The Fifth International Workshop on Analele Universit˘ a¸tii din Timi¸soara Differential Geometry and Its Applications Vol. XXXIX, 2001 September 18–22, 2001, Timisoara Seria Matematic˘a-Informatic˘ a Romania Fascicul˘ a special˘a-Matematic˘ a
Birkhoff and Iwasawa loop group splittings in the Willmore DPW framework by Vladimir Balan Abstract The DPW (Dorfmeister-Pedit-Wu) framework aims to determine various classes of harmonic maps from Riemann surfaces to reductive homogeneous spaces or Lie groups. As part of this, two essential steps of the associated DPW procedure are the Birkhoff and the Iwasawa decompositions of complex Lie loop groups. The particular symmetric space case, where the factorization of the complexified loop group is performed via the real form was intensively studied; still, the remaining general (noncompact) subcases reveal a large amount of open problems, some of them being outlined in the present paper. In the paper, the particular SO(4,1)/SO(3,1) and SO(4,1)/(SO(3)xSO(1,1)) cases are described as part of the DPW procedure for Willmore surfaces.
1
Basic facts on loop groups
Let G be a given Lie group, considered as subgroup of the matrix group GL(n, C), and let g = Lie(G) ⊂ Mn×n (C) be its Lie algebra. Consider the families of mappings L∞ (S 1 , G) = {g0 : S 1 → G | ∃C > 0, ||gλ || < C, a.e.} , H 1/2 (S 1 , G) = {g0 : S 1 → G | ||g0 ||H 1/2 < ∞},
(1)
AMS Subject Classification: 53A30, 22E67, 53C42, 58E20, 53C43, 22F30, 30F15 Keywords and phrases: loop group, Birkhoff decomposition, Iwasawa decomposition, meromorphic potential, conformal map, reductive homogeneous space, symmetric space, conformal Gauss map, Willmore surface
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where the two sets of maps are structured as Banach spaces with the norms ||g0 ||L∞ = inf{C > 0 | ||gλ || < C}, X √ X 1 + k 2 ||(g0 )k ||2 , gλ = (g0 )k λk . ||g0 ||H 1/2 = k∈ZZ
(2)
k∈ZZ
Then the set of bounded and measurable maps g0 ∈ ΛG = L∞ ∩ H 1/2 , g0 : S 1 → G,
(3)
is proved to be structured as a Banach space with the norm ||g0 || = ||g0 ||L∞ + ||g0 ||H 1/2
(4)
and also as (infinite dimensional) Lie group; it is called loop group and its elements, loops. Moreover, it can be shown that its Lie algebra Λg = L∞ (S 1 , g) ∩ H 1/2 (S 1 , g) consists of g-valued loops of finite norm. The type of loopification described in the DPW method leads to the consideration of twisted loop groups. This is produced by an (adjoint) involutive automorphism σ of G which canonically extends to its complexification G C and induces corresponding automorphisms of their Lie algebras g and g C. We note that the +1, −1 eigenspaces determined by σ in g and g C determine their splittings (Cartan decompositions) into Lie subalgebra and vector subspace direct sums g = k ⊕ p, g C = k C ⊕ p C, (5) with k C = k ⊗ C, p C = p ⊗ C. The subalgebras k, k C are shown to be the Lie algebras of the subgroups K = {g ∈ G | σgσ −1 = g} and K C respectively. The homogeneous space G/K is reductive. The associated twisted Lie loop groups and their corresponding Lie algebras are then Λσ G = {g ∈ ΛG | σg(λ)σ −1 = g(−λ)}, Λσ g = {ξ ∈ Λg | σξ(λ)σ −1 = ξ(−λ)}, Λσ G C = {g ∈ ΛG C | σg(λ)σ −1 = g(−λ)},
(6)
Λσ g C = {ξ ∈ Λg C | σξ(λ)σ −1 = ξ(−λ)}, and have the operational property that the Fourier series of a loop g ∈ Λσ g C has the even-power matrix coefficients in the subalgebra k C ⊂ g C and the odd ones in the vector space p C ⊂ g C ([13], [6]).
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2
Birkhoff and Iwasawa loop group decompositions
The two basic loop group splittings used in the DPW method are the Birkhoff and the Iwasawa decompositions. Denote by Λ+ GσC and Λ− GσC the subgroups of ΛGσC which admit a holomorphic extension inside the unit disk of C ∪ {∞} and outside this, respectively. For a subgroup H ⊂ G C having its Lie algebra h, we consider the subgroups + + C C C Λ+ H Gσ = {go ∈ Λ Gσ | gλ=0 ∈ H} ⊂ Λ Gσ , C − C − C Λ− H Gσ = {go ∈ Λ Gσ | gλ=∞ ∈ H} ⊂ Λ Gσ ,
(7)
+ C C − C Λ+ gσC and Λ− gσC their Lie algebras, and specify Λ+ ∗ Gσ = Λ1 Gσ and Λ∗ Gσ = h h C Λ− 1 Gσ .
Provided that there exists a solvable subgroup B of K C such that K C = K.B, the Iwasawa splitting of ΛGσC is then given by the preimages of the injective product map C C Λσ G × Λ+ B Gσ −→ ΛGσ ,
(8)
while the Birkhoff splitting of ΛGσC, the preimages of the injective map C + C C Λ− ∗ Gσ × Λ Gσ −→ ΛGσ .
(9)
Two open questions arise in the case when the Lie group G is noncompact: 1. Whether the Iwasawa map (8) is a diffeomorphism (true in the compact case); 2. If the Birkhoff map (9) is a diffeomorphism onto an open set of ΛG C, called the big cell (true in case when G C is a compactification of a compact Lie group). In the following we examine the noncompact G case within the Willmore DPW framework, and point out that the Iwasawa map satisfies 1 just only locally, and that the Birkhoff map satisfies 2.
3
The Willmore - DPW framework
The Willmore surfaces, more recently studied by F.Helein [6], on the basis of the works of J.Dorfmeister, F.Pedit and H.-Y. Wu [5], and R.Bryant [4],
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are the minimizers of the Willmore functional Z
Z
(k1 − k2 )2 1Z 2 2 dσ+4π(1−g) = (k +k )dσ+2π(1−g), 4 4 S 1 2 S S (10) 3 where the variation is made among the set S of surfaces S immersed in R , oriented and without boundary, H is the mean curvature, k1 and k2 the principal curvatures, g the genus and dσ the area element of S ∈ S. It is proved that, denoting by ∆ the Laplacian constructed using the first fundamental form of S, the Willmore surfaces are the solutions of the 4-th order PDE [19] ∆H + 2H(H 2 − K) = 0. (11) H 2 dσ =
WH (S) =
As particular cases of Willmore surfaces, one has the minimal surfaces and 3 the stereographic projection to R of compact minimal surfaces in S 3 . 2
Given a simply connected domain D 3 0 in R , a surface S = r(D) ∈ S, is given by a conformal immersion 3
r : D → R , r(u) = ~x(u), for all u = (u1 , u2 ) ∈ D. 3
(12)
4
The embedding of the ambient space R in S 3 ⊂ R via the stereographic 3 4 projection ψ : R → S 3 ⊂ R , 3
ψ(x) = (2x/ρ ; 1 − (2/ρ)) , for all x ∈ R , 3
(ρ = ||x||2R3 + 1),
(13)
¯ Φ
identifies R ∪ {∞}→ S 3 . Then S can be successively embedded into the ∼ 5-dimensional Minkowski space R 4,1 of S, ϕ : D → R ,
4,1
by the so-called conformal Gauss map
ϕ(u) = H(u)(ψ(r(u)), 1) + (n(u), 0), for all u ∈ D,
(14)
where n : D → S 3 is the normal unit vector field of S. The image ϕ(D) is contained in the hyperquadric S 3,1 ∼ = SO(4, 1)/SO(3, 1) 4,1 of R associated with the Lorentz metric 4
h(x, x), (y, y)i = hx, yi R4 − xy, for all (x, x), (y, y) ∈ R × R ≡ R
4,1
. (15)
Since the energy functional of ϕ is related to WH by Z D
Z
|| dϕ||2 dσ =
D
(H 2 − K) dσ = W(S) − 4π(1 − g), 24
(16)
it follows that the Willmore surfaces are characterized by the harmonicity of their conformal Gauss map, being minimizers of the Dirichlet functional of ϕ. 4,1 At the same time the surface S ∈ S can be embedded in R by the 4 4,1 mapping f : S 3 ⊂ R → R , Ã
f (q) =
1 + q4 1 2 3 1 − q4 √ ,q ,q ,q , √ 2 2
!
∈ R
4,1
, for all q = (q 1 , q 2 , q 3 , q 4 ) ∈ S 3 ,
(17) which identifies S ⊃ ψ( R ) ⊃ ψ(S) with the generators of the positive semicone of the Lorentz metric (15) and identifies the conformal transformations 3 of R ∪ {∞} ≡ S 3 with QSOo (4, 1)Q−1 [6], which is taken for the basic Lie group in the Willmore DPW approach, where we denoted √ √ 2/2 0 2/2 I3 Q = √0 . √0 2/2 0 − 2/2 3
3
Since the conformal Gauss map has values in the reductive homogeneous space S 3,1 , we search for shifted lifts F = F0 .g ∈ QSO(4, 1)Q−1 , where g ∈ Q(SO(3)×SO(1, 1))Q−1 is a convenient conformal transformation, such that F includes the conformal Gauss map of the surface [6]. Denoting by g and k the Lie algebras of G = SO(4, 1) and of the subgroup K = SO(3) × SO(1, 1), the Lie algebra of G is subject to the Cartan splitting g = k ⊕ p, where k = {ξ ∈ g | Ad diag(−1,I3 ,−1) ξ = ξ}, p = {ξ ∈ g | Ad diag(−1,I3 ,−1) ξ = −ξ}.
(18)
Then, introducing complex coordinates (z, z¯) on D, the Maurer-Cartan form ω = F −1 dF associated to the moving frame F splits ω = ωp0 + ωk + ωp00 ∈ g = k ⊕ p,
(19)
where ωp0 , ωp00 ∈ p are not necessary, in this case, the holomorphic/antiholomorphic parts of the p-part of ω and ωk0 ∈ k. This form is extended by ”loopifying” it ([5], [16], [8]), by introducing the complex parameter λ ∈ S 1 ⊂ C, ωλ = λ−1 ωp0 + ωk + λωp00 .
(20)
As in the case K = SO(3, 1), the conditions under which the equation Fλ−1 dFλ = ωλ 25
(21)
provides frames Fλ associated to Willmore immersions subject to Fλ |λ=1 = F ; Fλ (0) = I5 , are stated below [6]. Theorem. The first column of Fλ provides a Willmore immersion iff ωλ has zero curvature, i.e. it satisfies the integrability condition 1 dωλ + [ωλ ∧ ωλ ] = 0. 2
(22)
The frames associated to Willmore immersions can be obtained by the DPW loop group procedure, which builds the corresponding meromorphic potentials as follows. The extended frame Fλ resulting from the integration of (21) is split locally by means of the Birkhoff decomposition C C Fλ = Fλ− .Fλ+ ∈ Λ− · Λ+ σG σG .
(23)
Then ξ = (Fλ− )−1 d Fλ− is the meromorphic potential associated with the original map F . Then, after applying a ”dressing” which alters Fλ− and by re-splitting Birkhoff, the DPW procedure changes the potential ξ and integrates the new meromorphic potential, i.e. the equation ξ = F−−1 d F− , in order to provide a complex frame F− , which can be further split using the Iwasawa decomposition [5], [2], [13] F− = F∗ · Fλ+ . (24) Then, the resulting (real) frame F∗ includes the conformal Gauss map of a Willmore immersion, and the immersion itself within the extended frame, thus leading to the Willmore immersion.
4
The Willmore case decompositions In the Willmore DPW case we consider G = SO(4, 1),
G C = SO(5, C),
(25)
and the subgroup K can be chosen twofold. 1) For the involutive automorphism σ = (I3 , −1, 1), one gets K = SO(3, 1),
∗
K C = (SO(3, 1)) C = SO(3, 1) × C
and the decompositions are generally untractable [1] . Still, in this case, the Iwasawa decomposition provides a dense ”big cell” [10]. 26
2) In the case of the involutive automorphism σ = (−1, I3 , −1), we have [6]
∗
K = SO(3) × SO(1, 1) = SO(3) × R +
(26)
B = C × S 1, whence ∗
∗
K C = (SO(3) × SO(1, 1)) C = SO(3) C × C = (SO(3).C) · ( R + .S 1 ) = ∗
= (SO(3). R + ) · (C.S 1 ) ≡ K B, where the subgroup C of the solvable group B = C.S 1 is ∗
C = {A = diag(eiθ , R, e−iθ ) ∈ K C | ∃a ∈ R + , Ru = au; θ ∈ R }, 3
where u = t (1, i, 0) ∈ C is a fixed isotropic vector. In this second case, though the decompositions are feasible, the meromorphic potentials provide finally a frame which includes a ”roughly harmonic” conformal Gauss map, differing from a Willmore frame by a right multiplication with a K-element. Also, the terms ωp0 and ωp00 in the decomposition of the Maurer-Cartan form are no longer the (1,0)- and the (0,1)- parts of the component ωp = ωp0 + ωp00 ∈ p. The advantage in this case is that the meromorphic potential data provides directly the first partials of the immersion. Remarkable is the fact, that in both cases even if G is noncompact, G C is the compactification of the compact Lie group SO(5), whence the Birkhoff decomposition still holds on the big cell. A disadvantage is that, because of the need to incorporate a non-degenerate Gauss map into the moving frame, the Willmore DPW procedure removes from the domain D the umbilic points of the surface S, and hence the Iwasawa decomposition and the emerging results are mainly local. Acknowledgement. The author expresses his thanks to J.Dorfmeister and P.Kellersh for useful discussions on the current topic. Within the more general framework emerging from the study of the cases K = SO(3, 1) and K = SO(3) × SO(1, 1), recent results [2] were obtained, supported by the Fulbright Grant #21945-5/13/97, and by NSF Grant DMS - 9705479 and Sonderforschungsbereich 288, Differentialgeometrie und Quantenphysik, TU - Berlin, respectively.
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References [1] V. Balan, Willmore surfaces and loop groups, Mat. Bilt., 25 (2001), 1724. [2] V. Balan, J. Dorfmeister, Birkhoff decomposition and Iwasawa decomposition for general untwisted loop groups, Tˆohoku Math. J. (2) 53 (2001), 4, 593–615. [3] W. Blaschke, Vorlesungen u ¨ber Differentialgeometrie, Springer, Berlin, 1929. [4] R. Bryant, A duality theorem for Willmore surfaces, J.Diff.Geom., 20(1984), 25-53. [5] J. Dorfmeister, F. Pedit, H.-Y. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom., 6(1998), no. 4, 633-668. [6] F. Helein, Willmore immersions and loop groups, J.Diff.Geom., 50(1998), 331-385. [7] F. Helein, Applications harmoniques, lois de conservation et rep`eres mobiles, Diderot, Paris, 1996. [8] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. [9] H. Karcher, U. Pinkall and I. Sterling, New minimal surfaces in S 3 , MPI-86-27, preprint. [10] P. Kellersch, Eine Verallgemeinerung der Iwasawa Zerlegung in Loop Gruppen, Dissertation, TU-Munich, 1999. [11] H.B. Lawson, Jr., Complete minimal surfaces in S 3 , Ann. of Math., 92(1970), 335-374. [12] U. Pinkall, Hopf tori in S 3 , Invent. Math., 81(1985), 379-386. [13] A. Pressley, G. Segal, Loop Groups, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1986. [14] U. Pinkall, I. Sterling, Willmore surfaces, The Math. Intelligencer, 9(1987), no.2, 38-43.
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¨ [15] G. Thomsen, Uber konforme Geometrie, Abh.Math.Sem. Hamburg (1923), 31-56. [16] K. Uhlenbeck, Harmonic maps into Lie groups, J.Diff.Geom., 30(1989), 1-50. [17] H. Wente, Counterexample to a conjecture of H.Hopf, Pacific J. Math., 121(1986), no.1, 193-243. [18] T.J. Willmore, Note on embedded surfaces, An.St.Univ. “Al.I.Cuza” Iasi, Sect. Ia Mat., II(1965), 443-496. [19] T.J. Willmore, Riemannian Geometry, Oxford Science Publications, Oxford, 1993. [20] T.J. Willmore, A survey on Willmore immersions, in Geometry and Topology of Submanifolds, IV (1991, Leuven), (F.Dillen, L.Verstraelen, eds.), World Sci., 1992, pp 11-16. Vladimir Balan Universitatea Politehnica din Bucure¸sti Catedra Matematici I Splaiul Independent¸ei 313 RO-77206, Bucure¸sti, Romˆania
[email protected]
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