Constant mean curvature surfaces in Minkowski 3-space via loop groups

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1 Constant mean curvature surfaces in Minkowski 3-space via loop groups David Brander Now: Department of Mathematics Kobe University (From August 2008: Danish Technical University) Geometry, Integrability and Quantization - Varna 2008

2 Outline CMC Surfaces in Euclidean Space

3 Outline CMC Surfaces in Euclidean Space The loop group construction

4 Constant Mean Curvature Surfaces in Euclidean 3-space Soap films are CMC surfaces. Air pressure on both sides of surface the same mean curvature H = 0, minimal surface

5 Minimal Surfaces: H = 0 Gauss map of a minimal surface is holomorphic. Figure: Costa s surface

6 Minimal Surfaces: H = 0 Figure: Costa s surface Gauss map of a minimal surface is holomorphic. Weierstrass representation: pair of holomorphic functions minimal surface

7 CMC H 0 Surfaces Gauss map is a harmonic (not holomorphic) map into S 2 = SU(2)/K, K = {diagonal matrices}. Figure: A constant non-zero mean curvature surface

8 CMC H 0 Surfaces Gauss map is a harmonic (not holomorphic) map into S 2 = SU(2)/K, K = {diagonal matrices}. Loop group frame F λ. Figure: A constant non-zero mean curvature surface

9 CMC H 0 Surfaces Gauss map is a harmonic (not holomorphic) map into S 2 = SU(2)/K, Figure: A constant non-zero mean curvature surface K = {diagonal matrices}. Loop group frame F λ. Can recover f from the loop group map F λ via a simple formula.

10 Loop Group Methods ΛG C = {γ : S 1 G C γ smooth} F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a

11 Loop Group Methods ΛG C = {γ : S 1 G C γ smooth} F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a Example: flat surfaces in S 3. ω λβ λθ F 1 λ df λ = λβ t 0 0 = a 0 + a 1 λ, λθ t 0 0 order (0, 1).

12 Loop Group Methods F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a 1. AKS theory: 2. KDPW Method: 3. Dressing:

13 Loop Group Methods F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a 1. AKS theory: Constructs order (0, b) maps, b > 0, by solving ODE s. Related to inverse scattering. 2. KDPW Method: 3. Dressing:

14 Loop Group Methods F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a 1. AKS theory: Constructs order (0, b) maps, b > 0, by solving ODE s. Related to inverse scattering. 2. KDPW Method: Constructs order (a, b) maps, a < 0 < b, from a pair of (a, 0) and (0, b) maps. 3. Dressing:

15 Loop Group Methods F λ : M ΛG C is of connection order (a, b) if F 1 λ df λ = b a i λ i. a 1. AKS theory: Constructs order (0, b) maps, b > 0, by solving ODE s. Related to inverse scattering. 2. KDPW Method: Constructs order (a, b) maps, a < 0 < b, from a pair of (a, 0) and (0, b) maps. 3. Dressing: Any kind of connection order (a, b) maps. Produces families of new solutions from a given solution.

16 Krichever-Dorfmeister-Pedit-Wu (KDPW) Method Need Birkhoff factorization: ΛG C = Λ + G C Λ G C, where Λ ± G C consists of loops which extend holomorphically to D and Ĉ \ D resp.

17 Krichever-Dorfmeister-Pedit-Wu (KDPW) Method Need Birkhoff factorization: ΛG C = Λ + G C Λ G C, where Λ ± G C consists of loops which extend holomorphically to D and Ĉ \ D resp. If F λ is of order (a, b), a < 0 < b, decompose F = F + G = F G +.

18 Krichever-Dorfmeister-Pedit-Wu (KDPW) Method Need Birkhoff factorization: ΛG C = Λ + G C Λ G C, where Λ ± G C consists of loops which extend holomorphically to D and Ĉ \ D resp. If F λ is of order (a, b), a < 0 < b, decompose F = F + G = F G +. Then F + is of order (0, b) and F is of order (a, 0):

19 Krichever-Dorfmeister-Pedit-Wu (KDPW) Method Need Birkhoff factorization: ΛG C = Λ + G C Λ G C, where Λ ± G C consists of loops which extend holomorphically to D and Ĉ \ D resp. If F λ is of order (a, b), a < 0 < b, decompose F = F + G = F G +. Then F + is of order (0, b) and F is of order (a, 0): F+ 1 + = G (F 1 df)g 1 b = G ( a a i λ i )G 1 =. + G dg 1 + G dg 1

20 Krichever-Dorfmeister-Pedit-Wu (KDPW) Method Need Birkhoff factorization: ΛG C = Λ + G C Λ G C, where Λ ± G C consists of loops which extend holomorphically to D and Ĉ \ D resp. If F λ is of order (a, b), a < 0 < b, decompose F = F + G = F G +. Then F + is of order (0, b) and F is of order (a, 0): F+ 1 + = G (F 1 df)g 1 b = G ( a a i λ i )G 1 = c c b λ b. + G dg 1 + G dg 1

21 KDPW Method Conversely, given order (0, b) and (a, 0) maps, F + and F, we can construct an order (a, b) map F. After a normalization, both directions unique: { } F+ F (a, b) F (0, b) (a, 0)

22 Specific Case Harmonic Maps into Symmetric Spaces G/K symmetric space, K = G σ. On ΛG C, define involution ˆσ : (ˆσγ)(λ) := σ(γ( λ)).

23 Specific Case Harmonic Maps into Symmetric Spaces G/K symmetric space, K = G σ. On ΛG C, define involution ˆσ : Fixed point subgroup ΛGˆσ ΛG Cˆσ ΛGC.

24 F λ (z) a connection order ( 1, 1) map, C ΛGˆσ. KDPW:

25 F λ (z) a connection order ( 1, 1) map, C ΛGˆσ. KDPW: F {F +, F } In this case, F + determined by F, so F F

26 F λ (z) a connection order ( 1, 1) map, C ΛGˆσ. KDPW: F F Fix λ S 1 : then F λ : C G.

27 F λ (z) a connection order ( 1, 1) map, C ΛGˆσ. KDPW: F F Fix λ S 1 : then F λ : C G. Fact: Projection of F, to G/K, is a harmonic map C G/K if and only if F is holomorphic in z:

28 F λ (z) a connection order ( 1, 1) map, C ΛGˆσ. KDPW: F F Fix λ S 1 : then F λ : C G. Fact: Projection of F, to G/K, is a harmonic map C G/K if and only if F is holomorphic in z: order ( 1, 1) F F order ( 1, 1) harmonic holomorphic

29 Weierstrass Representation for CMC H 0 Surfaces a(z), b(z) arbitrary holomorphic. Set ( ) 0 a(z) α = λ 1 dz. b(z) 0

30 Weierstrass Representation for CMC H 0 Surfaces a(z), b(z) arbitrary holomorphic. Set ( ) 0 a(z) α = λ 1 dz. b(z) 0 Automatically, dα + α α = 0. Integrate to get F : Σ ΛG, connection order ( 1, 1).

31 Weierstrass Representation for CMC H 0 Surfaces a(z), b(z) arbitrary holomorphic. Set ( ) 0 a(z) α = λ 1 dz. b(z) 0 Automatically, dα + α α = 0. Integrate to get F : Σ ΛG, connection order ( 1, 1). Apply KDPW correspondence to get F, frame for harmonic map.

32 Weierstrass Representation for CMC H 0 Surfaces a(z), b(z) arbitrary holomorphic. Set ( ) 0 a(z) α = λ 1 dz. b(z) 0 Automatically, dα + α α = 0. Integrate to get F : Σ ΛG, connection order ( 1, 1). Apply KDPW correspondence to get F, frame for harmonic map. CMC surface obtained from F by Sym-Bobenko formula.

33 Weierstrass Representation for CMC H 0 Surfaces a(z), b(z) arbitrary holomorphic. Set ( ) 0 a(z) α = λ 1 dz. b(z) 0 Automatically, dα + α α = 0. Integrate to get F : Σ ΛG, connection order ( 1, 1). Apply KDPW correspondence to get F, frame for harmonic map. CMC surface obtained from F by Sym-Bobenko formula. All CMC surfaces in R 3 are obtained this way.

34 Iwasawa Decomposition In fact for harmonic maps, for the direction of KDPW, need Iwasawa splitting ΛG C = ΛG Λ + G C. This holds globally if G is compact. F is obtained from F via: F = FG +. More generally, for the direction, the holomorphic map F can be of order ( 1, b) where b 1.

35 Figure: CMC Unduloid Figure: CMC Nodoid

36 Figure: CMC 5-noid Figure: A Smyth Surface

37 CMC surfaces in Minkowski space, L 3 B.-, Rossman, Schmitt - Holomorphic representation of constant mean curvature surfaces in Minkowski space - Preprint Construction analogous to CMC in R 3. Change SU(2) SU(1, 1).

38 CMC surfaces in Minkowski space, L 3 B.-, Rossman, Schmitt - Holomorphic representation of constant mean curvature surfaces in Minkowski space - Preprint Construction analogous to CMC in R 3. Change SU(2) SU(1, 1). Only difference: SU(1, 1) non-compact Iwasawa decomposition not global. Iwasawa defined on an open dense set (the big cell )

39 CMC surfaces in Minkowski space, L 3 B.-, Rossman, Schmitt - Holomorphic representation of constant mean curvature surfaces in Minkowski space - Preprint Construction analogous to CMC in R 3. Change SU(2) SU(1, 1). Only difference: SU(1, 1) non-compact Iwasawa decomposition not global. Iwasawa defined on an open dense set (the big cell ) Surface has singularities at boundary of this set.

40 Classification of surfaces with rotational symmetry Figure: Examples from each of the eight families of surfaces with rotational symmetry in L 3. (Images made by Nick Schmitt s XLab.)

41 Smyth surfaces in L 3 Figure: Swallowtail singularity Figure: A Smyth surface in L 3 Figure: Swallowtail singularity

42 Outline CMC Surfaces in Euclidean Space The loop group construction

43 σ 1 := The loop group construction ( ) 0 1, σ := ( ) ( ) 0 i 1 0, σ i 0 3 := 0 1

44 σ 1 := The loop group construction ( ) 0 1, σ := G = SU(1, 1) iσ 1 SU(1, 1) G C = SL(2, C) ( ) ( ) 0 i 1 0, σ i 0 3 := 0 1

45 σ 1 := The loop group construction ( ) 0 1, σ := G = SU(1, 1) iσ 1 SU(1, 1) G C = SL(2, C) ΛG C = {γ : S 1 G C γ smooth} ( ) ( ) 0 i 1 0, σ i 0 3 := 0 1

46 σ 1 := The loop group construction ( ) 0 1, σ := G = SU(1, 1) iσ 1 SU(1, 1) G C = SL(2, C) ΛG C = {γ : S 1 G C γ smooth} ( ) ( ) 0 i 1 0, σ i 0 3 := 0 1 ΛG C σ := {x ΛG C σ(x) = x}, where, Λ ± G C σ := ΛG C σ Λ ± G C (σ(x))(λ) := Ad σ3 x( λ).

47 ΛG σ := ΛG ΛG C σ - real form". Note: ΛG σ = ΛSU(1, 1) σ iσ 1 ΛSU(1, 1) σ.

48 ΛG σ := ΛG ΛG C σ - real form". Note: ΛG σ = ΛSU(1, 1) σ iσ 1 ΛSU(1, 1) σ. Setting x (λ) := x( λ 1 ), Then { ( ) a b ΛSU(1, 1) σ = b a ΛGσ C { ( ) a b iσ 1 ΛSU(1, 1) σ = b a ΛGσ C }, },

49 Loop group characterization for CMC surfaces in L 3 Σ Riemmann surface F : Σ ΛG σ of connection order ( 1, 1)

50 Loop group characterization for CMC surfaces in L 3 Σ Riemmann surface F : Σ ΛG σ of connection order ( 1, 1) F : Σ Λ G C σ of order ( 1, 1), associated to F via (normalized) Birkhoff splitting: F = F G +.

51 Loop group characterization for CMC surfaces in L 3 Σ Riemmann surface F : Σ ΛG σ of connection order ( 1, 1) F : Σ Λ G C σ of order ( 1, 1), associated to F via (normalized) Birkhoff splitting: For λ 0 S 1, set F = F G +. f λ 0 = 1 2H S(F) λ=λ0, S(F) := Fiσ 3 F 1 + 2iλ λ F F 1.

52 Loop group characterization for CMC surfaces in L 3 Σ Riemmann surface F : Σ ΛG σ of connection order ( 1, 1) F : Σ Λ G C σ of order ( 1, 1), associated to F via (normalized) Birkhoff splitting: For λ 0 S 1, set F = F G +. f λ 0 = 1 2H S(F) λ=λ0, S(F) := Fiσ 3 F 1 + 2iλ λ F F 1. F holomorphic if and only if f λ 0 : Σ L 3 has constant mean curvature H.

53 SU(1, 1) Iwasawa decomposition Need: ω m = ( ) 1 0 λ m, m odd ; ω 1 m = ( ) 1 λ 1 m, m even. 0 1

54 Theorem (SU(1, 1) Iwasawa decomposition) ΛG C σ = B 1,1 n Z + P n, big cell: B 1,1 := ΛG σ Λ + G C σ, n th small cell: P n := ΛSU(1, 1) σ ω n Λ + G C σ. B 1,1, is an open dense subset of ΛG C σ. Any φ B 1,1 can be expressed as φ = FB, F ΛG σ, B Λ + G C σ, (1) F unique up to right multiplication by G σ := ΛG σ G. The map π : B 1,1 ΛG σ /G σ given by φ [F], derived from (1), is a real analytic projection.

55 Theorem (Holomorphic rep. for spacelike CMC surfaces in L 3 ) Let ξ = A i λ i dz i= 1 Lie(ΛG C σ) Ω 1 (Σ) be a holomorphic 1-form over a simply-connected Riemann surface Σ, with a 1 0, ( on Σ, where A 1 = 0 a 1 b 1 0 ). Let φ : Σ ΛGσ C be a solution of φ 1 dφ = ξ. On Σ := φ 1 (B 1,1 ), G-Iwasawa split: φ = FB, F ΛG σ, B Λ + G C σ. (2) Then for any λ 0 S 1, the map f λ 0 := ˆf λ 0 : Σ L 3, given by the Sym-Bobenko formula, is a conformal CMC H immersion, and is independent of the choice of F in (2).

56 Example 1: hyperboloid of two sheets. ( ) 0 λ 1 ξ = dz, Σ = C. 0 0 ( ) 1 zλ 1 φ = : Σ ΛG C 0 1 σ. Takes values in B 1,1 for z 1. G-Iwasawa: F = B = φ = F B, F : Σ \ S 1 ΛG, B : Σ \ S 1 Λ + G C σ, 1 ε(1 z 2 ) 1 ε(1 z 2 ) Sym-Bobenko formula gives ˆf 1 (z) = ( ε zλ 1 ), ε zλ 1 ( ) 1 0, ε = sign(1 z 2 ). ε zλ ε(1 z z) 1 H(x 2 + y 2 1) [2y, 2x, (1 + 3x 2 + 3y 2 )/2], two-sheeted hyperboloid {x x 2 2 (x 0 1 2H )2 = 1 H 2 }.

57 Hyperboloid: boundary of big cell behaviour In a small cell precisely when z = 1: There, have φ ΛSU(1, 1) σ ω 2 Λ + Gσ: C ( ) ( 1 zλ 1 p z λ 1 q ) ( z = 0 1 λq z 1 p (p + q) ) z 1 0 z 1 ω 2 λq z 1 (p q) z, where p 2 q 2 = 1 and p, q R. That is: φ P 2 for z = 1. Note: Surface blows up as z 1.

58 Example 2: numerical experiment ξ = λ 1 ( ) 0 1 dz, 100 z 0 Numerically: 1. Integrate with i.c. φ(0) = ω 1, to get φ : Σ ΛGσ C. 2. Iwasawa split to get F : Σ ΛG σ. 3. Compute Sym-Bobenko formula to get f 1 : Σ L Use XLab to view the surface.

59 Looks like Shcherbak surface singularity at z = 0. Since φ(0) = ω 1, the singularity occurs at P 1.

60 Results on boundary of big cell behaviour Proved: 1. The map f λ 0 : Σ L 3 always well defined (and real analytic) at z 0 φ 1 (P 1 ), but not immersed at such a point. 2. The map f λ 0 : Σ L 3 always blows up as z z 0 φ 1 (P 2 ).

61 Results on boundary of big cell behaviour Proved: 1. The map f λ 0 : Σ L 3 always well defined (and real analytic) at z 0 φ 1 (P 1 ), but not immersed at such a point. 2. The map f λ 0 : Σ L 3 always blows up as z z 0 φ 1 (P 2 ). Expect (have not proved yet): generic holomorphic data does not encounter P n for n > 2. Therefore: generic singularities of CMC surfaces occur only at points in P 1.

62 CMC Surfaces in Euclidean Space

JOSEF DORFMEISTER, MARTIN GUEST AND WAYNE ROSSMAN. 1. Introduction

THE tt STRUCTURE OF THE QUANTUM COHOMOLOGY OF CP FROM THE VIEWPOINT OF DIFFERENTIAL GEOMETRY JOSEF DORFMEISTER, MARTIN GUEST AND WAYNE ROSSMAN. Introduction The quantum cohomology of CP provides a distinguished