Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
43 (2010), pp. 27–32

RICCI SOLITONS AND LAGRANGIAN SUBMANIFOLDS IN
KÄHLER MANIFOLDSJONG TAEK CHO AND MAKOTO KIMURA(Received: January 23, 2010)
Abstract. In this paper we will study compact Lagrangian submanifold M in
Kähler manifolds and in particular complex space forms, such that the induced
metric on the Lagrangian submanifold is a Ricci soliton with respect to potential
vector ﬁeld given by mean curvature vector ﬁeld and complex structure.

1. Introduction
A Ricci soliton is deﬁned on a Riemannian manifold (M, g) by1(1.1)£V g + Ric −λg = 0,2where V is a vector ﬁeld (the potential vector ﬁeld), λ is a constant. Obviously, atrivial Ricci soliton is an Einstein metric with V zero or Killing. Compact Ricci
solitons are the ﬁxed points of the Ricci ﬂow: ∂g/∂t = −2Ric projected from thespace of metrics onto its quotient modulo diﬀeomorphisms and scalings, and often
arise as blow-up limits for the Ricci ﬂow on compact manifolds. The Ricci soliton
is said to be shrinking, steady, and expanding according as λ > 0, λ = 0 and λ < 0respectively. If the vector ﬁeld V is the gradient of a potential function f , then g
is called a gradient Ricci soliton. Due to Perelman’s result [13, Remark 3.2], weknow that in a compact Ricci soliton, the potential vector ﬁeld is written as thesum of a gradient and a Killing vector ﬁeld. We refer to [3] for details about Ricci
solitons or gradient Ricci solitons.
On the other hand, Lagrangian submanifolds have been important geometricobjects of study in symplectic geometry. The problem of minimizing the volumeof Lagrangian submanifolds under Hamiltonian deformations was proposed by Oh2000 Mathematics Subject Classification. 53C25, 53C40.
Key words and phrases. Lagrangian submanifolds, Ricci soliton, Hamiltonian minimal, complex space forms.
The ﬁrst author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2009-0071643).
The second author was partially supported by Grant-in-Aid for Scientiﬁc Research, The Ministry of Education, Science, Sports and Culture, Japan, No. 21540083.
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J. T. CHO AND M. KIMURA[10], and critical points of the problem, Hamiltonian minimal submanifolds are interesting and important objects among Lagrangian submanifolds (cf. [7]). Using
Oh’s result, we can see that compact Lagrangian submanifold M in a Kähler manf is Hamiltonian minimal if and only if tangent vector ﬁeld JH is divergenceifold Mf and mean curvature vector
free, where J and H denote complex structure of M
ﬁeld of M , respectively. In this paper, we investigate Lagrangian submanifolds,
whose induced metric is a Ricci soliton with potential vector ﬁeld JH, in Kähler
manifolds and in particular in complex space forms. Typical examples of such
Lagrangian submanifolds are: (i) Einstein minimal Lagrangian submanifolds, (ii)
irreducible Lagrangian submanifolds with parallel second fundamental form in Hermitian symmetric spaces (cf. [8], [9]) and (iii) Ricci-ﬂat Lagrangian submanifolds
with parallel mean curvature vector ﬁeld H (cf. [11]).
The primary result is concerned with compact oriented Lagrangian submanifold
f: Suppose that the induced metric g on M is a Ricci
M in a Kähler manifold M
soliton with potential vector ﬁeld JH. If M is Hamiltonian minimal, then M
is Einstein and either M is minimal or the mean curvature vector ﬁeld H of M
is non-zero and parallel with respect to the normal connection ∇⊥ (Theorem 5).
Furthermore if the scalar curvature ρ of (M, g) satisﬁes either ρ ≥ nλ or ρ ≤ nλ,then ρ = nλ and M is Hamiltonian minimal. Secondly, let M be a compact oriented
Lagrangian submanifold M in a complex space form, and suppose the induced
metric g is a Ricci soliton with potential vector ﬁeld JH. With respect to the Ricci
tensor of M , if Ric(JH, JH) ≤ 0, then ∇⊥ H = 0 and M is Ricci-ﬂat (Theorem 7).
Finally for either 2 or 3-dimensional case, using results of Hamilton [5] and Ivey[6], we obtain (Theorem 8): Let M n be a compact Lagrangian submanifold in acomplex space form with n = 2, 3 and let g be the induced metric on M . Suppose(M, g) is a Ricci soliton with V = JH. Then (M, g) is either totally geodesic or
ﬂat with ∇⊥ H = 0.
2. Ricci solitons
On Riemannian manifold (M, g), we have
(2.1)(£V g)(X, Y ) = g(∇X V, Y ) + g(X, ∇Y V ),where ∇ denotes the the Levi-Civita connection of M . Hence if (M, g) is a Ricci
soliton with potential vector ﬁeld V , then (1.1) and (2.1) imply(2.2)

div V + ρ − λn = 0,

where div V = traceg (X 7→ ∇X V ) and ρ denote divergence ∫of V and the scalar
curvature of M , respectively. By (2.2) and Green’s Theorem M div V µg = 0 for acompact oriented Riemannian manifold M (µg is the volume form of (M, g)), we
obtain:
Proposition 1. Let (M, g) be a compact oriented Ricci soliton satisfying (1.1) withrespect to potential vector field V on M . Then∫(i) M ρµg = λn vol(M ). Hence λ is uniquely determined by g.
(ii) The scalar curvature ρ of M is constant if and only if div V = 0.

RICCI SOLITONS AND LAGRANGIAN SUBMANIFOLDS IN KÄHLER MANIFOLDS

29(iii) If either ρ ≤ λn or ρ ≥ λn on M , then ρ = λn and div V = 0.
Now we recall the following results due to Hamilton and Ivey:
Theorem 2. [5], [6] Let (M, g) is either 2 or 3-dimensional compact Ricci soliton
with potential vector field V . Then sectional curvatures of M are constant.
Also the following is known:
Proposition 3. On compact Ricci soliton (M, g), if the scalar curvature is constant, then (M, g) is Einstein.
3. Lagrangian submanifolds in complex space forms
First we recall about Hamiltonian deformation of Lagrangian submanifolds inf be a complex n-dimensional Kähler
Kähler manifolds, deﬁned by Oh [10]. Let M
manifold with Kähler form ω, Riemannian metric ⟨ , ⟩, and complex structure J.
f be a Lagrangian immersion from a real n-dimensional manifold
Let x : M → M
f, i.e., ω|T M = 0. For a vector ﬁeld V along x, we deﬁne a 1-form αV
M to Mon M as αV = ⟨JV, ·⟩|T M . Smooth family of embeddings ιt : M → P is called
Hamiltonian deformation if for the variational vector ﬁeld V , the 1-form αV is
exact. A Lagrangian submanifold M is Hamiltonian minimal (or H-minimal ) if
M is stationary for any Hamiltonian deformation. Oh [10] showed that when M iscompact, M is H-minimal if and only if αH is co-closed, i.e., δαH = 0 where H is
the mean curvature vector ﬁeld of M . We have
(3.1)

M is Hamiltonian minimal ⇔ div JH = 0.

f and the induced
With respect to Lagrangian submanifold M in a Kähler manifold M
metric g on M , the following relations hold:(3.2)∇σ = 0 ⇒ ∇⊥ H = 0 ⇒ £JH g = 0 ⇒ div JH = 0,

where σ and ∇⊥ denote second fundamental form and normal connection of M inf respectively, and ∇σ is deﬁned by (∇X σ)(Y, Z) = ∇⊥ σ(Y, Z) − σ(∇X Y, Z) −M
X
σ(Y, ∇X Z) for tangent vector ﬁelds X, Y, Z on M . We note that ∇X (JH) =J∇⊥
X H.
fn (4c) be an n-dimensional complex space form with constant holomorphic
Let M
fn (4c).
sectional curvature 4c and let M = M n be a Lagrangian submanifold in M
Then Gauss equation isR(X, Y )Z = c{g(Y, Z)X − g(X, Z)Y } + Aσ(Y,Z) X − Aσ(X,Z) Y,
where R and A denote curvature tensor and shape operator of M with
⟨σ(X, Y ), ξ⟩ = g(Aξ X, Y ) for tangent vector ﬁeld X, Y and normal vector ﬁeld
ξ on M , respectively. The Ricci tensor of M is then given by(3.3)

Ric(Y, Z) = (n − 1)cg(Y, Z) + ⟨σ(Y, Z), H⟩ − traceg (X 7→ Aσ(X,Z) Y ).

The scalar curvature ρ of M is(3.4)

ρ = n(n − 1)c + ∥H∥2 − ∥σ∥2 .

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J. T. CHO AND M. KIMURALet T be a symmetric (0, 3)-tensor ﬁeld on M deﬁned by T (X, Y, Z) =⟨σ(X, Y ), JZ⟩. Then Codazzi equation(3.5)(∇X σ)(Y, Z) = (∇Y σ)(X, Z)

implies that
∇T is a symmetric (0, 4)-tensor ﬁeld on M .
Combining with (3.2), we can easily obtain the following local result:
Proposition 4. Let M be a Lagrangian submanifold in a complex space form of
constant holomorphic sectional curvature.Then the mean curvature vector field H
is parallel with respect to the normal connection ∇⊥ if and only if JH is a Killing
vector field on M .
4. Ricci soliton of Lagrangian submanifolds
fn , and let g
Let M = M n be a Lagrangian submanifolds in a Kähler manifold M
be the induced metric on M . Suppose (M, g) is a Ricci soliton (1.1) with V = JH.
By Proposition 1, Proposition 3, (2.2) and (3.1), we obtain:
Theorem 5. Let M n be a compact oriented Lagrangian submanifold in a Kähler
fn and let g be the induced metric on M . Suppose (M, g) is a Ricci
manifold M
soliton with V = JH. Then we have:(i) If M is Hamiltonian minimal, then M is Einstein and JH is a Killing
vector field on M .
(ii) If the scalar curvature ρ of (M, g) satisfies either ρ ≥ nλ or ρ ≤ nλ, then
ρ = nλ and M is Einstein and the same conclusion as (i) holds.
We note that Lagrangian submanifolds M n with parallel second fundamental
form in complex space forms are classiﬁed by Naitoh [8], [9]. By (3.1) and (3.2),
they are Hamiltonian minimal. Also they are locally symmetric and for irreducible
ones, they are Einstein. Hence irreducible Lagrangian submanifolds in complex
space forms satisfying ∇σ = 0 are considered as Ricci solitons with Killing potential
vector ﬁeld JH.
When dimR M = 2 or 3, by Theorem 2, (3.1) and (3.2), we get:
Proposition 6. Let M be a compact Lagrangian submanifold in a Kähler manifold
f with dimC M
f = 2 or 3, and let g be the induced metric on M . Suppose (M, g)
M
is a Ricci soliton with V = JH. Then (M, g) is constant curvature and ∇⊥ H = 0.
fn (4c) be an n-dimensional complex space form with constant holomorphic
Let M
fn (4c), using (2.1)
sectional curvature 4c. For Lagrangian submanifold (M, g) in M
and (3.3) we obtain that Ricci soliton equation (1.1) with V = JH is written as
1(4.1){g(∇X (JH), Y ) + g(X, ∇Y (JH))} − ⟨σ(X, Y ), H⟩2+ traceg (Z 7→ Aσ(Z,Y ) X) + ((n − 1)c − λ)g(X, Y ) = 0.

RICCI SOLITONS AND LAGRANGIAN SUBMANIFOLDS IN KÄHLER MANIFOLDS

31

By contracting with respect to X and Y , we obtain
div JH − ∥H∥2 + ∥σ∥2 + n((n − 1)c − λ) = 0,

(4.2)

which is clearly equivalent to (2.2) with V = JH by (3.4).
By taking covariant diﬀerentiation of (4.1) by W , we get
11g(∇W ∇X (JH) − ∇∇W X (JH), Y ) + g(X, ∇W ∇Y (JH) − ∇∇W Y (JH))
22−⟨(∇W σ)(X, Y ), H⟩ − ⟨σ(X, Y ), ∇⊥W H⟩+ traceg (Z 7→ (∇W A)σ(Z,Y ) X) + traceg (Z 7→ A(∇W σ)(Z,Y ) X) = 0.
Then Codazzi equation (3.5) yields that1
1g(R(W, X)JH, Y ) + g(X, ∇W ∇Y (JH) − ∇∇W Y (JH))
221⊥− g(W, ∇X ∇Y (JH) − ∇∇X Y (JH)) − ⟨σ(X, Y ), ∇⊥W H⟩ + ⟨σ(W, Y ), ∇X H⟩
2+ traceg (Z 7→ (∇W A)σ(Z,Y ) X) + traceg (Z 7→ A(∇W σ)(Z,Y ) X)
− traceg (Z 7→ (∇X A)σ(Z,Y ) W ) − traceg (Z 7→ A(∇X σ)(Z,Y ) W ) = 0.
By contracting with respect to W and Y , and using (3.5) and (4.2) we get
Ric(X, JH) + g(X, ∆(JH)) = 0.
Because of

and

∫
M

1
1
∆∥H∥2 = ∆∥JH∥2 = g(∆(JH), JH) + ∥∇(JH)∥22
2= − Ric(JH, JH) + ∥∇⊥ H∥2 ,∆∥H∥2 µg = 0 when M is compact and oriented, the following result holds:

Theorem 7. Let M n be a compact oriented Lagrangian submanifold in a complex
fn (4c), and let g be the induced metric on M . Suppose (M, g) is aspace form M
Ricci soliton with V = JH. If Ric(JH, JH) ≤ 0 on M , then ∇⊥ H = 0 and (M, g)
is Ricci-flat.
With respect to either 2 or 3-dimensional Lagrangian submanifolds in complex
space forms, Proposition 6 is reduced to:
Theorem 8. Let M n be a compact Lagrangian submanifold in a complex space
fn (4c) with n = 2, 3 and let g be the induced metric on M . Suppose (M, g)form Mis a Ricci soliton with V = JH. Then (M, g) is either totally geodesic or flat withparallel mean curvature vector field.
In fact the assumption yields that, by Propositions 4 and 6, ∇⊥ H = 0. WhenH = 0, Ejiri’s result [4] implies that (M, g) is either totally geodesic or ﬂat. WhenH ̸= 0, (M, g) is constant curvature and has parallel non-zero tangent vector ﬁeld
JH. Hence M is ﬂat.
f2 (4c) = CP2 : complex projective plane withFor Lagrangian ﬂat surfaces in M
c > 0, Ogata [11] proved the following: Let x : R2 → CP2 be an isometric32

J. T. CHO AND M. KIMURA

immersion with non-zero parallel mean curvature vector ﬁeld H. Then x(R2 ) is anorbit of the Abelian Lie subgroup G of U (3).
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Jong Taek Cho: Department of Mathematics, Chonnam National University,
CNU The Institute of Basic Sciences, Kwangju 500-757, Korea
E-mail address: jtcho@chonnam.ac.kr
Makoto Kimura: Department of Mathematics, Faculty of Science and Engineering, Shimane University Matsue, Shimane 690-8504, Japan
E-mail address: mkimura@riko.shimane-u.ac.jp