Dressing2011

Dressing actions on integrable surfaces

This is a joint work with Nick Schmitt at Tubingen University.Shimpei Kobayashi, Hirosaki University

6/21, 2011

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

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.

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.

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′.

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

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.

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.

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)}.

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.

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.

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.

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.

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 is 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.

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).

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.

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)} .

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)} .

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)

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

}

.

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+.

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).

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).

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.