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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dic. 2006
Stability of holomorphic-horizontal maps and Einstein metrics on flag manifolds
Caio J. C. Negreiros
Partialy supported by CNPq grant 303695/2005-6 and Fapesp grant 02/10246-2
Abstract: In this note we announce several results concerning the stability of certain families of harmonic maps that we call holomorphic-horizontal frames, with respect to families of invariant Hermitian structures on flag manifolds. Special emphasis is given to the Einstein case. See [23] for additional detail and the proofs of the results mentioned in this survey.
Let be a complex semi-simple Lie algebra and
a simple root system for
. If
is an arbitrary subset of
,
denotes the roots spanned by
. We have
| (1) |
where is a Cartan subalgebra of
and,
is the root space associated to the root
.
Let
| (2) |
the canonical parabolic subalgebra determined by . Thus,
| (3) |
The flag manifold is defined as , where
has Lie algebra
and
is the normalizer of
in
.
for
, is called the full flag manifold and is denoted by
. This case is nowadays well understood. Starting with the work of Borel (cf. [3]), the classification of all invariant Hermitian structures is known and it was described in [25].
The main purpose for this note is to announce some results discussing the stability phenomenon for the energy functional for to a special class of maps , called holomorphic-horizontal frames. The energy functional is taken with respect to several families of invariant Hermitian structures on
. These maps are deeply connected with the study of harmonic/minimal surfaces in
,
,
, Twistor Theory and so on (cf. [27], [8], [4], [15]).
The layout of the paper is as follows. In the first two sections, we state general results on the invariant Hermitian geometry of , and state the holomorphic and harmonic map equations. We give in this note, examples of families holomorphic-horizontal frames only in the
case, but in [23] we also discuss the cases
,
and
.
We generalize and give additional results to the approach initiated by Black in [2] and the author in [21]. As a reference for harmonic maps theory we suggest the Eells-Lemaire [14] article.
In Section 3 we compute the second variation of energy for an arbitrary harmonic map on . We state a basic perturbation lemma and a result for holomorphic-horizontal frames on
.
According to the classification results in [25], among all Hermitian invariant structures there are two main classes. Thus, in the last section we state results concerning the stability of holomorphic-horizontal frames regarding metrics in such classes and in particular, the case of metrics that are Einstein and non Kähler on the geometrical flag manifold .
2. Generalities on invariant Hermitian geometry of flag manifolds
Let be a root system and
a simple root system for a simple Lie algebra
. If
is a subset of
,
denotes the roots generated by
. We have the root decomposition :
where is a Cartan subalgebra of
and
is the root space associated to the root
.
Let
| (6) |
The space is called a flag manifold, where
and
are the Lie algebras of
and
, respectively.
Each manifold has families of complex geometries, i.e., families of invariant Hermitian structures denoted by
.
We denote by the Cartan-Killing form of
, and fix once and for all a Weyl basis of
, which amounts to take
such that
, and
with
,
, and
if
is not a root.
We define the compact real form of , as the real subalgebra
|
where and
.
Let be the origin of
.
is identified with
where . Complexifying
we obtain
, which can be identified with
|
We denote the irreducible components of as
, where
is the set of roots
with
, thus
.
Let be the collection of sets
originating the irreducible components. We write
|
Each defines a field of complex subspaces
such that
for each
.
A -invariant almost complex structure
on
is completely determined by a linear map
. The map
satisfies
and commutes with the adjoint action of
on
. We denote also by
its complexification to
.
The invariance of entails that
for all
. The eigenvalues of
are
, and the eigenvectors in
are
,
. Hence, in each irreducible component
,
with
satisfying
. A
-invariant almost complex structure on
is completely determined by the numbers
,
.
A -invariant Riemannian metric
on
is completely determined by the following inner product
on
| (7) |
with definite-positive with respect to the Cartan-Killing form. On each irreducible component of
,
with
.
Consider the conjugation of
relatively to
. Hence,
is a Hermitian form on
, that originates a
-invariant Hermitian form on
.
If denotes the corresponding Kähler form then
| (8) |
We recall that an almost-Hermitian manifold is said -symplectic if
when one of the vectors
,
,
is of type
, and the other two are of type
. If
is integrable and
, we say
is a Kähler manifold.
From now on, for abuse of notation we will denote a map by
where
, despite
and
being completely different objects.
Black for the cases ,
and
(see [2]) and the author in the case
(see ([21]), obtained the Cauchy-Riemann equations in our situation:
Proposition 3.1. A map is
-holomorphic on
if and only if for every
,
implies
.
We define the energy of as:
To deduce the harmonic map equations, a basic remark is that
, for any perturbations
. In fact, every map
:
satisfies
.
We now consider the map :
.
. According to Stokes' Theorem we have
. Thus,
.
We now perturb the map in the following natural way
We are considering here the natural action of on
and taking an arbitrary
map
.
In [23] we deduce the following Euler-Lagrange equations for our variational problem
Proposition 3.2. A map is harmonic if and only if
if and only if
| (9) |
We will use a generalization of to an
-structure following Yano ([29]). An
-structure
on
is a section of
such that
.
An invariant -structure is given by the matrix
with
,
or
, according to the eigenvalues of
.
We now state the Cauchy-Riemann equations in the case of -structures
Proposition 3.3. A map is
-holomorphic if and only if it is subordinate to
.
Definition 3.1. Consider an invariant -structure
on
. Let
|
and
|
is said horizontal if
.
The following theorem due to Black ([2]) is essential in our study
Theorem 3.4. Let be subordinate to a horizontal
-structure
. Then
is equi-harmonic.
Consider now . We can prove that
is a horizontal
-structure, and we will call it by
-structure associated to
.
We will now exhibit families of equi-harmonic and holomorphic maps subordinate to an horizontal
-structure, thus all of them are equi-harmonic according to Theorem 3.4. Any map in these families, is called a holomorphic and horizontal frame.
Let
be a holomorphic and non-degenerate map. We consider its associate curve
, where
and
.
We define the map . We can prove that
, with
if
, where
denotes a simple root system for
.
We can prove that any such map is holomorphic and subordinate to
, thus, again according to Theorem 3.4, it is an equi-harmonic map.
More generally, we will now construct families of holomorphic and equi-harmonic maps .
Let be the geometric flag manifold
, where
,
and
.
A root system of height one with respect to , is given by:
Let any holomorphic and nondegenerate map, and
as we have defined above.
We define the map by:
.
We prove in [23] that any such is a holomorphic-horizontal frame.
4. The second variation of energy and stability on
We compute now the second variation of the energy in our situation.
Theorem 4.1. Consider a harmonic map .
Thus,
where the map is defined by
.
Definition 4.1. A harmonic map is said stable if
, for any variation
. Otherwise,
is said unstable.
The following Theorem due to Lichnerowicz ([19]) is fundamental in our study of stability on flags.
Theorem 4.2. Let be a
-holomorphic map and
a Kähler structure. Then
is stable.
Definition 4.2. We say that is a
-perturbation of
subordinate to
if
-
;
-
if
;
-
,
if
;
-
if
.
Regarding the families of the holomorphic and horizontal frames we have just defined, we can simply consider .
Using the above definition of perturbation we derive the following basic lemma.
Lemma 4.3. Let a holomorphic and horizontal frame. Then,
According to Gray-Hervella ([16]) the almost Hermitian structures can be decomposed into four irreducible components. For instance, corresponds to Kähler metrics,
to the
-symplectic ones and, so on. See [25] and [26].
Lemma 4.4. A necessary and sufficient condition for to be in
is:
if
is a
-triple.
As an immediate consequence of this lemma we notice that the Cartan-Killing structure is in . We will now consider perturbations of the Cartan-Killing structure.
We consider a and denote by
the subset of roots
such that there exists a
-triple
.
Let given by
for each
, and
otherwise. According to Lemma 4.4,
. We can prove the following theorem.
Theorem 4.5. Let be an arbitrary holomorphic-horizontal frame. Then,
is unstable.
According to the results obtained in [25], among all the invariant Hermitian structures, the main cases are and
. We will now discuss the stability phenomenon of holomorphic-horizontal frames in these two main classes.
Based on a crucial result derived in [25] we present the following definition.
Definition 5.1. Let ,
and
. We fix a Kähler structure
. The metric
is said a perturbation of type (1,2)-symplectic of
if
- for each
with
and,
we have
.
- for each
with
and if
, where
is the highest root, and each
, then
.
We now are ready to discuss the case. We consider
equipped with an invariant Hermitian structure that comes from a perturbation of type (1,2)-symplectic of a Kähler structure
. Thus, in [23] we prove
Theorem 5.1. Let be a holomorphic-horizontal frame. Then
is stable.
We will now concentrate our attention on the family of invariant Hermitian structures on that are in
. We begin our discussion mentioning the classification of Einstein metrics on
and
derived in [17] and [22], exploiting these results and obtaining (see [23])
Theorem 5.2. Let a holomorphic and horizontal frame with
or
and
a Einstein and non-Kähler metric. Then, the map
is unstable.
A basic result due to Arvanitoyeorgos [1] and Kimura [17] is the following one.
Theorem 5.3. The space for
admits as Einstein metrics only the normal and the Kähler-Einstein metrics. If
it admits at least
Einstein metrics. The
metrics are the already mentioned Kähler-Einstein metrics described by Borel, one is the usual normal metric and the remaining n are given explicitly as follows:
.
More generally, in his Ph.D. thesis ([12]), dos Santos has found new families of Einstein non-Kähler metrics on arbitrary . See also [9] and [13] for additional details.
We notice that any known invariant Einstein metric on has a common feature: either it is Kähler or is in
. In fact, we believe that this fact is true for any Einstein metric on
.
Using an appropriate Cartan-Killing perturbation (as in Theorem 4.5) we can prove.
Theorem 5.4. Let equipped with any of the known Einstein non-Kähler metrics above described, and
be any arbitrary holomorphic-horizontal frame. Then,
is unstable.
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Caio J. C. Negreiros
Department of Mathematics.
Universidade Estadual de Campinas.
Cx. Postal 6065 13081-970, Campinas-SP, Brasil
Recibido: 5 de octubre de 2005
Aceptado: 19 de septiembre de 2006