(18)
Now, if we ombine (12) with (16), (17), (18), and (6), we get (8).
P r o o f o f L e m m a 2. For
16k6q−1
from the expressions for the
Riemannian onnetion we get
n
X
i=1
n
X
j=1 h∇YiYn+1, Yjih∇YiYj, Xki=
q−1
X
i=1 h1
2[Xi, Xn+1],−Zqi
+h−1
2J(Zn+1)Xi, Xqi!h1
2J(Zq)Xi, Xki
+
q−1
X
i=1
n
X
j=q+1h1
2[Xi, Xn+1], Zjih−1
2J(Zj)Xi, Xki
+
q−1
X
j=1 h−1
2J(Zn+1)Xq+1
2J(Zq)Xn+1, Xjih1
2J(Zq)Xj, Xki
+ h1
2J(Zq)Xn+1 −1
2J(Zn+1)Xq, Xqi+h1
2[Xq, Xn+1],−Zqi!hJ(Zq)Xq, Xki
+
n
X
j=q+1h1
2[Xq, Xn+1], Zjih−1
2J(Zj)Xq, Xki
+
n
X
i=q+1
q−1
X
j=1 h−1
2J(Zi)Xn+1, Xjih−1
2J(Zi)Xj, Xki
10

+
n
X
i=q+1h−1
2J(Zi)Xn+1, Xqih−1
2J(Zi)Xq, Xki.
The skew-symmetry of
J
implies
hJ(Z)Xq, Xn+1i=−hJ(Z)Xn+1, Xqi= 0
for all
Z∈ Z
, and
[Xq, Xn+1] = 0
. Hene we an rewrite the above expression
in the form
n
X
i=1
n
X
j=1 h∇YiYn+1, Yjih∇YiYj, Xki
=1
2
n
X
j=qX
16i6q, i=n+1h−J(Zj)Xn+1, XiihJ(Zj)Xk, Xii
=−1
2
n
X
j=qhJ(Zj)Xn+1, J (Zj)Xki=1
2
n
X
j=qhJ(Zj)2Xn+1, Xki
= Ric(Xk, Xn+1) + 2hR(Xk, Zn+1)Zn+1, Xn+1i.
This implies the rst equality in (17).
For
q+ 1 6k6n
we have
n
X
i=1
n
X
j=1 h∇YiYn+1, Yjih∇YiYj, Zki=
q−1
X
i=1
q−1
X
j=1 h−1
2J(Zn+1)Xi, Xjih1
2[Xi, Xj], Zki
+1
2
q−1
X
i=1 h−1
2J(Zn+1)Xi, Xqi+h1
2[Xi, Xn+1],−Zqi!h1
2[Xi, Xq], Zki
+
q−1
X
j=1 h−1
2J(Zn+1)Xq+1
2J(Zq)Xn+1, Xjih1
2[Xq, Xj], Zki
=1
4
q
X
i=1
q
X
j=1 h−J(Zn+1)Xi, Xjih[Xi, Xj], Zki
=1
4X
16i6q, i=n+1 X
16j6q, j =n+1h−J(Zn+1)Xi, XjihJ(Zk)Xi, Xji
−1
2X
16i6q, i=n+1h−J(Zn+1)Xn+1, XiihJ(Zk)Xn+1 , Xii
11

=1
4X
16i6q, i=n+1h−J(Zn+1)Xi, J(Zk)Xii − 1
2h−J(Zn+1)Xn+1, J(Zk)Xn+1i
=1
4X
16i6q, i=n+1hJ(Zk)J(Zn+1)Xi, Xii − 1
2hJ(Zk)J(Zn+1)Xn+1, Xn+1i
=−Ric(Zk, Zn+1) + 2hR(Xn+1, Zk)Zn+1, Xn+1i.
This ompletes the proof of (17) and of the lemma, as
Yq=Xq−Zq
, and
Yn+1 =Xn+1 +Zn+1
.
P r o o f o f L e m m a 3. Let on
U
Ei=
n+1
X
j=1
cij Yj,
(19)
where
cij
,
16i6n
,
16j6n+ 1
are salar funtions on
U
. Note that
Ei(p) = Yi
, so
cij (p) = δij
. Using (19), we get
∇EiYn+1 =
n+1
P
j=1
cij∇YjYn+1 =1
2
q
P
j=1
cij [Xj, Xn+1]−J(Zn+1)Xj
+1
2ciqJ(Zq)Xn+1 −1
2
n
P
j=q+1
cij J(Zj)Xn+1 −ci n+1J(Zn+1)Xn+1.
(20)
Also, for
16k6n
at
p
we have
∇EkEi=
n+1
X
j=1 Yk(cij)Yj+cij (p)∇YkYj!=
n+1
X
j=1
Yk(cij)Yj+∇YkYi.
(21)
In partiular, at
p
bki(p) = h∇EkEi, ηi(p) = Yk(ci n+1) + h∇YkYi, Yn+1i.
(22)
Considering (21) for
k=i
, pro jeting both sides of it to
TpM
, and using the
properties of the geo desi frame, we get
0 =
n
X
j=1
Yi(cij)Yj+ (∇YiYi)T.
12

For
16i6q−1
and
q+ 1 6i6n∇YiYi= 0
, and
∇YqYq=J(Zq)Xq=
∇YqYqT
, sine
hJ(Zq)Xq, Xn+1 +Zn+1i= 0
. Then, for
16j6q
we obtain
Yi(cij ) = 

0,16i6q−1;
−hJ(Zq)Xq, Yji, i =q;
0, q + 1 6i6n.
We an dedue from (22) and the above onsiderations that for
16i6n
bii(p) = Yi(ci n+1)
. Dierentiate (20) with respet to
Ei
at
p
. For
16i6q−1
we get
∇Ei∇EiYn+1 =−Yi(ci n+1)J(Zn+1)Xn+1 +1
2∇Xi [Xi, Xn+1]−J(Zn+1)Xi!
=−bii(p)J(Zn+1)Xn+1 −1
4J([Xi, Xn+1])Xi−1
4[Xi, J(Zn+1)Xi].
For
i=q
we have
∇Eq∇EqYn+1 =−Yq(ci n+1)J(Zn+1)Xn+1 +
n
X
j=1
Yq(cqj )∇YjYn+1
+1
2∇Yq [Xq, Xn+1]−J(Zn+1)Xq+J(Zq)Xn+1!=−bqq (p)J(Zn+1 )Xn+1
−1
2
q−1
X
j=1 hJ(Zq)Xq, Xji [Xj, Xn+1]−J(Zn+1)Xj!
−1
4[Xq, J(Zn+1)Xq] + 1
4[Xq, J(Zq)Xn+1]−1
4J(Zq)J(Zn+1)Xq+1
4J(Zq)2Xn+1.
For
q+ 1 6i6n
we obtain
∇Ei∇EiYn+1 =−Yi(ci n+1)J(Zn+1)Xn+1 −1
2∇Zi(J(Zi)Xn+1)
=−bii(p)J(Zn+1)Xn+1 +1
4J(Zi)2Xn+1.
Summing up these expressions, we get for
16k6q−1
−
n
X
i=1 h∇Ei∇EiYn+1, Xki=nH(p)hJ(Zn+1)Xn+1, Xki
13

+1
4
q−1
X
i=1 hJ([Xi, Xn+1])Xi, Xki+1
2
q−1
X
j=1 hJ(Zq)Xq, Xjih−J(Zn+1)Xj, Xki
+1
4hJ(Zq)J(Zn+1)Xq, Xki − 1
4hJ(Zq)2Xn+1, Xki − 1
4
n
X
i=q+1hJ(Zi)2Xn+1, Xki
=nH(p)hJ(Zn+1)Xn+1, Xki − 1
2X
16i6q, i=n+1hJ([Xn+1, Xi])Xi, Xki
=nH(p)hJ(Zn+1)Xn+1, Xki − Ric(Xk, Xn+1).
Here we use the equation
J(Zq)J(Zn+1)Xq=J(Zn+1)2Xn+1
, whih follows
from the onstrution of the frame. Thus we obtain the rst expression in
(18).
For
q+ 1 6k6n
we have
−
n
X
i=1 h∇Ei∇EiYn+1, Zki=1
4
q−1
X
i=1 h[Xi, J(Zn+1)Xi], Zki
+1
2
q−1
X
j=1 hJ(Zq)Xq, Xjih[Xj, Xn+1], Zki
+1
4h[Xq, J(Zn+1)Xq], Zki − 1
4h[Xq, J(Zq)Xn+1], Zki
=−1
4X
16i6q, i=n+1hJ(Zk)J(Zn+1)Xi, Xii+hJ(Zk)J(Zn+1)Xn+1, Xn+1i
= Ric(Zk, Zn+1)−4hR(Xn+1 , Zk)Zn+1, Xn+1 i.
In the above alulation we used the fat that
J(Zq)Xq=J(Zn+1)Xn+1
and
[Xq, J(Zq)Xn+1] = [Xn+1 , J(Zn+1)Xn+1]
. As
Yq=Xq−Zq
and
Yn+1 =
Xn+1 +Zn+1
, we get the last three equalities in (18).
3 Mean urvature and harmoniity
Consider the tangent bundle
T N
and the distribution in
T N
formed
by left invariant vetor elds from
Z
. Sine
Z
is an abelian ideal, we an
integrate this distribution and obtain a foliation. Denote this foliation by
FZ
. Let G b e harmoni. Sine by (8), in this ase
Yk(nH) = 0
for all
q+ 1 6k6n
, we have
14

Corollary 4.
If the Gauss map of
M
is harmoni, then for eah leaf
M′
of
FZ
the mean urvature of the immersion is onstant on
M∩M′
.
Now we obtain some analogues of the results for Lie groups with bi-
invariant metris that were stated in [6℄.
Let
ν
be a vetor eld on
M
dened by
ν(p) = Yq
, for
p∈M
. In other
words, we obtain
ν(p)
rotating the unit normal vetor
η(p)
by the angle
π
2
in
the 2-plane ontaining
η(p)
and orthogonal to both
dLp(V)
and
dLp(Z)
.
Proposition 5.
Let
M
be a ompat smooth oriented hypersurfae in a
2
-step
nilpotent Lie group
N
. Assume that
(i). the mean urvature of
M
is onstant on the integral urves of
ν
;
(ii). the Gauss map of
M
is harmoni;
(iii).
kBk2+ Ric(η, η)>0
on
M
and
kBk2+ Ric(η, η)>0
in some point of
M
;
(iv). the set of points
p∈M
suh that
η(p)/∈dLp(V)
is dense in
M
.
Then
G(M)
is ontained in a losed hemisphere of
Sn
if and only if
G(M)
is ontained in a great sphere of
Sn
.
P r o o f. One of the impliations in the proposition is obvious. Suppose that
some losed hemisphere of
Sn
ontains
G(M)
, i.e., there exists a unit vetor
v∈Rn+1
suh that for all
p∈MhG(p), vi
is nonp ositive. Consider a smo oth
funtion
f=hG, vi
on
M
. The oeient of
Yq(e)
in (8) vanishes. For all
points from some dense set of
M
we have
Xq6= 0
and thus
Xn+1 =|Xn+1|
|Xq|Xq
.
This, together with
Yq(nH) = 0
, implies that the oeient of
Yn+1(e)
is
equal to
−kBk2−Ric(η, η)
on the dense subset of
M
and hene on the whole
M
beause both the o eient and
−kBk2−Ric(η, η)
are ontinuous. Taking
the salar produt of (8) with
v
, we obtain
∆f=−kBk2+ Ric(η, η)f>0.
Then
f
is a subharmoni funtion on the ompat manifold
M
. Thus
f
is
onstant, and
kBk2+ Ric(η, η)f=−∆f= 0
. From the hypothesis, this
implies
f= 0
, hene
G(M)
is ontained in the equator
v⊥
. This ompletes
the pro of.
15

Proposition 6.
Suppose that a smooth oriented hypersurfae
M
in a
2
-
step nilpotent Lie group
N
is CMC, its Gauss map is harmoni, for all
p
from some dense set of
M
the normal vetor
η(p)/∈dLp(V)
, and
G(M)
is
ontained in an open hemisphere of
Sn
. Then
M
is stable.
P r o o f. From the hypothesis, there exists
v∈Rn+1
suh that for all
p∈M
hG(p), vi>0
. As in the proof of Proposition 5, onsider a salar funtion
w(p) = hG(p), vi
on
M
. This funtion is smooth and positive. As above, (8)
implies the Jaobi equation
(∆ + kBk2+ Ric(η, η)) w= 0
. Now [7, Theorem
1℄ implies the stability of
M
.
4 Groups of Heisenberg type
Let
N
be a group of Heisenberg type. Then from (5), for all
X, Y ∈ V
,
Ric(X, Y ) = 1
2
n+1
X
k=qhJ(Zk)2X, Y i=−1
2(n+ 1 −q)hX, Y i.
Also, we an rewrite the oeients in (8) for
16k6q
and for
k=n+ 1
in the form
q−1
X
j=1 hJ([Xk, Xj])Xj, Xn+1i+ 4hR(Xk, Zn+1)Zn+1, Xn+1i
=

0,16k6q−1;
|Zn+1||Xn+1|q−n−1 + |Zn+1 |2, k =q;
|Xn+1|2q−n−1 + |Zn+1|2, k =n+ 1.
Moreover,
Ric(Zn+1, Zn+1) = −1
4Tr J(Zn+1)2=q
4|Zn+1|2,
and thus
Ric(Yn+1, Yn+1) = q
4|Zn+1|2−1
2(n+ 1 −q)|Xn+1|2.
16

Equation (8) now takes the form
∆G(p) =
q−1
P
k=1 −Yk(nH)−2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Xki
+ 2
q
P
i=1
biq (p)hJ(Zq)Xi, Xki+nH(p)hJ(Zn+1)Xn+1, XkiYk(e)
+−Yq(nH) + |Zn+1||Xn+1|q−n−1 + |Zn+1|2
−2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Xqi
+2
q
P
i=1
biq (p)hJ(Zq)Xi, Xqi+nH(p)hJ(Zn+1)Xn+1, XqiYq(e)
+
n
P
k=q+1 −Yk(nH)Yk(e)
+ −2
q
P
i=1
n
P
j=q+1
bij (p)hJ(Zj)Xi, Xn+1i
+ 2
q
P
i=1
biq (p)hJ(Zq)Xi, Xn+1i − kBk2(p)−q
4|Zn+1|2
+|Xn+1|21
2(q−n−1) + |Zn+1|2Yn+1(e).
(23)
Consider the ase
n=q
, i.e.,
dim Z= 1
. It is easy to see that
n
is then
even,
n= 2m
, where
m
is a positive integer, and
N
is isomorphi to the
2m+ 1
-dimensional Heisenberg group (reall that
N
is onneted and simply
onneted).
In this ase at
p
we an hoose
X1,...,X2m+1
so that
J(Z)Xi=Xm+i,16i6m−1;
J(Z)Xm=X2m
|X2m|=X2m
|Z2m+1|
if
Z2m+1 6= 0;
or
X2m+1
|X2m+1|
if
X2m+1 6= 0;
J(Z)Xm+i=−Xi,16i6m−1;
J(Z)X2m=−|X2m|Xm=−|Z2m+1|Xm;
J(Z)X2m+1 =−|X2m+1|Xm.
17

Choose
Z2m=|X2m+1|Z
and
Z2m+1 =|Z2m+1|Z
. Then (23) has the form
∆G(p) = −
m−1
P
k=1 Yk(2mH) + 2b2m m+k(p)|X2m+1|Yk(e)
−Ym(2mH) + 2mH(p)|X2m+1|
+ 2b2m2m(p)|X2m+1||Z2m+1|Ym(e)
−
m−1
P
k=1 Ym+k(2mH)−2b2m k(p)|X2m+1 |Yk(e)
−Y2m(2mH) + |X2m+1|3|Z2m+1|
−2b2m m(p)|X2m+1||Z2m+1|Y2m(e)
−kBk2(p) + m
2|Z2m+1|2−1
2|X2m+1|2
+|X2m+1|4−2b2m m (p)|X2m+1|2Y2m+1(e).
(24)
Consider an example of the three-dimensional Heisenberg group
Nil
. In
the spae
R3
with Cartesian oordinates
(x, y, z)
, dene vetor elds
X=∂
∂x , Y =∂
∂y +x∂
∂z , Z =∂
∂z .
Then
Span(X, Y, Z)
is a Lie algebra (with the only nonzero braket
[X, Y ] =
Z
), whih is the Lie algebra of
Nil
. Introdue a salar pro dut in suh a way
that the vetors
X, Y
and
Z
are orthonormal. Consider the following unit
vetor eld:
η=xY +Z
√1 + x2,
and vetor elds
F1=X, F2=Y−xZ
√1 + x2,
whih are orthogonal to
η
. In the notation of setion 2, in eah
p F1=X1
,
F2=X2−Z2
,
η=X3+Z3
. By diret omputation of ovariant derivatives
it an be shown that the distribution spanned by
F1
and
F2
is integrable and
form the tangent bundle of some two-dimensional foliation
F
in
Nil
. From
the omputation of the seond fundamental form we obtain
kBk2=(x2−1)2
2(1+x2)2
,
18

and
H= 0
. Thus the leaves of this foliation are minimal surfaes. The
Laplaian on
G=η
is
∆G=F1F1+F2F2−(∇F1F1)T−(∇F2F2)TG
=−x
(1 + x2)2F2−1
(1 + x2)2η.
We obtain the same result onsidering (24) at some
p
. In fat,
2b22 |X3||Z3|= 0;
|X3|3|Z3| − 2b21 |X3||Z3|=x
(1 + x2)2;
kBk2+1
2|Z3|2−1
2|X3|2+|X3|4−2b21 |X3|2=1
(1 + x2)2.
In partiular, foliation
F
gives an example of a CMC-surfae in
Nil
suh
that its Gauss map is not harmoni.
Proposition 7.
Suppose that
M
is a smooth oriented
2m
-dimensional man-
ifold immersed in the
2m+ 1
-dimensional Heisenberg group. If any two of
the following three laims are true, then the third one is also true.
(i).
M
is CMC;
(ii). the Gauss map of
M
is harmoni;
(iii). at every point of
M
, the fol lowing holds:













b2m k = 0,16k6m−1, m + 1 6k62m−1;
|Z2m+1||X2m+1|2−2b2m m= 0;
|Z2m+1|(b1 1 +···+b2m−1 2m−1+ 3b2m2m) = 0.
(25)
Here
bij
,
16i, j 62m
are the oeients of the seond fundamental
form of
M
in the basis hosen as above.
19

P r o o f. If
(iii)
is true, then the equivaleny of
(i)
and
(ii)
immediately
follows from 24. Supp ose
(i)
and
(ii)
are true. Let
A
be a set of suh p oints
of
M
that
|X2m+1| 6= 0
. At the points of
A
24 implies the expressions in
(25) . Sine the distribution orthogonal to
Z
is non-integrable,
A
is dense in
M
. Now the ontinuity of the left hand sides of the equations (25) implies
(iii)
.
In the ase
m= 1
the next theorem shows that the restritions for
M
arising from (25) are rather strit.
Theorem 8.
Let
M
be a smooth oriented CMC-surfae in the Heisenberg
group
Nil
whose Gauss map is harmoni. Then
M
is a ylinder, that is,
its position vetor in the oordinates
x
,
y
,
z
has the form
r(s, t) = (f1(s), f2(s), t),
(26)
where
f1
and
f2
are some smooth funtions.
P r o o f. For eah
p∈M
denote
a(p) = |X3|
,
b(p) = |Z3|
. Then
a
and
b
are
smooth salar funtions on
M
, and
a2+b2= 1
. Consider an arbitrary p oint
p
of
M
. Choose
X1
as above and put
X2=J(Z)X1
. Denote by
T1
and
T2
the vetor elds that at eah
p∈M
are equal to
X1
and
X2
respetively.
Consider unit tangent vetor elds
F1
and
F2
, and a unit normal vetor eld
η
of
M
of the form
F1=T1, F2=bT2−aZ, η =aT2+bZ.
Denote by
κ1
and
κ2
the geodesi urvatures of the integral urves of
F1
and
F2
respetively. In other words,
∇F1F1=κ1F2,∇F1F2=−κ1F1,∇F2F1=−κ2F2,∇F2F2=κ2F1,
(27)
where
∇
is the Riemannian onnetion on
M
indued by the immersion. The
Gaussian urvature of the surfae is
K=F1(κ2) + F2(κ1)−(κ1)2−(κ2)2.
(28)
Assume that for some
p∈M a(p)6= 0
and
b(p)6= 0
. Then
ab 6= 0
on
some neighborhood
U
of
p
. Then (25) implies that on
U
the matrix of the
seond fundamental form of
M
is
3H1
2a2
1
2a2−H.
(29)
20

In partiular, the extrinsi urvature
Kext
of the surfae is
−3H2−1
4a4
.
Denote by
B
the seond fundamental form of the immersion. Then the
Codazzi equations for
M
are
(∇F1B) (F2, F1)−(∇F2B) (F1, F1) = hR(F1, F2)F1, ηi=ab;
(∇F2B) (F1, F2)−(∇F1B) (F2, F2) = hR(F2, F1)F2, ηi= 0.
Computing the ovariant derivatives of the seond fundamental form, we
obtain for
UaF1(a) + 4Hκ1−a2κ2−ab = 0,
aF2(a)−4Hκ2−a2κ1= 0.
(30)
The Gauss equation has the form
K=Kext +hR(F1, F2)F2, F1i=−3H2−1
4a4−3
4b2+1
4a2.
From (28) we obtain
F1(κ2) + F2(κ1)−(κ1)2−(κ2)2=−3H2−1
4a4−3
4b2+1
4a2.
(31)
Using (30) and the form of
F2
and
η
, we an derive
h∇F1F2, ηi=−hF2,∇F1ηi=−hF2,∇F1a
bF2+a2
b+bZi
=−F1a
bhF2, F2i − F11
bhF2, Zi − a
bhF2,∇F1F2i − 1
bhF2,∇F1Zi
=−F1a
b+aF11
b−1
bhF2,−1
2T2i=−1
b−4Hκ1
a+aκ2+b+1
2
=−1
2+4Hκ1
ab −a
bκ2;
h∇F2F1, ηi=−hF1,∇F2ηi=−hF1,∇F2a
bF2+a2
b+bZi
=−a
bhF1,∇F2F2i − 1
bhF1,∇F2Zi=−a
bκ2−1
bhT1,1
2bT1i=−a
bκ2−1
2.
21

In the above equations we used the fat that
Z
is left invariant and the
expressions (2) for the ovariant derivative. Sine
ab 6= 0
, the integrability
ondition
h[F1, F2], ηi= 0
takes the form
Hκ1= 0
. Besides, (29) imply
3H=b11 =h∇F1F1, ηi=−hF1,∇F1ηi
=−hF1,∇F1a
bF2+a2
b+bZi=−a
bhF1,∇F1F2i−1
bhF1,∇F1Zi=a
bκ1.
Thus
H=κ1= 0
. In partiular,
∇F1F1= 0
, hene
T1=F1
is a geodesi
vetor eld in the ambient manifold. Note that
T1
belongs to the distri-
bution that spans the left invariant vetor elds of
V
. Considering the set
of geodesis in
Nil
(see [3, proposition (3.1), proposition (3.5)℄), we obtain
that
T1=cX +dY
, where
c, d ∈R
are some onstants, i.e.,
T1=X1
and
T2=X2
are left invariant. Note that the seond equation of (30) implies
F2(a) = F2(b) = 0
. Thus we obtain
∇F2F2=∇F2(bX2−aZ) = b∇bX2−aZ X2−a∇bX2−aZZ=−abX1.
Therefore
κ2=−ab
. It follows from this equation, from the omputations
above in this proof, and from (29) that
1
2a2=b12 =h∇F1F2, ηi=−a
bκ2−1
2=a2−1
2,
and
a2=b2=1
2
. But then
a=b=√2
2
, and the rst equation in (30) implies
aκ2+b= 0
, whih leads to a ontradition.
Thus
ab = 0
at eah point of
M
. Sine
a2+b2= 1
and
a
,
b
are ontinuous,
a= 1
or
b= 1
identially. The latter ase is impossible beause
Z⊥
is not
integrable; then the normal vetor of
M
is orthogonal to
Z
, and
F2=−Z
.
Therefore
M
is invariant under the ation of
Z
by left translations, and
M
is formed by integral urves of
Z
, whih are geo desis
(0,0, t)
. Then
M
has
the form (26).
Note that similar result for another denition of the Gauss map was
obtained in [13℄. Also, in [13℄ the equations of the CMC-surfaes of the form
(26) were obtained. Proposition 7 then implies that the Gauss maps of all
these surfaes are harmoni.
22

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23

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24