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# Igv2008

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### Igv2008

1. 1. Harmonic trinoids in complex projective spaces Shimpei Kobayashi, Hirosaki University 12/12, 2008
2. 2. Introduction Harmonic maps into complex projective spacesPreliminaries Harmonic spheres Harmonic toriEquivariant harmonic maps in CPn Isomorphisms between loop algebras Potentials for equivariant harmonic mapsHarmonic trinoids in CPn DPW method System of ODEs and a scalar ODE Hypergeometric equations Unitarizability and interlace on the unit circle Open problems
3. 3. Let (M, g) and (N, h) be Riemannian manifolds and Ψ : (M, g) → (N, h)a C∞ map.Deﬁne E(Ψ) = |dΨ|2 dVg , Mwhere the norm is deﬁned by g and h, and dV g is the volume formof M.
4. 4. Let (M, g) and (N, h) be Riemannian manifolds and Ψ : (M, g) → (N, h)a C∞ map.Deﬁne E(Ψ) = |dΨ|2 dVg , Mwhere the norm is deﬁned by g and h, and dV g is the volume formof M.Consider the variation Ψt for Ψ. def d Ψ is harmonic ⇔ E(Ψt )|t=0 = 0 ⇔ τ (Ψ) = 0, dtwhere τ (Ψ) = trace dΨ is the tension ﬁeld.
5. 5. In particular, if dim M = 2, then the harmonicity can be writtenas Ψ ∂ ∂ dΨ( ) = 0, (1) ∂¯ z ∂zwhere z = x + iy and (x, y) is a conformal coordinate.
6. 6. Harmonic spheres If M = S2 , the followings (N, h) were studied in details: Sn (RPn ) (Calabi, Chern) CPn (D. Burns, Eells-Wood, Din-Zakrzewski, Glaser-Stora) Gr2 (Cn ) (Chern-Wolfson, Burstall-Wood) Grk (Cn ) (Wolfson, Wood) These are based on 1) Holomorphic diﬀerential on S2 is zero 2) Techniques of Hermitian vector bundles.
7. 7. Harmonic tori If M = T2 , the followings (N, h) were studied in details : S2 (Pinkall-Sterling) S3 (Hitchin) S4 (Pinkall-Ferus-Sterling-Pedit) Sn , CPn (Burstall, McIntosh) Gr2 (C4 ), HP3 (Udagawa) Rank 1 compact symmetric spaces (Burstall-Ferus-Pedit-Pinkall) These are based on integrable system methods.
8. 8. Goal of this talk I would like to discuss harmonic maps from M = S 1 × R or M = CP1 {0, 1, ∞} into N = CPn .
9. 9. Goal of this talk I would like to discuss harmonic maps from M = S 1 × R or M = CP1 {0, 1, ∞} into N = CPn . Consider a C∞ map Ψ from a Riemann surface M into a symmetric space G/K: ∂ 1 Ψ dαk + 2 [αk ∧ αk ] = −[αp ∧ αp ] = 0, ∂ dΨ( )=0 ⇔ ∂¯ z ∂z dαp + [αk ∧ αp ] = 0, 1 dαλ + 2 [αλ ∧ αλ ] = 0, ⇔ αλ = λ−1 αp + αk + λαp , λ ∈ S1 . where α = F−1 dF is the Maurer-Cartan form of a lift F : M → G, g = k ⊕ p and TMC = T M + T M.
10. 10. Equivariant harmonic maps in k-symmetric spaces Deﬁnition A map Ψ : R2 → G/K is called R-equivariant if Ψ(x, y) = exp(xA0 )Φ(y), for some A0 ∈ g and Φ : R → G/K. Theorem (Burstall-Kilian) All equivariant primitive harmonic maps in k-symmetric spaces G/K (with an order k-automorphism τ ) are constructed from degree one potentials: ξ = λ−1 ξ−1 + ξ0 + λξ1 ∈ Λgτ , (2) where Λgτ = {ξ : S1 → g | ξ(e2πi/k λ) = τ ξ(λ)} is the loop algebra of the Lie algebra g of G, ξj ∈ gj and ξj = ξ−j with the eigenspace decomposition of gC = i∈Zk gi .
11. 11. Equivariant harmonic maps in CPn For CPn case, G = SU(n + 1) with the involution σ = Ad diag [1, −1, . . . , −1], thus K = S(U(1) × U(n)) and GC = SL(n + 1, C). It is known that harmonic maps in CPn can be classiﬁed into isotropic, non-isotropic weakly conformal with isotropic dimension r ∈ {1, . . . , n − 1}, non-conformal. Problem: Which degree one potentials are corresponding to the above cases?
12. 12. Isomorphism Lemma (Pacheco) Let g be a Lie algebra, τ : g → g an automorphism of order k and σ : g → g an involution. Deﬁne Γ as a map between Λgτ and Λgσ Γ(ξ)(λ) = s(λ)t(λ−2/k )ξ(λ2/k ) ∈ Λgσ for ξ ∈ Λgτ , (3) where t : S1 → Aut g and s : S1 → Aut g are automorphism such that t(e2πi/k ) = τ and s(−1) = σ respectively. Then Γ is an isomorphism.
13. 13. Isomorphism Lemma (Pacheco) Let g be a Lie algebra, τ : g → g an automorphism of order k and σ : g → g an involution. Deﬁne Γ as a map between Λgτ and Λgσ Γ(ξ)(λ) = s(λ)t(λ−2/k )ξ(λ2/k ) ∈ Λgσ for ξ ∈ Λgτ , (3) where t : S1 → Aut g and s : S1 → Aut g are automorphism such that t(e2πi/k ) = τ and s(−1) = σ respectively. Then Γ is an isomorphism. Let t and s be t(λ) = Ad diag[1, λ, . . . , λ k−2 , λk−1 , . . . , λk−1 ] and s(λ) = Ad diag[1, λ, . . . , λ] respectively. Then it is easy to see t(e2πi/k ) = τ and s(−1) = σ. Deﬁne Γ as in (3), and let t ξ = λ−1 ξ−1 + ξ0 + λξ−1 ∈ Λsu(n + 1)τ be the degree one potential.
14. 14. PropositionA harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)where the order k of τ and the degree one potential ξ are given asfollows:
15. 15. PropositionA harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)where the order k of τ and the degree one potential ξ are given asfollows:(a) if it is isotropic: k = n + 1 and ξ−1 is principal nilpotent.
16. 16. PropositionA harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)where the order k of τ and the degree one potential ξ are given asfollows:(a) if it is isotropic: k = n + 1 and ξ−1 is principal nilpotent.(b) if it is non-isotropic weakly conformal with the isotropic dimension r ∈ {1, 2, · · · , n − 1}: k = r + 2 ∈ {3, 4, · · · , n + 1} and ξ−1 is semisimple.
17. 17. PropositionA harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)where the order k of τ and the degree one potential ξ are given asfollows:(a) if it is isotropic: k = n + 1 and ξ−1 is principal nilpotent.(b) if it is non-isotropic weakly conformal with the isotropic dimension r ∈ {1, 2, · · · , n − 1}: k = r + 2 ∈ {3, 4, · · · , n + 1} and ξ−1 is semisimple.(c) if it is non-conformal: k = 2 and ξ−1 is semisimple.
18. 18. Equivariant harmonic maps in CP1 Figure: These ﬁgures are created by Nick Schmitt.
19. 19. Loop groups Deﬁnition G : A compact simple Lie group, g : Lie algebra of G, GC : The complexiﬁcation of G, gC : Lie algebra of GC , σ : A involution of G, K : The ﬁxed point set of σ k : Lie algebra of K, g = k ⊕ p : Direct sum B : The solvable part of an Iwasawa decomposition KC = K · B, K ∩ B = e
20. 20. Loop groups ΛGσ := {H : S1 → G | σH(λ) = H(−λ)}, Λgσ := {h : S1 → g | σh(λ) = h(−λ)} ΛgC σ := {h : S1 → gC | σh(λ) = h(−λ)} ΛGC := {H : S1 → GC | σH(λ) = H(−λ)} σ H+ can be extend holomorphically Λ+ GC := B σ H+ ∈ ΛGC | σ to D1 and H+ (0) ∈ B
21. 21. Loop groups ΛGσ := {H : S1 → G | σH(λ) = H(−λ)}, Λgσ := {h : S1 → g | σh(λ) = h(−λ)} ΛgC σ := {h : S1 → gC | σh(λ) = h(−λ)} ΛGC := {H : S1 → GC | σH(λ) = H(−λ)} σ H+ can be extend holomorphically Λ+ GC := B σ H+ ∈ ΛGC | σ to D1 and H+ (0) ∈ B We assume that the coeﬃcients of all g ∈ Λg σ are in the Wiener algebra A= f(λ) = fn λn : C r → C ; |fn | < ∞ . n∈Z n∈Z The Wiener algebra is a Banach algebra relative to the norm f = |fn |, and A consists of continuous functions.
22. 22. DPW method Step1 η(z, λ) = ∞ λk ξk (z) : ΛgC -valued 1-form on Σ ⊂ C, k=−1 σ where ξeven (z) ∈ Ω1,0 (kC ) and ξodd (z) ∈ Ω1,0 (pC ). Step2 Solve ODE dC = Cη. Step3 Iwasawa decomposition : C = FW + , F : Σ → ΛGσ and W+ : Σ → Λ + GC . B σ Step4 Projection π ◦ F|λ∈S1 : Σ → G/K, where π : G → G/K. Theorem (Dorfmeister-Pedit-Wu) Multiplication ΛGσ × Λ+ GC → ΛGC is a diﬀeomorphism onto. B σ σ Theorem (Dorfmeister-Pedit-Wu, 1998) Every harmonic map from a simply connected domain Σ into G/K can be constructed in this way.
23. 23. DPW method Step1 η(z, λ) = ∞ λk ξk (z) : ΛgC -valued 1-form on Σ ⊂ C, k=−1 σ where ξeven (z) ∈ Ω1,0 (kC ) and ξodd (z) ∈ Ω1,0 (pC ). Step2 Solve ODE dC = Cη. Step3 Iwasawa decomposition : C = FW + , F : Σ → ΛGσ and W+ : Σ → Λ + GC . B σ Step4 Projection π ◦ F|λ∈S1 : Σ → G/K, where π : G → G/K. Theorem (Dorfmeister-Pedit-Wu) Multiplication ΛGσ × Λ+ GC → ΛGC is a diﬀeomorphism onto. B σ σ Theorem (Dorfmeister-Pedit-Wu, 1998) Every harmonic map from a simply connected domain Σ into G/K can be constructed in this way. From now on, CPn is represented as the symmetric space U(n + 1)/U(1) × U(n) with the involution σ = Ad diag [1, −1, . . . , −1].
24. 24. System of ODEs and a scalar ODE Consider ∞ ν, τj ∈ λ2k−3 g2k−3 g2k−3 is a holomorphic function on M , k=1 where i = 1, . . . , n and dn+1 ν dn dn−1 − − ντ1 n−1 − · · · − ντn u = 0. (5) dzn+1 ν dzn dz
25. 25. System of ODEs and a scalar ODE Consider ∞ ν, τj ∈ λ2k−3 g2k−3 g2k−3 is a holomorphic function on M , k=1 where i = 1, . . . , n and dn+1 ν dn dn−1 − − ντ1 n−1 − · · · − ντn u = 0. (5) dzn+1 ν dzn dz Set u1 , . . . un+1 : A fundamental solutions of (5),  (n)  u1 (n−1) (0)  ν u1 · · · u1   . . . . .. .  . C :=  . . . . ,  (n)  un+1 (n−1) (0) ν un+1 · · · un+1 (0) (k) dk u where uj = uj and uj = dzk , (k > 0).
26. 26. Lemma(1)   0 ν 0 ··· 0   τ1 0 1 · · · 0   −1  . . .. . . .. . .  η := C dC =  . . . . . . (6) . . ..    . . ..   . . . . 1  τn 0 · · · · · · 0 ∞(2) η = λk ξk is a holomorphic potential on M, where k −1 ξeven ∈ Ω1,0 (kC ) and ξodd ∈ Ω1,0 (pC ).Fact: Monodromy representations of (5) and (6) are the same.
27. 27. Hypergeometric functions n+1 Fn (α1 , . . . , αn+1 ; β1 , . . . , βn |z) ∞ (α1 )k · · · (αn+1 )k k = z , (7) k=0 (β1 )k · · · (βn )k k! where α1 , . . . , αn+1 , β1 , . . . , βn ∈ C, (x)k is the Pochhammer symbol or rising factorial Γ(x + k) (x)k = = x(x + 1) · · · (x + k − 1). Γ(x) n+1 Fn (α1 , . . . , αn+1 ; β1 , . . . , βn |z) is called the hypergeometric function n+1 Fn . 2 F1 (α1 , α2 ; β1 |z) is the Gauß’s hypergeometric function.
28. 28. Let D(α; β) = D(α1 . . . αn+1 ; β1 . . . βn+1 ) be the diﬀerentialoperatorD(α; β) = (θ+β1 −1) . . . (θ+βn+1 −1)−z(θ+α1 ) . . . (θ+αn+1 ) dfor α1 , . . . , αn+1 , β1 , . . . , βn+1 ∈ C, where θ = z dz . Thehypergeometric equation is deﬁned by D(α; β)u = 0.
29. 29. Let D(α; β) = D(α1 . . . αn+1 ; β1 . . . βn+1 ) be the diﬀerentialoperatorD(α; β) = (θ+β1 −1) . . . (θ+βn+1 −1)−z(θ+α1 ) . . . (θ+αn+1 ) dfor α1 , . . . , αn+1 , β1 , . . . , βn+1 ∈ C, where θ = z dz . Thehypergeometric equation is deﬁned by D(α; β)u = 0.Local exponents around the points z = 0, ∞, 1 are    z=0  z=∞ z=1    1−β α1 0     1     1 − β2   α2 1      1 − β3 α3 2   . . . . . .     . . .        n+1 n+1     1 − βn+1 αn+1 γ =   βj − αj − 1      1 1Fact: D(α; β)u = 0 is well-deﬁned on CP 1 {0, 1, ∞}.
30. 30. If βi are distinct mod Z, n + 1 independent solutions ofD(α; β)u = 0 are given byz1−βi n+1 Fn (1+α1 −βi , . . . , 1+αn+1 −βi ; 1+β1 −βi , . ∨ ., 1+βn+1 −βi |z), .where i = 1, . . . , n + 1 and ∨ denotes omission of 1 + β i − βi . V(α; β):The local solution space of D(α; β)u = 0 around z 0 . G : The fundamental group π1 (CP1 {0, 1, ∞}, z0 ). M(α, β) : G → GL(V(α; β)) : Monodromy representation of D(α; β)u = 0.
31. 31. Theorem (Beukers-Heckman, 1989)Let M(α; β) be the Monodromy group of D(α; β)u = 0. Then M(α; β) are simultaneously conjugated into U(n + 1). iﬀ 0 < α1 < β1 < α2 < β2 < . . . < αn+1 < βn+1 1 or 0 < β1 < α1 < β2 < α2 < . . . < βn+1 < αn+1 1 .
32. 32. Remarkαj and βj are determined by solving the indicial equations, whichare n-th order algebraic equations.There are several problems for an application to harmonic maps inCPn . αj and βj depend on the additional parameter λ ∈ C. αj and βj need to be real and satisfy the inequality for almost all λ ∈ S1 . Products and sums of αj and βj are ν and τj as in the holomorphic potential of (6).
33. 33. The case n = 1 (Gauß’s hypergeometric equation) Local exponents    z=0  z=∞ z=1    1−β α1 0   1  2 2  1 − β2   α2 γ= βj − αj − 1      1 1 Set α1 = 1 − v 1 − v 2 − v 3 , α 2 = 1 − v 1 − v 2 + v 3 , and β1 = 1 − 2v1 , β2 = 1, where 1 1 vj = − 1 + wj (λ − λ−1 )2 2 2
34. 34. Spherical triangle inequality   v1 + v 2 + v 3 < 1  v1 < v 2 + v 3  0 < α 1 < β1 < α2 < β2 1⇔ (8)  v2 < v 1 + v 3  v3 < v 1 + v 2  It is not diﬃcult to show that the above inequality are satisﬁed for some choices of wj . Moreover all problems can be solved (Kilian-Kobayashi-Rossman-Schmitt, Dorfmeister-Wu). Remark Umehara-Yamada considered the similar inequality for CMC H=1 in H3 . (No λ dependence!)
35. 35. Examples of CMC trinoids in space forms Figure: These ﬁgures are created by Nick Schmitt.
36. 36. The case n > 1 Example For the isotropic case, αj and βj do not depend on λ. Thus there exist isotropic harmonic trinoids in CP n . For n = 2, 3, the indicial equation can be solved explicitly. We can show that there exist examples of harmonic trinoids in CP2 and CP3 .
37. 37. Open problem What are behaviors around the punctures? Are they asymptotically converge to equivariant ones? Prove the existence of non-isotropic harmonic trinoids for n 4.