Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

0.11 MB·

by  Nobumitsu Nakauchi

BIHARMONIC HYPERSURFACES IN A RIEMANNIAN
MANIFOLD WITH NON-POSITIVE RICCI
CURVATURE
2
1
0 ∗
NOBUMITSU NAKAUCHI AND HAJIME URAKAWA
2

n
a
J
1 Abstract. In this paper, we show that, for a biharmonic hyper-
3 surface (M,g) of a Riemannian manifold (N,h) of non-positive
] Riccicurvature,if M|H|2vg <∞,whereH isthemeancurvature
G of (M,g) in (N,h)R, then (M,g) is minimal in (N,h). Thus, for a
counter example (M,g) in the case of hypersurfaces to the gener-
D
alizedChen’sconjecture(cf. Sect.1),itholdsthat |H|2v =∞.
. M g
h R
t
a 1. Introduction and statement of results.
m
[ In this paper, we consider an isometric immersion ϕ : (M,h) →

(N,h), of a Riemannian manifold (M,g) of dimension m, into another
2
v Riemannian manifold (N,h) of dimension n = m+1. We have
8
4 ∇Nϕ∗Xϕ∗Y = ϕ∗(∇XY)+k(X,Y)ξ,
1
3 for vector fields X and Y on M, where ∇, ∇N are the Levi-Civita
.
1 connections of (M,g) and (N,h), respectively, ξ is the unit normal
0
vector field along ϕ, and k is the second fundamental form. Let A :
1
1 TxM → TxM (x ∈ M) be the shape operator defined by g(AX,Y) =
: k(X,Y), (X,Y ∈ T M), and H, the mean curvature defined by H :=
v x
i 1Tr (A). Then, let us recall the following B.Y. Chen’s conjectrure
X m g
(cf. [3], [4]):
r
a
Let ϕ : (M,g) → (Rn,g ) be an isometric immersion into the stan-
0
dard Euclidean space. If ϕ is biharmonic (see Sect. 2), then, it is
minimal.
This conjecture is still open up to now, and let us recall also the
following generalized B.Y. Chen’s conjecture (cf. [3], [2]):
2000 Mathematics Subject Classification. 58E20.
Key words and phrases. harmonic map, biharmonic map, isometric immersion,
minimal, Ricci curvature.
∗
Supportedbythe Grant-in-Aidforthe Scientific Research,(C), No. 21540207,
Japan Society for the Promotionof Science.
1
2 NOBUMITSU NAKAUCHIAND HAJIMEURAKAWA∗
Let ϕ : (M,g) → (N,h) be an isometric immersion, and the sec-
tional curvature of (N,h) is non-positive. If ϕ is biharmonic, then, it
is minimal.
Oniciuc ([9]) and Ou ([11]) showed this is true if H is constant.
In this paper, we show
Theorem 1.1. Assume that (M,g) is complete and the Ricci tensor
RicN of (N,h) satisfies that
RicN(ξ,ξ) ≤ |A|2. (1.1)
If ϕ : (M,g) → (N,h) is biharmonic (cf. Sect. 2) and satisfies that
H2v < ∞, (1.2)
Z g
M
then, ϕ has constant mean curvature, i.e., H is constant.
As a direct corollary, we have
Corollary 1.2. Assume that (M,g) is a complete Riemannian mani-
fold of dimension m and (N,h) is a Riemannian manifold of dimension
m+1 whose Ricci curvature is non-positive. If an isometric immersion
ϕ : (M,g) → (N,h) is biharmonic and satisfies that H2v < ∞,
M g
then, ϕ is minimal. R
ByourCorollary1.2, iftherewould exist a counter example (cf. [11])
in the case dimN = dimM +1, then it must hold that
H2v = ∞, (1.3)
Z g
M
which imposes the strong condition on the behaviour of the boundary
of M at infinity. Indeed, (1.3) implies that either H is unbounded on
M, or it holds that H2 ≥ C on an open subset Ω of M with infinite
volume, for some constant C > 0.
Acknowledgement. We express our thanks to thereferee(s) who
informed relevant references and gave useful comments to us.
BIHARMONIC HYPERSURFACES 3
2. Preliminaries.
In this section, we prepare general materials about harmonic maps
and biharmonic maps of a complete Riemannian manifold into another
Riemannian manifold (cf. [5]).
Let (M,g) be anm-dimensional complete Riemannian manifold, and
the target space (N,h) is an n-dimensional Riemannian manifold. For
every C∞ map ϕ of M into N, and relatively compact domain Ω in
M, the energy functional on the space C∞(M,N) of all C∞ maps of
M into N is defined by
1
E (ϕ) = |dϕ|2v ,
Ω 2 Z g
Ω
and for a C∞ one parameter deformation ϕ ∈ C∞(M,N) (−ǫ < t < ǫ)
t
of ϕ with ϕ = ϕ, the variation vector field V along ϕ is defined by
0
V = d ϕ . Let Γ (ϕ−1TN) be the space of C∞ sections of the
dt(cid:12)t=0 t Ω
induced(cid:12)bundle ϕ−1TN of the tangent bundle TN by ϕ whose sup-
(cid:12)
ports are contained in Ω. For V ∈ Γ (ϕ−1TN) and its one-parameter
Ω
deformation ϕ , the first variation formula is given by
t
d
E (ϕ ) = − hτ(ϕ),Viv .
dt(cid:12)(cid:12) Ω t ZΩ g
(cid:12)t=0
(cid:12)
The tension field τ(ϕ(cid:12)) is defined globally on M by
m
τ(ϕ) = B(ϕ)(e ,e ), (2.1)
i i
X
i=1
where
B(ϕ)(X,Y) = ∇N ϕ (Y)−ϕ (∇ Y)
ϕ∗(X) ∗ ∗ X
for X,Y ∈ X(M). Then, a C∞ map ϕ : (M,g) → (N,h) is harmonic
if τ(ϕ) = 0. For a harmonic map ϕ : (M,g) → (N,h), the second
variation formula of the energy functional E (ϕ) is
Ω
d2
E (ϕ ) = hJ(V),Viv
dt2(cid:12)(cid:12) Ω t ZΩ g
(cid:12)t=0
(cid:12)
where (cid:12)
J(V) := ∆V −R(V),
m
∗
∆V := ∇ ∇V = − {∇ (∇ V)−∇ V},
ei ei ∇eiei
X
i=1
m
R(V) := RN(V,ϕ (e ))ϕ (e ).
∗ i ∗ i
X
i=1
4 NOBUMITSU NAKAUCHIAND HAJIMEURAKAWA∗
Here, ∇ is the induced connection on the induced bundle ϕ−1TN,
and RN is the curvature tensor of (N,h) given by RN(U,V)W =
[∇N ,∇N ]W −∇N W (U,V,W ∈ X(N)).
U V [U,V]
The bienergy functional is defined by
1
E (ϕ) = |τ(ϕ)|2v ,
2,Ω 2 Z g
Ω
and the first variation formula of the bienergy is given (cf. [7]) by
d
E (ϕ ) = − hτ (ϕ),Viv
dt(cid:12)(cid:12) 2,Ω t ZΩ 2 g
(cid:12)t=0
(cid:12)
where the bitension fi(cid:12)eld τ (ϕ) is defined globally on M by
2
τ (ϕ) = J(τ(ϕ)) = ∆τ(ϕ)−R(τ(ϕ)), (2.2)
2
and a C∞ map ϕ : (M,g) → (N,h) is called to be biharmonic if
τ (ϕ) = 0. (2.3)
2
3. Some Lemma for the Schro¨dinger type equation
In this section, we prepare some simple lemma of the Schro¨dinger
type equation of the Laplacian ∆ on an m-dimensional non-compact
g
complete Riemannian manifold (M,g) defined by
m
∞
∆ f := e (e f)−∇ e f (f ∈ C (M)), (3.1)
g i i ei i
X
i=1
where {e }m is a locally defined orthonormal frame field on (M,g).
i i=1
Lemma 3.1. Assume that (M,g) is a complete non-compact Riemann-
ian manifold, and L is a non-negative smooth function on M. Then,
every smooth L2 function f on M satisfying the Schr¨odinger type equa-
tion
∆ f = Lf (on M) (3.2)
g
must be a constant.
Proof. Take any point x in M, and for every r > 0, let us consider the
0
following cut-off function η on M:
0 ≤ η(x) ≤ 1 (x ∈ M),

η(x) = 1 (x ∈ B (x )),
η(x) = 0 (x 6∈ B2rr(x00)), (3.3)
2
|∇η| ≤ r (on M),
BIHARMONIC HYPERSURFACES 5
where B (x ) = {x ∈ M : d(x,x ) < r}, and d is the distance of
r 0 0
(M,g). Multiply η2f on (3.2), and integrale it over M, we have
(η2f)∆ f v = Lη2f2v . (3.4)
Z g g Z g
M M
By the integration by part for the left hand side, we have
(η2f)∆ f v = − g(∇(η2f),∇f)v . (3.5)
Z g g Z g
M M
Here, we have
g(∇(η2f),∇f) = 2ηf g(∇η,∇f)+η2g(∇f,∇f)
= 2ηf h∇η,∇fi+η2|∇f |2, (3.6)
where we use h·, ·i and | · | instead of g(·, ·) and g(u,u) = |u|2
(u ∈ T M), for simplicity. Substitute (3.6) into (3.5), the right hand
x
side of (3.5) is equal to
RHSof (3.5) = − 2ηf h∇η,∇fiv − η2|∇f|2v
Z g Z g
M M
= −2 hf ∇η,η∇f iv − η2|∇f|2v .
Z g Z g
M M (3.7)
Here, applying Young’s inequality: for every ǫ > 0, andevery vectors
X and Y at each point of M,
1
±2hX,Yi ≤ ǫ|X|2 + |Y|2, (3.8)
ǫ
to the first term of (3.7), we have
1
RHSof (3.7) ≤ ǫ |η∇f|2v + |f ∇η|2v − η2|∇f|2v
Z g ǫ Z g Z g
M M M
1
= −(1−ǫ) η2|∇f|2v + f2|∇η|2v .
Z g ǫ Z g
M M (3.9)
Thus, by (3.5) and (3.9), we obtain
1
Lη2f2v +(1−ǫ) η2|∇f|2v ≤ f2|∇η|2v .
Z g Z g ǫ Z g
M M M (3.10)
Now, puttig ǫ = 1, (3.10) implies that
2
1
Lη2f2v + η2|∇f|2v ≤ 2 f2|∇η|2v .
Z g 2 Z g Z g
M M M (3.11)
6 NOBUMITSU NAKAUCHIAND HAJIMEURAKAWA∗
Since η = 1 on B (x ) and |∇η| ≤ 2, and L ≥ 0 on M, we have
r 0 r
1 8
0 ≤ Lf2v + |∇f|2v ≤ f2v .
Z g 2 Z g r2 Z g
Br(x0) Br(x0) M (3.12)
Since (M,g) is non-compact and complete, r can tend to infinity, and
B (x ) goes to M. Then we have
r 0
1
0 ≤ Lf2v + |∇f|2v ≤ 0 (3.13)
Z g 2 Z g
M M
since f2v < ∞. Thus, we have Lf2 = 0 and |∇f| = 0 (on M)
M g
whichRimplies that f is a constant. (cid:3)
4. Biharmonic isometric immersions.
Inthissection, weconsiderahypersurfaceM ofan(m+1)-dimensional
Riemannian manifold (N,h). Recently, Y-L. Ou showed (cf. [10])
Theorem 4.1. Let ϕ : (M,g) → (N,h) be an isometric immersion
of an m-dimensional Riemannian manifold (M,g) into another (m +
1)-dimensional Riemannian manifold (N,h) with the mean curvature
vector field η = Hξ, where ξ is the unit normal vector field along ϕ.
Then, ϕ is biharmonic if and only if the following equations hold:
∆ H −H |A|2 +HRicN(ξ,ξ) = 0,
g
 m (4.1)
2A(∇H)+ ∇(H2)−2H(RicN(ξ))T = 0,
2
where RicN: T N → T N is the Ricci transform which is defined by
y y
h(RicN(Z),W) = RicN(Z,W) (Z,W ∈ T N), (·)T is the tangential
y
component corresponding to the decomposition of T N = ϕ (T M)⊕
ϕ(x) ∗ x
Rξ (x ∈ M), and ∇f is the gradient vector field of f ∈ C∞(M) on
x
(M,g), respectively.
Due to Theorem 4.1 and Lemma 3.1, we can show immediately our
Theorem 1.1.
(Proof of Theorem 1.1.)
Let us denote by L := |A|2 − RicN(ξ,ξ) which is a smooth non-
negativefunctiononM duetoourassumption. Then, thefirstequation
is reduced to the following Schro¨dinger type equation:
∆ f = Lf, (4.2)
g
where f := H is a smooth L2 function on M by the assumption (1.2).
BIHARMONIC HYPERSURFACES 7
Assume that M is compact. In this case, by (4.2)andthe integration
by part, we have
0 ≤ Lf2v = f (∆ f)v = − g(∇f,∇f)v ≤ 0,
Z g Z g g Z g
M M M (4.3)
which implies that g(∇f,∇f)v = 0, that is, f is constant.
M g
Assume that M iRs non-compact. In this case, we can apply Lemma
3.1 to (4.2). Then, we have that f = H is a constant. (cid:3)
(Proof of Corollary 1.2.)
Assume that RicN is non-positive. Since L = |A|2 − RicN(ξ,ξ) is
non-negative, H is constant due to Theorem 1.1. Then, due to (4.1),
we have that HL = 0 and H(RicN(ξ))T = 0. If H 6= 0, then L = 0,
i.e.,
RicN(ξ,ξ) = |A|2. (4.4)
By our assumption, RicN(ξ,ξ) ≤ 0, and the right hand side of (4.4) is
non-negative, so we have |A|2 = 0, i.e., A ≡ 0. This contradicts H 6= 0.
We have H = 0. (cid:3)
References
[1] R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of S3, In-
tern. J. Math., 12 (2001), 867–876.
[2] R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps, Contemp. Math.
288 (2001), 286–290.
[3] B.Y.Chen,Someopenproblems andconjecturesonsubmanifolds offinitetype,
Soochow J. Math., 17 (1991), 169–188.
[4] B.Y. Chen, A report on submanifolds of finite type , Soochow J. Math., 22
(1996), 117–337.
[5] J. Eells and L. Lemaire, Selected topic in harmonic maps, C.M.M.S. Regional
Conf. Series Math., 50 Amer. Math. Soc., Providence, 1983.
[6] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press,
New York and London, 1962.
[7] G. Y. Jiang, 2-harmonic maps and their first and second variation formula,
Chinese Ann. Math 7A (1986), 388–402.
[8] S. KobayashiandK. Nomizu, Foundation of Differential Geometry, Vol. I, II,
(1963), (1969), John Wiley and Sons, New York.
[9] C.Oniciuc,BiharmonicmapsbetweenRiemannianmanifolds,An.St.Al.Univ.
“Al. I. Cuza”, Iasi, Vol. 68, (2002), 237–248.
[10] Ye-Lin Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J.
Math., 248 (2010), 217–232.arXiv: 0901.1507v.3,(2009).
8 NOBUMITSU NAKAUCHIAND HAJIMEURAKAWA∗
[11] Ye-Lin Ou and Liang Tang, The generalized Chen’s conjecture on biharmonic
submanifolds is false, a preprint, arXiv: 1006.1838v.1,(2010).
Graduate School of Science and Engineering,
Yamaguchi University, Yamaguchi, 753-8512, Japan.
E-mail address: [email protected]
Division of Mathematics, Graduate School of Information Sciences,
Tohoku University, Aoba 6-3-09, Sendai, 980-8579, Japan.
Current Address: Institute for International Education,
Tohoku University, Kawauchi 41, Sendai, 980-8576, Japan.
E-mail address: [email protected]