By Joyce D.
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Extra info for Constructing special Lagrangian m-folds in mathbbCmCm by evolving quadrics
S. Then in [11, Sect. 6] we work through the integrable systems framework for the corresponding family of harmonic maps Ψ : R2 → CP2 , showing that they are generically superconformal of ﬁnite type, and determining their harmonic sequences, Toda solutions, algebras of polynomial Killing ﬁelds, and spectral curves. 4 are interesting because they construct large families of superconformal harmonic tori in CP2 . 7. Examples from evolving non-centred quadrics We will now apply the construction of Sect.
J + u j =a+1 αj − u Then u and θ satisfy Q(u)1/2 sin θ ≡ A for some A ∈ R. Suppose that αj +u > 0 for j = 1, . . , a and αj − u > 0 for j = a +1, . . , m−1 and t ∈ (− , ). Deﬁne a subset N of Cm to be x1 eiθ1(t) α1 +u(t), . . , xa eiθa(t) αa +u(t), xa+1 eiθa+1(t) αa+1 −u(t), . . , xm−1 eiθm−1 (t) αm−1 −u(t), xm + 21 u(t)−iAt : 2 2 −· · ·−xm−1 +2xm = 0 . t ∈ (− , ), xj ∈ R, x12 +· · ·+xa2 −xa+1 Then N is a special Lagrangian submanifold in Cm . As in Sect. 3, if we assume that θ (t) ∈ (−π/2, π/2) for t ∈ (− , ) then u is an increasing function of t, and we can choose to regard everything as a function of u rather than of t.
Case (a) above, with A = 0, corresponds to Im(C D) 3 ¯ = 0 then is a subset of an afﬁne special Lagrangian 3-plane R in C3 . If Im(C D) 3 N is an embedded submanifold diffeomorphic to R , with coordinates (x1 , x2 , t). 5. 1. 3) become dw2 dβ dw1 = w¯ 2 , = −w¯ 1 and = w1 w2 . 8) dt dt dt The ﬁrst two equations have solutions ¯ it − i Ce ¯ −it , w1 = Ceit + De−it and w2 = i De where C = 1 2 w1 (0) − iw2 (0) and D = 1 2 w1 (0) + iw2 (0) . 8) gives ¯ 2it + 1 CDe ¯ −2it + i |C|2 − |D|2 t + E, β(t) = 21 C De 2 ¯ where E = β(0) − Re(CD).