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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