Adding a Uniton via the DPW Method

Correia, N.; Pacheco, Rui

doi:10.1142/s0129167x09005613

Cited by 5 publications

(6 citation statements)

References 11 publications

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“…Moreover, as in [7], Φ = sΨΨ(1, •) −1 is an extended solution, and a short calculation shows that the original map is recovered via the Cartan embedding by evaluating Φ at λ = ω, that is, ι • ϕ = Φ(ω, •). Observe that this extended solution takes values in (5.23)…”

Section: Primitive Harmonic Maps Into K-symmetric Spacesmentioning

confidence: 99%

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Symmetric shift-invariant subspaces and harmonic maps

2021

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Section: Primitive Harmonic Maps Into K-symmetric Spacesmentioning

confidence: 99%

“…where Λ + GL(n, C) is the subgroup of loops γ ∈ ΛGL(n, C) which extend holomorphically to |λ| < 1. We can decompose g µ = Φ µ b µ according to the Iwasawa decomposition; then Φ µ : M → Ω U(n) is an extended solution (see [7,8]).…”

Section: Holomorphic Potentialsmentioning

confidence: 99%

Symmetric shift-invariant subspaces and harmonic maps

2021

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“…Moreover, as in [6], Φ = sΨΨ(1, •) −1 is an extended solution and a short calculation shows that the original map is recovered via the Cartan embedding by evaluating Φ at λ = ω, that is, ι • ϕ = Φ ω . Observe that this extended solution takes values in…”

Section: Primitive Harmonic Maps Into K-symmetric Spacesmentioning

confidence: 99%

Section: Holomorphic Potentialsmentioning

confidence: 99%

Symmetric shift-invariant subspaces and harmonic maps

Aleman¹,

Pacheco²,

Wood³

2019

Preprint

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The Grassmannian model represents harmonic maps from Riemann surfaces by families of shift-invariant subspaces of a Hilbert space. We impose a natural symmetry condition on the shift-invariant subspaces that corresponds to considering an important class of harmonic maps into symmetric and k-symmetric spaces. In particular, we obtain new general forms for such symmetric shift-invariant subspaces and for the corresponding extended solutions.

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“…The maps ϕ may or may not be of finite type. For example, if α = G ′ (f ) where f = span{F } so that ϕ is given by (5.9), then ϕ is of finite type by arguments similar to those in [12,Theorem 9.2] and [25, §4]. However, for most choices of α, ϕ is not of finite type.…”

Section: 3mentioning

confidence: 99%

Harmonic maps into the exceptional symmetric space G₂ /SO(4)

Svensson¹,

Wood

2014

Journal of the London Mathematical Society

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We show that a harmonic map from a Riemann surface into the exceptional symmetric space G 2 /SO(4) has a J 2 -holomorphic twistor lift into one of the three flag manifolds of G 2 if and only if it is 'nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into G 2 /SO(4) which are not of finite uniton number, and which have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal; for such maps, we show that our lifts can be constructed explicitly from extended solutions.2000 Mathematics Subject Classification. 53C43, 58E20.HARMONIC MAPS INTO THE EXCEPTIONAL SYMMETRIC SPACE G 2 /SO(4) 27 2 isotropic subspace of C 7 which is not complex-coassociative, justifying our alternative description of T 2 as the set of all such.That the map (4.14)(s=2) intertwines both J 1 and J 2 is proved as in case (i); this time,. Now T is the stabilizer in G 2 of the flags (4.12)(s = 3) and (4.13)(s = 3); as in case (ii), for convenience we work with the latter. Thenthe proof that the map (4.14)(s=3) intertwines both J 1 and J 2 proceeds as before.7.4. Proof of Lemma 3.4. We need the following facts. As in §4.1, we identify smooth maps ϕ : M → G k (C n ) and rank k subbundles of C n . As described in that section, we give C n the connection D ϕ and the Koszul-Malgrange structure with∂-operator given over each coordinate domain (U, z) by D φ z . From §4.1, the connection D ϕ is the direct sum of the connections ∇ ϕ on ϕ and ∇ ϕ ⊥ on ϕ ⊥ ; those connections are given by projection of the flat connection on C n .We now prove Lemma 3.4 for s = 1. The cases s = 2 and s = 3 are similar. Suppose that ψ : M → T 1 is J 2 -holomorphic. We need to show:(7.2) W is a holomorphic subbundle of (C n , D φ z ) which lies in ker A ϕ z . By J 2 -holomorphicity, the horizontal component of ψ z has values in H 1,0 J 2 T 1 = H 1,0 J 1 T 1 = α(ξ)/i >0, odd g α = g α 2 + g α 1 +α 2 + g 2α 1 +α 2 + g 3α 1 +α 2 , and the vertical component has values in V 1,0 J 2 T 1 = V 0,1 J 1 T 1 = g −3α 1 −2α 2 . Hence the horizontal component of ψ z maps W = ψ 1 = ℓ α 1 +α 2 ⊕ ℓ 2α 1 +α 2 to 0, and the vertical component has zero component in End(ψ i , ψ j ) for j > i so that A ′ ϕ ⊥ ⊖W,W = 0. As in [10], this means W is a holomomorphic subbundle of (C n , D φ z ) and (7.2) follows. Conversely, suppose that ψ : M → T 1 is not J 2 -holomorphic. Then (i) either the horizontal component of ψ z has a non-zero component in one of the components of H 0,1is non-zero: for example, if ψ z has a non-zero component in g −α 2 , then A ϕ z (W ) ⊃ A ϕ z (ℓ α 1 +α 2 ) = ℓ α 1 = 0. Similarly, in case (ii), A ′ W ,W = A ′ ψ −1 ,ψ 1 is non-zero, so that W is not a holomorphic subbundle of (C n , D φ z ). Hence one of the conditions in (7.2) is violated.

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Adding a Uniton via the DPW Method

Cited by 5 publications

References 11 publications

Symmetric shift-invariant subspaces and harmonic maps

Symmetric shift-invariant subspaces and harmonic maps

Symmetric shift-invariant subspaces and harmonic maps

Harmonic maps into the exceptional symmetric space G₂ /SO(4)

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Adding a Uniton via the DPW Method

Cited by 5 publications

References 11 publications

Symmetric shift-invariant subspaces and harmonic maps

Symmetric shift-invariant subspaces and harmonic maps

Symmetric shift-invariant subspaces and harmonic maps

Harmonic maps into the exceptional symmetric space G2 /SO(4)

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Harmonic maps into the exceptional symmetric space G₂ /SO(4)