Lagrangian and Hamiltonian Mechanics Solutions to the Exercises
Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori...
Transcript of Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori...
![Page 1: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/1.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Hamiltonian stationary Lagrangian tori in C2,revisited
Katrin Leschke
University of Leicester
”Riemann Surfaces, Harmonic Maps and Visualization”Osaka 2008
Katrin Leschke HSL tori
![Page 2: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/2.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian surfaces
Consider R4 with canonical complex structure J such thatω(., .) =< J., . > where < ., . > is the scalar product onR4 and ω the standard symplectic form.
V ∈ Lag(R4) Lagrangian subspace if and only if ω|V = 0.
An immersion f : M → R4 of a Riemann surface M intoR4 = C2 is called Lagrangian if f ∗ω = 0.
The Gauss map γ of a Lagrangian immersion has values inthe space of Lagrangian subspaces Lag(R4):
γ : M → Lag(R4) .
Katrin Leschke HSL tori
![Page 3: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/3.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian surfaces
Consider R4 with canonical complex structure J such thatω(., .) =< J., . > where < ., . > is the scalar product onR4 and ω the standard symplectic form.
V ∈ Lag(R4) Lagrangian subspace if and only if ω|V = 0.
An immersion f : M → R4 of a Riemann surface M intoR4 = C2 is called Lagrangian if f ∗ω = 0.
The Gauss map γ of a Lagrangian immersion has values inthe space of Lagrangian subspaces Lag(R4):
γ : M → Lag(R4) .
Katrin Leschke HSL tori
![Page 4: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/4.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian surfaces
Consider R4 with canonical complex structure J such thatω(., .) =< J., . > where < ., . > is the scalar product onR4 and ω the standard symplectic form.
V ∈ Lag(R4) Lagrangian subspace if and only if ω|V = 0.
An immersion f : M → R4 of a Riemann surface M intoR4 = C2 is called Lagrangian if f ∗ω = 0.
The Gauss map γ of a Lagrangian immersion has values inthe space of Lagrangian subspaces Lag(R4):
γ : M → Lag(R4) .
Katrin Leschke HSL tori
![Page 5: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/5.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian surfaces
Consider R4 with canonical complex structure J such thatω(., .) =< J., . > where < ., . > is the scalar product onR4 and ω the standard symplectic form.
V ∈ Lag(R4) Lagrangian subspace if and only if ω|V = 0.
An immersion f : M → R4 of a Riemann surface M intoR4 = C2 is called Lagrangian if f ∗ω = 0.
The Gauss map γ of a Lagrangian immersion has values inthe space of Lagrangian subspaces Lag(R4):
γ : M → Lag(R4) .
Katrin Leschke HSL tori
![Page 6: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/6.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian angle and Maslov form
U(2) operates on Lag(R4)
Lag(R4) = U(2)/S0(2)
thus we can define
s = det γ : M → S1.
The Lagrangian angle β is the lift of s to the universal cover:
s = e iβ .
Moreover, when M = T 2 = C/Γ is a 2-torus,
β(z) = 2π < β0, z >
where β0 ∈ Γ∗ ⊂ C is called the Maslov form.
Katrin Leschke HSL tori
![Page 7: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/7.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian angle and Maslov form
U(2) operates on Lag(R4)
Lag(R4) = U(2)/S0(2)
thus we can define
s = det γ : M → S1.
The Lagrangian angle β is the lift of s to the universal cover:
s = e iβ .
Moreover, when M = T 2 = C/Γ is a 2-torus,
β(z) = 2π < β0, z >
where β0 ∈ Γ∗ ⊂ C is called the Maslov form.
Katrin Leschke HSL tori
![Page 8: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/8.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian angle and Maslov form
U(2) operates on Lag(R4)
Lag(R4) = U(2)/S0(2)
thus we can define
s = det γ : M → S1.
The Lagrangian angle β is the lift of s to the universal cover:
s = e iβ .
Moreover, when M = T 2 = C/Γ is a 2-torus,
β(z) = 2π < β0, z >
where β0 ∈ Γ∗ ⊂ C is called the Maslov form.
Katrin Leschke HSL tori
![Page 9: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/9.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Lagrangian angle and Maslov form
U(2) operates on Lag(R4)
Lag(R4) = U(2)/S0(2)
thus we can define
s = det γ : M → S1.
The Lagrangian angle β is the lift of s to the universal cover:
s = e iβ .
Moreover, when M = T 2 = C/Γ is a 2-torus,
β(z) = 2π < β0, z >
where β0 ∈ Γ∗ ⊂ C is called the Maslov form.Katrin Leschke HSL tori
![Page 10: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/10.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Variational problems
Consider Hamiltonian stationary Lagrangians (HSL), that isimmersions f : M → C2 which are critical points of the areafunctional
A(f ) =
∫M|df |2
under variations by Hamiltonian vector fields.
Fact: f : M → C2 is Hamiltonian stationary Lagrangian if andonly if its Lagrangian angle map β is harmonic.
Katrin Leschke HSL tori
![Page 11: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/11.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Variational problems
Consider Hamiltonian stationary Lagrangians (HSL), that isimmersions f : M → C2 which are critical points of the areafunctional
A(f ) =
∫M|df |2
under variations by Hamiltonian vector fields.
Fact: f : M → C2 is Hamiltonian stationary Lagrangian if andonly if its Lagrangian angle map β is harmonic.
Katrin Leschke HSL tori
![Page 12: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/12.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Results
Oh: first and second variational formulae of the areafunctional
Oh’s conjecture: Clifford torus minimizes the area in itsHamiltonian isotopy class
Ilmanen, Anciaux: if there exists a smooth minimizer, ithas to be the Clifford torus.
Castro, Chen, Urbano: non-trivial examples.
Helein-Romon: complete description of HSL tori byFourier polynomials; frequencies lie on a circle whoseradius is governed by the Maslov class.
Katrin Leschke HSL tori
![Page 13: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/13.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Results
Oh: first and second variational formulae of the areafunctional
Oh’s conjecture: Clifford torus minimizes the area in itsHamiltonian isotopy class
Ilmanen, Anciaux: if there exists a smooth minimizer, ithas to be the Clifford torus.
Castro, Chen, Urbano: non-trivial examples.
Helein-Romon: complete description of HSL tori byFourier polynomials; frequencies lie on a circle whoseradius is governed by the Maslov class.
Katrin Leschke HSL tori
![Page 14: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/14.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Results
Oh: first and second variational formulae of the areafunctional
Oh’s conjecture: Clifford torus minimizes the area in itsHamiltonian isotopy class
Ilmanen, Anciaux: if there exists a smooth minimizer, ithas to be the Clifford torus.
Castro, Chen, Urbano: non-trivial examples.
Helein-Romon: complete description of HSL tori byFourier polynomials; frequencies lie on a circle whoseradius is governed by the Maslov class.
Katrin Leschke HSL tori
![Page 15: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/15.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Results
Oh: first and second variational formulae of the areafunctional
Oh’s conjecture: Clifford torus minimizes the area in itsHamiltonian isotopy class
Ilmanen, Anciaux: if there exists a smooth minimizer, ithas to be the Clifford torus.
Castro, Chen, Urbano: non-trivial examples.
Helein-Romon: complete description of HSL tori byFourier polynomials; frequencies lie on a circle whoseradius is governed by the Maslov class.
Katrin Leschke HSL tori
![Page 16: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/16.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Results
Oh: first and second variational formulae of the areafunctional
Oh’s conjecture: Clifford torus minimizes the area in itsHamiltonian isotopy class
Ilmanen, Anciaux: if there exists a smooth minimizer, ithas to be the Clifford torus.
Castro, Chen, Urbano: non-trivial examples.
Helein-Romon: complete description of HSL tori byFourier polynomials; frequencies lie on a circle whoseradius is governed by the Maslov class.
Katrin Leschke HSL tori
![Page 17: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/17.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Left normal of a HSL surface
Let f : M → R4 be a conformal immersion. Then the Gaussmap of f is given by
(N,R) : M → S2 × S2 = Gr2(R4) .
Helein-Romon: f Hamiltonian stationary Lagrangian iff the leftnormal N : M → S1 of f takes values in S1 and is harmonic.In fact, identifying R4 = H we can write
df = ejβ2 dz g
and the left normalN = e jβ i
satisfies ∗df = Ndf .
Katrin Leschke HSL tori
![Page 18: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/18.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Left normal of a HSL surface
Let f : M → R4 be a conformal immersion. Then the Gaussmap of f is given by
(N,R) : M → S2 × S2 = Gr2(R4) .
Helein-Romon: f Hamiltonian stationary Lagrangian iff the leftnormal N : M → S1 of f takes values in S1 and is harmonic.
In fact, identifying R4 = H we can write
df = ejβ2 dz g
and the left normalN = e jβ i
satisfies ∗df = Ndf .
Katrin Leschke HSL tori
![Page 19: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/19.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Left normal of a HSL surface
Let f : M → R4 be a conformal immersion. Then the Gaussmap of f is given by
(N,R) : M → S2 × S2 = Gr2(R4) .
Helein-Romon: f Hamiltonian stationary Lagrangian iff the leftnormal N : M → S1 of f takes values in S1 and is harmonic.In fact, identifying R4 = H we can write
df = ejβ2 dz g
and the left normalN = e jβ i
satisfies ∗df = Ndf .
Katrin Leschke HSL tori
![Page 20: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/20.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Left normal of a HSL surface
Let f : M → R4 be a conformal immersion. Then the Gaussmap of f is given by
(N,R) : M → S2 × S2 = Gr2(R4) .
Helein-Romon: f Hamiltonian stationary Lagrangian iff the leftnormal N : M → S1 of f takes values in S1 and is harmonic.In fact, identifying R4 = H we can write
df = ejβ2 dz g
and the left normalN = e jβ i
satisfies ∗df = Ndf .
Katrin Leschke HSL tori
![Page 21: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/21.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
Helein-Romon: family of flat connections
dλ = λ−2α−2 + λ−1α−1 + α0 + λα1 + λ2α2
where αj lie in the eigenspaces of an order 4 autmorphismof the Lie algebra of the group of symplectic isometries ofR4.
(McIntosh-Romon) Associate minimal polynomial Killingfield to define a spectral curve of f .
Gives all weakly conformal Hamiltonian stationaryLagrangian tori
Gives Hamiltonian stationary Lagrangian tori with branchpoints and ”no” control on the branch locus.
Katrin Leschke HSL tori
![Page 22: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/22.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
Helein-Romon: family of flat connections
dλ = λ−2α−2 + λ−1α−1 + α0 + λα1 + λ2α2
where αj lie in the eigenspaces of an order 4 autmorphismof the Lie algebra of the group of symplectic isometries ofR4.
(McIntosh-Romon) Associate minimal polynomial Killingfield to define a spectral curve of f .
Gives all weakly conformal Hamiltonian stationaryLagrangian tori
Gives Hamiltonian stationary Lagrangian tori with branchpoints and ”no” control on the branch locus.
Katrin Leschke HSL tori
![Page 23: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/23.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
Helein-Romon: family of flat connections
dλ = λ−2α−2 + λ−1α−1 + α0 + λα1 + λ2α2
where αj lie in the eigenspaces of an order 4 autmorphismof the Lie algebra of the group of symplectic isometries ofR4.
(McIntosh-Romon) Associate minimal polynomial Killingfield to define a spectral curve of f .
Gives all weakly conformal Hamiltonian stationaryLagrangian tori
Gives Hamiltonian stationary Lagrangian tori with branchpoints and ”no” control on the branch locus.
Katrin Leschke HSL tori
![Page 24: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/24.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
Helein-Romon: family of flat connections
dλ = λ−2α−2 + λ−1α−1 + α0 + λα1 + λ2α2
where αj lie in the eigenspaces of an order 4 autmorphismof the Lie algebra of the group of symplectic isometries ofR4.
(McIntosh-Romon) Associate minimal polynomial Killingfield to define a spectral curve of f .
Gives all weakly conformal Hamiltonian stationaryLagrangian tori
Gives Hamiltonian stationary Lagrangian tori with branchpoints and ”no” control on the branch locus.
Katrin Leschke HSL tori
![Page 25: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/25.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
HSL tori f : T 2 → R4 are conformal: have multiplierspectral curve (Schmidt, Taimanov, Bohle-L-Pedit-Pinkall)
The left normal of a HSL torus N : T 2 → S1 is harmonic:have spectral curve of the harmonic left normal (Hitchin).
Katrin Leschke HSL tori
![Page 26: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/26.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Spectral curves
HSL tori f : T 2 → R4 are conformal: have multiplierspectral curve (Schmidt, Taimanov, Bohle-L-Pedit-Pinkall)
The left normal of a HSL torus N : T 2 → S1 is harmonic:have spectral curve of the harmonic left normal (Hitchin).
Katrin Leschke HSL tori
![Page 27: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/27.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The family of flat connections
A map N : M → S2 ⊂ Im H is harmonic if and only if thefamily of complex connections
dµ = d + (µ− 1)A1,0 + (µ−1 − 1)A0,1
on the trivial bundle H is flat, where A = 14 (∗dN + NdN) and
µ ∈ C∗.
Here C = span1, I where the complex structure I on H isdefined by right multiplication by i .Moreover, for ω ∈ Ω1
ω1,0 =1
2(ω − I ∗ ω), ω0,1 =
1
2(ω + I ∗ ω)
denote the (1, 0) and (0, 1) parts with respect to the complexstructure I .
Katrin Leschke HSL tori
![Page 28: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/28.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The family of flat connections
A map N : M → S2 ⊂ Im H is harmonic if and only if thefamily of complex connections
dµ = d + (µ− 1)A1,0 + (µ−1 − 1)A0,1
on the trivial bundle H is flat, where A = 14 (∗dN + NdN) and
µ ∈ C∗.Here C = span1, I where the complex structure I on H isdefined by right multiplication by i .
Moreover, for ω ∈ Ω1
ω1,0 =1
2(ω − I ∗ ω), ω0,1 =
1
2(ω + I ∗ ω)
denote the (1, 0) and (0, 1) parts with respect to the complexstructure I .
Katrin Leschke HSL tori
![Page 29: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/29.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The family of flat connections
A map N : M → S2 ⊂ Im H is harmonic if and only if thefamily of complex connections
dµ = d + (µ− 1)A1,0 + (µ−1 − 1)A0,1
on the trivial bundle H is flat, where A = 14 (∗dN + NdN) and
µ ∈ C∗.Here C = span1, I where the complex structure I on H isdefined by right multiplication by i .Moreover, for ω ∈ Ω1
ω1,0 =1
2(ω − I ∗ ω), ω0,1 =
1
2(ω + I ∗ ω)
denote the (1, 0) and (0, 1) parts with respect to the complexstructure I .
Katrin Leschke HSL tori
![Page 30: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/30.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of a harmonic map [Hitchin]
Let N : M → S2 and dµ the associated family of flatconnections.
If M = C/Γ is a 2-torus, the parallel sections α ∈ Γ(H) ofdµ with multiplier, that is
γ∗α = αhγ , γ ∈ Γ, hγ ∈ C∗,C = span1, i,
are the eigenvectors of the monodromy of dµ.
The spectral curve Σe of N : T 2 → S2 is thenormalization of
Eig := (µ, h) | ∃α : dµα = 0, γ∗α = αhγ , γ ∈ Γ
Katrin Leschke HSL tori
![Page 31: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/31.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of a harmonic map [Hitchin]
Let N : M → S2 and dµ the associated family of flatconnections.
If M = C/Γ is a 2-torus, the parallel sections α ∈ Γ(H) ofdµ with multiplier, that is
γ∗α = αhγ , γ ∈ Γ, hγ ∈ C∗,C = span1, i,
are the eigenvectors of the monodromy of dµ.
The spectral curve Σe of N : T 2 → S2 is thenormalization of
Eig := (µ, h) | ∃α : dµα = 0, γ∗α = αhγ , γ ∈ Γ
Katrin Leschke HSL tori
![Page 32: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/32.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of a harmonic map [Hitchin]
Let N : M → S2 and dµ the associated family of flatconnections.
If M = C/Γ is a 2-torus, the parallel sections α ∈ Γ(H) ofdµ with multiplier, that is
γ∗α = αhγ , γ ∈ Γ, hγ ∈ C∗,C = span1, i,
are the eigenvectors of the monodromy of dµ.
The spectral curve Σe of N : T 2 → S2 is thenormalization of
Eig := (µ, h) | ∃α : dµα = 0, γ∗α = αhγ , γ ∈ Γ
Katrin Leschke HSL tori
![Page 33: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/33.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The eigenline bundle [Hitchin]
Let N : M → S2 and dµ the associated family of flatconnections.
Generically, the space of parallel sections of dµ with agiven multiplier is 1-dimensional, and one obtains theeigenline bundle E → Σe .
The harmonic map can be reconstructed by linear flow inthe Jacobian of Σe .
Katrin Leschke HSL tori
![Page 34: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/34.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The eigenline bundle [Hitchin]
Let N : M → S2 and dµ the associated family of flatconnections.
Generically, the space of parallel sections of dµ with agiven multiplier is 1-dimensional, and one obtains theeigenline bundle E → Σe .
The harmonic map can be reconstructed by linear flow inthe Jacobian of Σe .
Katrin Leschke HSL tori
![Page 35: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/35.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Let f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N and family of flat connections dµ.
Theorem (L-Romon, Moriya)
All parallel sections with multiplier can be computed explicitely:
αµ± = e j β2 (1∓ k
√µ−1)e±2π(<Aµ,.>+i<Cµ,.>)
with Aµ = iβ04 (√µ−1 −√µ), Cµ = β0
4 (√µ−1 +
õ).
Katrin Leschke HSL tori
![Page 36: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/36.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Let f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N and family of flat connections dµ.
Theorem (L-Romon, Moriya)
All parallel sections with multiplier can be computed explicitely:
αµ± = e j β2 (1∓ k
√µ−1)e±2π(<Aµ,.>+i<Cµ,.>)
with Aµ = iβ04 (√µ−1 −√µ), Cµ = β0
4 (√µ−1 +
õ).
Katrin Leschke HSL tori
![Page 37: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/37.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 38: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/38.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 39: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/39.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 40: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/40.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 41: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/41.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 42: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/42.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The spectral curve of the left normal
Lef f : T 2 → C2 be a Hamiltonian stationary Lagrangian toruswith spectral curve Σe of its harmonic left normal N.
Theorem (L-Romon)
Σe compactifies with Σe = CP1.
µ : Σe → CP1, (µ, h) 7→ µ is a 2-fold covering, branchedover 0,∞.
The eigenline bundle E extends holomorphically to Σe .
Let J ∈ Γ(End(H)), J2 = −1, be the complex structuregiven by the quaternionic extension of
J|Ex∞ = I |Ex∞ , µ(x∞) =∞ .
Then J is in fact the complex structure given by leftmultiplication by N.
Katrin Leschke HSL tori
![Page 43: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/43.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : M → C2, be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N.
Theorem (L-Romon)
Let α ∈ Γ(H) be a parallel section of dµ and put
T−1 =1
2(Nα(a− 1)α−1 + αbα−1)
a = µ+µ−1
2 , b = i µ−1−µ
2 .
Katrin Leschke HSL tori
![Page 44: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/44.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : M → C2, be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N.
Theorem (L-Romon)
Let α ∈ Γ(H) be a parallel section of dµ and put
T−1 =1
2(Nα(a− 1)α−1 + αbα−1)
a = µ+µ−1
2 , b = i µ−1−µ
2 .Then
N = −TNT−1
is a harmonic map N : M → S2 of M into the 2-sphere.
Katrin Leschke HSL tori
![Page 45: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/45.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : M → C2, be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N.
Theorem (L-Romon)
Let α be a parallel section with multiplier and put
T−1 =1
2(Nα(a− 1)α−1 + αbα−1)
a = µ+µ−1
2 , b = i µ−1−µ
2 .
Katrin Leschke HSL tori
![Page 46: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/46.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : M → C2, be a Hamiltonian stationary Lagrangian toruswith harmonic left normal N.
Theorem (L-Romon)
Let α be a parallel section with multiplier and put
T−1 =1
2(Nα(a− 1)α−1 + αbα−1)
a = µ+µ−1
2 , b = i µ−1−µ
2 .
Then T−1 is again globally defined and
N = −TNT−1
is a harmonic map N : M → S2 from M into the 2-sphere.
Katrin Leschke HSL tori
![Page 47: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/47.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : T 2 → C2, be a Hamiltonian stationary Lagrangian torus
with harmonic left normal N and df = ejβ2 dzg .
Theorem (L-Romon)
If α ∈ Γ(H) is a parallel section with multiplier, than N is theleft normal of a HSL torus
f = f + TH−1,
where H = πg−1β0ejβ2 k.
We call f a µ-Darboux transform of f .
Katrin Leschke HSL tori
![Page 48: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/48.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Let f : T 2 → C2, be a Hamiltonian stationary Lagrangian torus
with harmonic left normal N and df = ejβ2 dzg .
Theorem (L-Romon)
If α ∈ Γ(H) is a parallel section with multiplier, than N is theleft normal of a HSL torus
f = f + TH−1,
where H = πg−1β0ejβ2 k.
We call f a µ-Darboux transform of f .
Katrin Leschke HSL tori
![Page 49: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/49.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Remark
Locally, a µ-Darboux transform is always at least
constrained Willmore .
A similar theorem holds both for µ-Darboux transforms ofCMC tori (Carberry-L-Pedit), and (constrained) Willmoretori (Bohle).
Katrin Leschke HSL tori
![Page 50: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/50.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Remark
Locally, a µ-Darboux transform is always at least
constrained Willmore .
A similar theorem holds both for µ-Darboux transforms ofCMC tori (Carberry-L-Pedit), and (constrained) Willmoretori (Bohle).
Katrin Leschke HSL tori
![Page 51: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/51.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Remark
The µ-Darboux transformation is a generalization of theclassical Darboux transformation.
f ] is called a classical Darboux transformation of f if thereexists a sphere congruence enveloping f and f ].
The µ-Darboux transformation satisfies a weakerenveloping condition.
Katrin Leschke HSL tori
![Page 52: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/52.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Remark
The µ-Darboux transformation is a generalization of theclassical Darboux transformation.
f ] is called a classical Darboux transformation of f if thereexists a sphere congruence enveloping f and f ].
The µ-Darboux transformation satisfies a weakerenveloping condition.
Katrin Leschke HSL tori
![Page 53: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/53.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
µ-Darboux transforms
Remark
The µ-Darboux transformation is a generalization of theclassical Darboux transformation.
f ] is called a classical Darboux transformation of f if thereexists a sphere congruence enveloping f and f ].
The µ-Darboux transformation satisfies a weakerenveloping condition.
Katrin Leschke HSL tori
![Page 54: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/54.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial
:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 55: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/55.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 56: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/56.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 57: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/57.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 58: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/58.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 59: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/59.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Special cases
For special parameter µ the transform on the left normal istrivial:
if µ ∈ S1 then N = −N.
if µ > 0 then N = N.
Question: Is f for µ > 0 the orginal HSL torus?
More generally, does the Lagrangian angle β determine f ?
What is the condition for the existence of a HSL torus withLagrangian angle β?
Katrin Leschke HSL tori
![Page 60: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/60.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Recall: A Hamiltonian stationary Lagrangian immersion f hasLagrangian angle β ⇐⇒ ∗df = Ndf with N = e jβ i .
The operator D : Γ(H)→ Γ(KH)
D :=1
2(d + J ∗ d)
is a (quaternionic) holomorphic structure where the complexstructure J on H is given by left multiplication by N.
Note: dµα = 0 =⇒ Dα = 0.
Katrin Leschke HSL tori
![Page 61: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/61.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Recall: A Hamiltonian stationary Lagrangian immersion f hasLagrangian angle β ⇐⇒ ∗df = Ndf with N = e jβ i .
The operator D : Γ(H)→ Γ(KH)
D :=1
2(d + J ∗ d)
is a (quaternionic) holomorphic structure where the complexstructure J on H is given by left multiplication by N.
Note: dµα = 0 =⇒ Dα = 0.
Katrin Leschke HSL tori
![Page 62: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/62.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Recall: A Hamiltonian stationary Lagrangian immersion f hasLagrangian angle β ⇐⇒ ∗df = Ndf with N = e jβ i .
The operator D : Γ(H)→ Γ(KH)
D :=1
2(d + J ∗ d)
is a (quaternionic) holomorphic structure where the complexstructure J on H is given by left multiplication by N.
Goal: given N = e jβ i find all holomorphic sections α ∈ ker D.
Note: dµα = 0 =⇒ Dα = 0.
Katrin Leschke HSL tori
![Page 63: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/63.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Recall: A Hamiltonian stationary Lagrangian immersion f hasLagrangian angle β ⇐⇒ ∗df = Ndf with N = e jβ i .
The operator D : Γ(H)→ Γ(KH)
D :=1
2(d + J ∗ d)
is a (quaternionic) holomorphic structure where the complexstructure J on H is given by left multiplication by N.
Goal: given N = e jβ i find all holomorphic sections α ∈ ker D.
Note: dµα = 0 =⇒ Dα = 0.
Katrin Leschke HSL tori
![Page 64: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/64.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Recall: A Hamiltonian stationary Lagrangian immersion f hasLagrangian angle β ⇐⇒ ∗df = Ndf with N = e jβ i .
The operator D : Γ(H)→ Γ(KH)
D :=1
2(d + J ∗ d)
is a (quaternionic) holomorphic structure where the complexstructure J on H is given by left multiplication by N.
Goal: given N = e jβ i find all holomorphic sections α withmultiplier, that is α ∈ ker D with γ∗α = αhγ , γ ∈ Γ.
Note: dµα = 0 =⇒ Dα = 0.
Katrin Leschke HSL tori
![Page 65: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/65.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Let f : C/Γ→ R4 be a Hamiltonian stationary torus. For(A,B) ∈ C2 consider
|δ − B|2 − |A|2 =|β0|2
4, < A, δ − B >= 0 (1)
with δ ∈ Γ∗ + β02 .
Denote by
Γ∗A,B = δ ∈ Γ∗ +β0
2| δ satisfies (1)
the set of admissible frequencies.
Katrin Leschke HSL tori
![Page 66: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/66.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Let f : C/Γ→ R4 be a Hamiltonian stationary torus. For(A,B) ∈ C2 consider
|δ − B|2 − |A|2 =|β0|2
4, < A, δ − B >= 0 (1)
with δ ∈ Γ∗ + β02 . Denote by
Γ∗A,B = δ ∈ Γ∗ +β0
2| δ satisfies (1)
the set of admissible frequencies.
Katrin Leschke HSL tori
![Page 67: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/67.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Let f : C/Γ→ R4 be a Hamiltonian stationary torus, and Dthe quaternionic holomorphic structure given by the complexstructure J.
Theorem (L-Romon)
Multipliers of holomorphic sections are exactly given by
hA,B = e2π(<A,·>−i<B,·>)
with Γ∗A,B 6= ∅.
For δ ∈ Γ∗A,B
αδ = ejβ2 (1− kλδ)e2π(<A,·>+<δ−B,·>)
with λδ ∈ C∗ is a (monochromatic) holomorphic section.
Katrin Leschke HSL tori
![Page 68: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/68.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Let f : C/Γ→ R4 be a Hamiltonian stationary torus, and Dthe quaternionic holomorphic structure given by the complexstructure J.
Theorem (L-Romon)
Multipliers of holomorphic sections are exactly given by
hA,B = e2π(<A,·>−i<B,·>)
with Γ∗A,B 6= ∅.For δ ∈ Γ∗A,B
αδ = ejβ2 (1− kλδ)e2π(<A,·>+<δ−B,·>)
with λδ ∈ C∗ is a (monochromatic) holomorphic section.
Katrin Leschke HSL tori
![Page 69: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/69.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
HSL tori with prescribed Lagrangian angle[Helein-Romon, L-Romon]
Let Γ be a lattice in C, and let β0 ∈ Γ∗. Then β = 2π < β0, >is a Lagrangian angle of a Hamiltonian stationary torus f if andonly if
Γ∗0,0 ) ±β0
2
In this case, all HSL tori with Lagrangian angle β are (up totranslation) of the form
f =∑
δ∈Γ∗0,0\±β02
αδmδ, mδ ∈ C .
Katrin Leschke HSL tori
![Page 70: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/70.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
HSL tori with prescribed Lagrangian angle[Helein-Romon, L-Romon]
Let Γ be a lattice in C, and let β0 ∈ Γ∗. Then β = 2π < β0, >is a Lagrangian angle of a Hamiltonian stationary torus f if andonly if
Γ∗0,0 ) ±β0
2
In this case, all HSL tori with Lagrangian angle β are (up totranslation) of the form
f =∑
δ∈Γ∗0,0\±β02
αδmδ, mδ ∈ C .
Katrin Leschke HSL tori
![Page 71: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/71.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
HSL tori with prescribed Lagrangian angle[Helein-Romon, L-Romon]
Let Γ be a lattice in C, and let β0 ∈ Γ∗. Then β = 2π < β0, >is a Lagrangian angle of a Hamiltonian stationary torus f if andonly if
Γ∗0,0 ) ±β0
2
In this case, all HSL tori with Lagrangian angle β are (up totranslation) of the form
f =∑
δ∈Γ∗0,0\±β02
αδmδ, mδ ∈ C .
Katrin Leschke HSL tori
![Page 72: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/72.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Theorem (L-Romon)
Every holomorphic section with multiplier hA,B is given by
α =∑δ∈Γ∗A,B
αδmδ
mδ ∈ C.
|Γ∗A,B | = 1 away from a discrete set of pairs (A,B).
Katrin Leschke HSL tori
![Page 73: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/73.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Holomorphic sections with multiplier
Theorem (L-Romon)
Every holomorphic section with multiplier hA,B is given by
α =∑δ∈Γ∗A,B
αδmδ
mδ ∈ C.
|Γ∗A,B | = 1 away from a discrete set of pairs (A,B).
Katrin Leschke HSL tori
![Page 74: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/74.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The multiplier spectral curve
Let Spec := h | ∃α ∈ ker D : γ∗α = αhγ , γ ∈ Γ, and Σ itsnormalization.
Then there exists a line bundle L such that
Lσ = H0σ
for generic points σ ∈ Σ.(see Bohle-L-Pedit-Pinkall for general conformal tori).
Theorem (L-Romon)
The spectral curves Σe and Σ of a Hamiltonian stationarytorus are biholomorphic, and the eigenline bundle E and Lcoincide.
In particular, the multiplier spectral curve of aHamiltonian stationary torus can be compactified and hasgeometric genus 0.
Katrin Leschke HSL tori
![Page 75: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/75.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The multiplier spectral curve
Let Spec := h | ∃α ∈ ker D : γ∗α = αhγ , γ ∈ Γ, and Σ itsnormalization.Then there exists a line bundle L such that
Lσ = H0σ
for generic points σ ∈ Σ.(see Bohle-L-Pedit-Pinkall for general conformal tori).
Theorem (L-Romon)
The spectral curves Σe and Σ of a Hamiltonian stationarytorus are biholomorphic, and the eigenline bundle E and Lcoincide.
In particular, the multiplier spectral curve of aHamiltonian stationary torus can be compactified and hasgeometric genus 0.
Katrin Leschke HSL tori
![Page 76: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/76.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The multiplier spectral curve
Let Spec := h | ∃α ∈ ker D : γ∗α = αhγ , γ ∈ Γ, and Σ itsnormalization.Then there exists a line bundle L such that
Lσ = H0σ
for generic points σ ∈ Σ.(see Bohle-L-Pedit-Pinkall for general conformal tori).
Theorem (L-Romon)
The spectral curves Σe and Σ of a Hamiltonian stationarytorus are biholomorphic, and the eigenline bundle E and Lcoincide.
In particular, the multiplier spectral curve of aHamiltonian stationary torus can be compactified and hasgeometric genus 0.
Katrin Leschke HSL tori
![Page 77: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/77.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
The multiplier spectral curve
Let Spec := h | ∃α ∈ ker D : γ∗α = αhγ , γ ∈ Γ, and Σ itsnormalization.Then there exists a line bundle L such that
Lσ = H0σ
for generic points σ ∈ Σ.(see Bohle-L-Pedit-Pinkall for general conformal tori).
Theorem (L-Romon)
The spectral curves Σe and Σ of a Hamiltonian stationarytorus are biholomorphic, and the eigenline bundle E and Lcoincide.
In particular, the multiplier spectral curve of aHamiltonian stationary torus can be compactified and hasgeometric genus 0.
Katrin Leschke HSL tori
![Page 78: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/78.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Again, we can use holomorphic sections with multiplier todefine a Darboux transform
f = f + TH−1
of a HSL torus f where T = αβ−1, dα = −dfHβ and
H = πg−1β0ejβ2 k .
Theorem (L-Romon)
If α = αδ is a monochromatic holomorphic section then fis HSL with (after reparametrization) Lagrangian angle β.
f is obtained as limit of Darboux transforms withmultiplier σ → σ∞ ∈ Σ.
Katrin Leschke HSL tori
![Page 79: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/79.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Again, we can use holomorphic sections with multiplier todefine a Darboux transform
f = f + TH−1
of a HSL torus f where T = αβ−1, dα = −dfHβ and
H = πg−1β0ejβ2 k .
Theorem (L-Romon)
If α = αδ is a monochromatic holomorphic section then fis HSL with (after reparametrization) Lagrangian angle β.
f is obtained as limit of Darboux transforms withmultiplier σ → σ∞ ∈ Σ.
Katrin Leschke HSL tori
![Page 80: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/80.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Again, we can use holomorphic sections with multiplier todefine a Darboux transform
f = f + TH−1
of a HSL torus f where T = αβ−1, dα = −dfHβ and
H = πg−1β0ejβ2 k .
Theorem (L-Romon)
If α = αδ is a monochromatic holomorphic section then fis HSL with (after reparametrization) Lagrangian angle β.
f is obtained as limit of Darboux transforms withmultiplier σ → σ∞ ∈ Σ.
Katrin Leschke HSL tori
![Page 81: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/81.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
The Darboux transformation is a further generalization of theµ-Darboux transformation:
Theorem (L-Romon)
The monochromatic holomorphic sections are exactly thedµ-parallel sections for some µ ∈ C∗.
In other words, the monochromatic Darboux transforms areexactly the µ-Darboux transforms.
Katrin Leschke HSL tori
![Page 82: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/82.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
The Darboux transformation is a further generalization of theµ-Darboux transformation:
Theorem (L-Romon)
The monochromatic holomorphic sections are exactly thedµ-parallel sections for some µ ∈ C∗.
In other words, the monochromatic Darboux transforms areexactly the µ-Darboux transforms.
Katrin Leschke HSL tori
![Page 83: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/83.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Theorem (L-Romon)
If |Γ∗0,0| = 4 then all monochromatic Darboux transformsare after reparametrization f .
However, there exist examples where |Γ∗0,0| > 4, and theresulting monochromatic Darboux transforms are notMobius transformations of f .
Moreover, there exists HSL tori with polychromaticholomorphic sections α so that the corresponding Darbouxtransforms are not Lagrangian in C2.
Katrin Leschke HSL tori
![Page 84: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/84.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Theorem (L-Romon)
If |Γ∗0,0| = 4 then all monochromatic Darboux transformsare after reparametrization f .
However, there exist examples where |Γ∗0,0| > 4, and theresulting monochromatic Darboux transforms are notMobius transformations of f .
Moreover, there exists HSL tori with polychromaticholomorphic sections α so that the corresponding Darbouxtransforms are not Lagrangian in C2.
Katrin Leschke HSL tori
![Page 85: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/85.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Theorem (L-Romon)
If |Γ∗0,0| = 4 then all monochromatic Darboux transformsare after reparametrization f .
However, there exist examples where |Γ∗0,0| > 4, and theresulting monochromatic Darboux transforms are notMobius transformations of f .
Moreover, there exists HSL tori with polychromaticholomorphic sections α
so that the corresponding Darbouxtransforms are not Lagrangian in C2.
Katrin Leschke HSL tori
![Page 86: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/86.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Darboux transforms
Theorem (L-Romon)
If |Γ∗0,0| = 4 then all monochromatic Darboux transformsare after reparametrization f .
However, there exist examples where |Γ∗0,0| > 4, and theresulting monochromatic Darboux transforms are notMobius transformations of f .
Moreover, there exists HSL tori with polychromaticholomorphic sections α so that the corresponding Darbouxtransforms are not Lagrangian in C2.
Katrin Leschke HSL tori
![Page 87: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/87.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Homogeneous torus
Katrin Leschke HSL tori
![Page 88: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/88.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Clifford torus
Katrin Leschke HSL tori
![Page 89: Hamiltonian stationary Lagrangian tori in C2, revisited · Hamiltonian stationary Lagrangian tori in C2, ... if there exists a smooth minimizer, it has to be the Cli ord torus. Castro,](https://reader031.fdocuments.in/reader031/viewer/2022021712/5b9ef74409d3f2d0208c787b/html5/thumbnails/89.jpg)
HSL tori
KatrinLeschke
HSL in C2
HSL tori
The Hitchinspectral curve
µ-Darbouxtransforms
The multiplierspectral curve
Darbouxtransforms
Thanks!
Katrin Leschke HSL tori