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Dressing actions on integrable surfacesThis is a joint work with Nick Schmitt at T¨bingen University. u Shimpei Kobayashi, Hirosaki University 6/21, 2011
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Introduction Overview (Bianchi) B¨cklund transformation a Harmonic maps into symmetric spaces Loop groupsDressing actions and Bianchi-B¨cklund transformations a Factorizations and dressing actions Dressing actions and Bianchi-B¨cklund transformations aComplex CMC surfaces and real forms Complex CMC surfaces Real forms Integrable surfacesDressing action on complex extended frames Double loop group decomposition Dressing action on complex extended frames Dressing action on integrable surfaces
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Overview1: Motivation ◮ Understand transformation theory of surfaces in terms of modern language; ﬂat connections, harmonic maps and loop groups. ◮ More speciﬁcally: Why do constant positive Gauss curvature surfaces have only Bianchi-B¨cklund transformation? Note! a Constant negative Gauss curvature surfaces have also B¨cklund transformation. a Remark ◮ B¨cklund transformation is a transformation by tangential line a congruences and Bianchi-B¨cklund transformation is an a extension of the B¨cklund transformation by Bianchi. a ◮ Classical theorem by B¨cklund says that if two surfaces are a related by B¨cklund transformation, then they are constant a negative Gauss curvature surfaces.
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Overview 2: Previous works Remark ◮ Uhlenbeck considered dressing action on extended frame of harmonic maps into Lie groups, J. Diﬀ. Geom. 1989. ◮ Uhlenbeck proved that the simple factor dressing action on extended frames of negative CGC surfaces is equivalent to the B¨cklund transformation, J. Geom. Phys. 1992. a ◮ Terng and Uhlenbeck generalized the dressing action to U/K-system, Comm. Pure Appl. Math. 2000.
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B¨cklund transformation a Theorem (Backlund) S, S ′ ⊂ R3 : a surface in Euclidean three space S and S ′ are related by the tangential line congruences with constant angle and distance. ⇓ S and S ′ are constant negative Gauss curvature surfaces. Remark ◮ Tnagential line congruences ℓ are line congruences which tangent to both surfaces. ◮ ˜ Angle between S and S ′ are determined by N, N = c, ˜ where N and N are the unit normal ﬁelds of S and S ′ .
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Bianchi-B¨cklund transformation a Theorem (Bianchi) Let S ⊂ R3 be a CGC surface. There exits a surface S ′ in complex Euclidean three space such that S and S ′ are related by tangential line congruences with complex constant angle. This transformation is called a Bianchi-B¨cklund transformation. a Moreover, twice of the Bianchi-B¨cklund transformation with a suitable angle conditions gives a CGC surface in R3
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Permutability and surperposition formula ∗ β,β = S β ∗ ,β Sκ κ . β θ β∗ u u ˆ β∗ β θ∗ The superposition formula: u−u ˆ β − β∗ θβ − θβ∗ tanh = tanh tanh , 2 2 2 ˆ ˆ where u = θβ,β∗ = θβ∗ ,β . ˆ
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A family of ﬂat connections Theorem (Pohlmeyer 1976) Let M be a simply connected open Riemann surface and G/K a symmetric space. The followings are equivalent. 1. Φ : M → G/K is a harmonic map. 2. There exist a Fλ : M → ΛGσ such that F−1 dFλ = λ−1 α′ + αk + λα′′ and π ◦ F|λ=1 = Φ. λ p p Corollary The set of extended frames, Fλ. The family of CMC surfaces, fλ.
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Loop groups factorizations Theorem (Birkhoﬀ and Iwasawa decompositions) 1. Birkhoﬀ decomposition: Λ+ GC × Λ−GC → ΛGC ∗ σ σ σ is a diﬀeomorphism onto the open dense subset Λ+ GC · Λ−GC of ΛGC . ∗ σ σ σ 2. Iwasawa decomposition: Assume that G is compact. ΛGσ × Λ−GC → ΛGC σ σ C is a diﬀeomorphism onto ΛGσ . Remark The Iwasawa decomposition is obtained from the Birkhoﬀ decomposition and a real from.
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Dressing actions and Bianchi-B¨cklund transformations 1 a Let F be an extended framing and g an element in Λ+ GC . σ Decompose gF according to the Iwasawa decomposition as ˜ ΛGC ∋ gF = FV+ ∈ ΛGσ × Λ+ GC . σ σ Then ˜ ◮ F is again the extended framing. Thus Λ+ GC acts, the σ dressing action, that is, Id#F = F and ˜ −1 g(#(f#F)) = (gf)#F, where g#F = F = gFV+ . ◮ Bianchi permutability theorem is just associativity of the action: g2 #(g1 #F) = g2 #(ˆ1 #F), ˆ g where g2 g1 = ˆ2 g1 . g ˆ ◮ The dressing action by rational loops with simple pole is called the simple type dressing action.
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Dressing actions and Bianchi-B¨cklund transformations 2 a Theorem (Inoguchi-Kobayashi, 2005) The simple type dressing action and Bianchi-B¨cklund a transformation are equivalent, where the Bianchi-B¨cklund a transformation is a transformation of a CMC surface by line congruences. Figure: A twizzler, its Bianchi-Backlund transformation and a bubbleton.
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Recall that a simple factor dressing by g: −1 g#F = gFV+ .This action corresponds to the twice of Bianchi-B¨cklund atransformation.The g has two simple poles at λ1 , λ2 , which are related ¯λ2 = 1/λ1 ∈ C× S1 . ◮ Is is possible to factor g to g = g2 g1 so that each gj corresponds to once Bianchi-B¨cklund? a ◮ Answer: Yes, but one needs an extension of the dressing action.
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Constant negative Gauss curvature surfaces Fact There exists a similar dressing action on constant negative Gauss curvature surfaces. Try to unify negative and positive Gauss curvature surfaces. ⇓ ◮ Complex CMC surfaces (Dorfmeister-Kobayashi-Pedit 2010). ◮ Real form surfaces (Kobayashi 2011).
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Ruh-Vilms type theorem Theorem The following two conditions are equivalent: 1. The complex mean curvature H is constant. 2. There exist a Fλ : D2 → ΛGσ (= ΛSL2 Cσ ) such that C F−1 dFλ = λ−1 α′ + αk + λα′′ and π ◦ F|λ=1 = Φ, where λ p p Φ is the unit normal to f. Here ′ (resp. ′′) denotes dz-part and (resp. dw-part). ◮ Fλ (F in short) is called the complex extended framing. ◮ The complex CMC surfaces are given by Sym formula for F.
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Almost compact real forms Theorem (Kobayashi, 2011) Let cj for j ∈ {1, 2, 3, 4} be the following involutions on Λsl2 Cσ : t ¯ c1 : g(λ) → −g(−1/λ) , ¯ c2 : g(λ) → g −1/λ , t √ t ¯ 1/ i 0 ¯ c3 : g(λ) → −g 1/λ , c4 : g(λ) → −Ad √ g(i/λ) , 0 i where g ∈ Λsl2 Cσ . Then, the almost compact real forms of Λsl2 Cσ are the following real Lie subalgebras of Λsl2 Cσ : Λsl2 C(c,j) = {g ∈ Λsl2 Cσ | cj ◦ g(λ) = g(λ) } . σ
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Almost split real forms Theorem (Kobayashi, 2011) Let sj for j ∈ {1, 2, 3} be the following involutions on Λsl2 Cσ : t ¯ s1 : g(λ) → −g(−λ) , ¯ s2 : g(λ) → g −λ , t s3 : g(λ) → −Ad λ 0 ¯ g λ , 0 λ−1 where g ∈ Λsl2 Cσ . Then, the almost split real forms of Λsl2 Cσ are the following real Lie subalgebras of Λsl2 Cσ : Λsl2 C(s,j) = {g ∈ Λsl2 Cσ | sj ◦ g(λ) = g(λ) } . σ
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Double loop group decomposition The following r-loop group and its loop subgroups will be used. Hr,R = Λr SL2 C × ΛR SL2 C, H+ = Λ+ SL2 C × Λ−SL2 C ⊂ Hr,R , r,R r R g1 and g2 extends holomorphically H− = (g1 , g2 ) ∈ Hr,R . r,R to Ar,R and g1 |Ar,R = g2 |Ar,R
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TheoremThe multiplication map H− × H+ → His a diﬀeomorphism on an open dense subset of H, which is calledthe big cell. On the big cell, an element (gr , gR ) ∈ H can bedecomposed as (gr , gR ) = (F, F)(h+ , h−), r Rwhere (F, F) ∈ H− denotes the boundary values on Cr and CR ofthe map F : Ar,R → ΛSL2 Cσ and (hr , h−) is an element in H+ . + R
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Dressing action on complex extended frames Let F be a diagonal set of complex extended frames: F = {(F, F) | F is a complex extended frames.} Then H acts F as follows: ˜ ˜ Let (gr , gR ) ∈ H and (F, F) ∈ F. Deﬁne (F, F) ˜ ˜ (F, F) = (gr F, gR F)Ar,R , where the subscript Ar,R denotes the H− part of the double loop ˜ ˜ group decomposition. Denote (F, F) by ˜ ˜ (F, F) = (gr , gR )#(F, F). It is easy to see that (Id, Id)#(F, F) = (F, F) and (˜r , gR )#((gr , gR )#(F, F)) = ((˜r , gR ) · (gr , gR ))#(F, F). g ˜ g ˜
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ρ dressing action on complex extended frames We generalized H to K as ˜ K = {(gr , gR ) | gr ∈ Λr SL2 C and gR ∈ ΛR SL2 C.}. Let (gr , gR ) ∈ K and √ 0 λ ρ= √ −1 . − λ 0 ˜ ˜ Deﬁne (F, F) by ˜ ˜ (F, F) = (gr F, gR Fρj )Ar,R , where j = 4 if gR is single valued and j = 1 if gR is double valued. ˜ ˜ part of the double loop group decomposition. Denote (F, F) by ˜ ˜ (F, F) = (gr , gR )#ρ(F, F).
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Theorem (Kobayashi-Schmitt) 1. #ρ deﬁnes an action and it is an extension of the dressing action. 2. If F and g satisfy CMC (positive CGC) reality condition, then the dressing action reduces to the CMC dressing action. 3. If F and g satisfy negative CGC reality condition, then the dressing action reduces to the negative CGC dressing action.Remark ◮ ¯ The CMC case, the simple poles must be pair λ1 , 1/λ1 . Thus Bianchi-B¨cklund transformation can only be applied. a ◮ The CMC case, the simple poles need not to be pair. Thus B¨cklund and Bianchi-B¨cklund transformation can be a a applied.
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