dressing2011
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Dressing actions on integrable surfaces
This is a joint work with Nick Schmitt at Tubingen University.Shimpei Kobayashi, Hirosaki University
6/21, 2011
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IntroductionOverview(Bianchi) Backlund transformationHarmonic maps into symmetric spacesLoop groups
Dressing actions and Bianchi-Backlund transformationsFactorizations and dressing actionsDressing actions and Bianchi-Backlund transformations
Complex CMC surfaces and real formsComplex CMC surfacesReal formsIntegrable surfaces
Dressing action on complex extended framesDouble loop group decompositionDressing action on complex extended framesDressing action on integrable surfaces
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Overview1: Motivation
◮ Understand transformation theory of surfaces in terms ofmodern language; flat connections, harmonic maps and loopgroups.
◮ More specifically: Why do constant positive Gauss curvaturesurfaces have only Bianchi-Backlund transformation? Note!Constant negative Gauss curvature surfaces have alsoBacklund transformation.
Remark
◮ Backlund transformation is a transformation by tangential linecongruences and Bianchi-Backlund transformation is anextension of the Backlund transformation by Bianchi.
◮ Classical theorem by Backlund says that if two surfaces arerelated by Backlund transformation, then they are constantnegative Gauss curvature surfaces.
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Overview 2: Previous works
Remark
◮ Uhlenbeck considered dressing action on extended frame ofharmonic maps into Lie groups, J. Diff. Geom. 1989.
◮ Uhlenbeck proved that the simple factor dressing action onextended frames of negative CGC surfaces is equivalent to theBacklund transformation, J. Geom. Phys. 1992.
◮ Terng and Uhlenbeck generalized the dressing action toU/K-system, Comm. Pure Appl. Math. 2000.
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Backlund transformation
Theorem (Backlund)
S,S′ ⊂ R3 : a surface in Euclidean three space
S and S′ are related by the tangential line congruences withconstant angle and distance.
⇓S and S′ are constant negative Gauss curvature surfaces.
Remark
◮ Tangential line congruences ℓ are line congruences whichtangent 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-Backlund transformation
Theorem (Bianchi)
Let S ⊂ R3 be a CGC surface. There exits a surface S′ incomplex Euclidean three space such that S and S′ are related bytangential line congruences with complex constant angle. Thistransformation is called a Bianchi-Backlund transformation.Moreover, twice of the Bianchi-Backlund transformation withsuitable angle conditions gives a CGC surface in R3
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Permutability and superposition formula
Sβ,β∗κ = Sβ∗ ,β
κ .
θ
u u
θ∗
β β∗
β∗ β
The superposition formula:
tanh
(
u − u
2
)
= tanh
(
β − β∗
2
)
tanh
(
θβ − θβ∗
2
)
,
where u = θβ,β∗ = θβ∗,β.
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A family of flat connections
Theorem (Pohlmeyer 1976)
Let M be a simply connected open Riemann surface and G/K asymmetric space. The followings are equivalent.
1. Φ : M → G/K is a harmonic map.
2. There exist a Fλ : M → ΛGσ such thatF−1λ dFλ = λ−1α′
p + αk + λα′′p and π ◦ F|λ=1 = Φ.
Corollary
The set of extended frames, Fλ.m
The family of CMC surfaces, fλ.
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Loop groups
ΛGCσ := {H : S1 → GC | σH(λ) = H(−λ)},
ΛGσ := {H ∈ ΛGCσ | H(λ) = H(λ) on λ ∈ S1},
Λ±GCσ :=
{
H± ∈ ΛGCσ |
H± can be extend holomorphicallyto D (or E).
}
,
Λ±∗ GC
σ :={
H± ∈ Λ±GCσ | H+(0) = id (or H−(∞) = id)
}
,
ΛgCσ := {h : S1 → gC | σh(λ) = h(−λ)}.
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Loop groups
ΛGCσ := {H : S1 → GC | σH(λ) = H(−λ)},
ΛGσ := {H ∈ ΛGCσ | H(λ) = H(λ) on λ ∈ S1},
Λ±GCσ :=
{
H± ∈ ΛGCσ |
H± can be extend holomorphicallyto D (or E).
}
,
Λ±∗ GC
σ :={
H± ∈ Λ±GCσ | H+(0) = id (or H−(∞) = id)
}
,
ΛgCσ := {h : S1 → gC | σh(λ) = h(−λ)}.
Fourier expansions of H ∈ ΛGσ and H± ∈ Λ±GCσ:
H = · · · + λ−2H−2 + λ−1H−1 + H0 + λH1 + λ2H2 + · · · ,
H± = H±,0 + λ±1H±,1 + λ±2H±,2 + · · · ,
where Hj = H−j, σHj = (−1)jHj, and σH±,j = (−1)jH±,j.
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Loop groups factorizations
Theorem (Birkhoff and Iwasawa decompositions)
1. Birkhoff decomposition:
Λ+∗G
Cσ × Λ−GC
σ → ΛGCσ
is a diffeomorphism onto the open dense subsetΛ+∗G
Cσ · Λ−GC
σ of ΛGCσ.
2. Iwasawa decomposition: Assume that G is compact.
ΛGσ × Λ−GCσ → ΛGC
σ
is a diffeomorphism onto ΛGCσ.
RemarkThe Iwasawa decomposition is obtained from the Birkhoffdecomposition and a real from.
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Dressing actions and Bianchi-Backlund transformations 1
Let F be an extended framing and g an element in Λ+GCσ.
Decompose gF according to the Iwasawa decomposition as
ΛGCσ ∋ gF = FV+ ∈ ΛGσ × Λ+GC
σ.
Then
◮ F is again the extended framing. Thus Λ+GCσ acts, the
dressing action, that is, Id#F = F andg(#(f#F)) = (gf)#F, where g#F = F = gFV−1
+ .
◮ Bianchi permutability theorem is just associativity of theaction:
g2#(g1#F) = g2#(g1#F),
where g2g1 = g2g1.
◮ The dressing action by rational loops with simple pole is calledthe simple type dressing action.
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Dressing actions and Bianchi-Backlund transformations 2
Theorem (Inoguchi-Kobayashi, 2005)
The simple type dressing action and Bianchi-Backlundtransformation are equivalent, where the Bianchi-Backlundtransformation is a transformation of a CMC surface by linecongruences.
Figure: A twizzler, its Bianchi-Backlund transformation and a bubbleton.
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Recall that a simple factor dressing by g:
g#F = gFV−1+ .
This action corresponds to the twice of Bianchi-Backlundtransformation.The g has two simple poles at λ1, λ2, which are relatedλ2 = 1/λ1 ∈ C× \ S1.
◮ Is it possible to factor g to
g = g2g1
so that each gj corresponds to once Bianchi-Backlund?
◮ Answer: Yes, but one needs an extension of the dressingaction.
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Constant negative Gauss curvature surfaces
FactThere exists a similar dressing action on constant negative Gausscurvature surfaces. Try to unify negative and positive Gausscurvature surfaces.
⇓
◮ Complex CMC surfaces (Dorfmeister-Kobayashi-Pedit 2010).
◮ Real form surfaces (Kobayashi 2011).
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Ruh-Vilms type theorem
TheoremThe following two conditions are equivalent:
1. The complex mean curvature H is constant.
2. There exist a Fλ : D2 → ΛGCσ(= ΛSL2Cσ) such that
F−1λ dFλ = λ−1α′
p + αk + λα′′p and π ◦ F|λ=1 = Φ, where
Φ is the unit normal to f. Here ′ (resp. ′′) denotes dz-partand (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 Λsl2Cσ:
c1 : g(λ) 7→ −g(−1/λ)t, c2 : g(λ) 7→ g
(
−1/λ)
,
c3 : g(λ) 7→ −g(
1/λ)t
, c4 : g(λ) 7→ −Ad
(
1/√
i 0
0√
i
)
g(i/λ)t,
where g ∈ Λsl2Cσ. Then, the almost compact real forms ofΛsl2Cσ are the following real Lie subalgebras of Λsl2Cσ:
Λsl2C(c,j)σ = {g ∈ Λsl2Cσ | cj ◦ g(λ) = g(λ)} .
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Almost split real forms
Theorem (Kobayashi, 2011)
Let sj for j ∈ {1, 2, 3} be the following involutions on Λsl2Cσ:
s1 : g(λ) 7→ −g(−λ)t, s2 : g(λ) 7→ g
(
−λ)
,
s3 : g(λ) 7→ −Ad(
λ 00 λ−1
)
g(
λ)t
,
where g ∈ Λsl2Cσ. Then, the almost split real forms of Λsl2Cσ
are the following real Lie subalgebras of Λsl2Cσ:
Λsl2C(s,j)σ = {g ∈ Λsl2Cσ | sj ◦ g(λ) = g(λ)} .
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Integrable surfaces
Theorem (Kobayashi, 2011)
The real forms induce the following integrable surfaces.
Surfaces class Gauß curvature Gauß curvature : Parallel CMC
R3 K(s,3) = −4|H|2 K(c,3) = 4|H|2 : H(c,3) = |H|
Spacelike R1,2 K(s,1) = 4|H|2 K(c,1) = −4|H|2 : H(c,1) = |H|
Timelike R1,2 K(c,2) = −4|H|2 K(s,2) = 4|H|2 : H(s,2) = |H|
H3 * : H(c,4) = tanh(q)
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Double loop group decomposition
The following r-loop group and its loop subgroups will be used.
Hr,R = ΛrSL2C × ΛRSL2C,
H+r,R = Λ+
r SL2C × Λ−R SL2C ⊂ Hr,R,
H−r,R =
{
(g1, g2) ∈ Hr,R
∣
∣
∣
∣
g1 and g2 extends holomorphicallyto Ar,R and g1|Ar,R = g2|Ar,R
}
.
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TheoremThe multiplication map
H− × H+ → H
is 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+r , h−R ),
where (F, F) ∈ H− denotes the boundary values on Cr and CR ofthe map F : Ar,R → ΛSL2Cσ and (h+r , h
−R ) is an element in H+.
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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) = (grF, gRF)Ar,R,
where the subscript Ar,R denotes the H− part of the double loopgroup decomposition. Denote (F, F) by
(F, F) = (gr, gR)#(F, F).
It is easy to see that (Id, Id)#(F, F) = (F, F) and(gr, gR)#((gr, gR)#(F, F)) = ((gr, gR) · (gr, gR))#(F, F).
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ρ dressing action on complex extended frames
We generalized H to K as
K = {(gr, gR) | gr ∈ ΛrSL2C and gR ∈ ˜ΛRSL2C.}.
Let (gr, gR) ∈ K and
ρ =(
0√
λ
−√
λ−1
0
)
.
Define (F, F) by
(F, F) = (grF, gRFρj)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 dressingaction.
2. If F and g satisfy CMC (positive CGC) reality condition, thenthe dressing action reduces to the CMC dressing action.
3. If F and g satisfy negative CGC reality condition, then thedressing action reduces to the negative CGC dressing action.
Remark
◮ The CMC case, the simple poles must be pair λ1, 1/λ1. ThusBianchi-Backlund transformation can only be applied.
◮ The CMC case, the simple poles need not to be pair. ThusBacklund and Bianchi-Backlund transformation can beapplied.