Special Lagrangian 3-folds and integrable systems
By D. Joyce. 2001
59 citations recorded by the source corpus.
This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-folds in C^3 involving three commuting o.d.e.s. Now special Lagrangian cones in C^3 are linked to the theory of harmonic maps and integrable systems. Harmonic maps from a Riemann surface into complex projective space CP^n are an integrable system, and can be studied and classified using loop group techniques. If N is a special Lagrangian cone in C^3, then N is the cone on the image of a conformal harmonic map \psi : S --> S^5 for some Riemann surface S, and the projection of \psi to CP^2 is also conformal harmonic. Our examples of special Lagrangian cones in C^3 yield conformal harmonic maps \psi : R^2 --> CP^2. We work through the integrable systems theory for these examples, showing that they are superconformal of finite type, and calculating their harmonic sequences, Toda and Tzitzeica solutions, algebra of polynomial Killing fields and spectral curves. We also study the double periodicity conditions for \psi, and so find families of superconformal tori in CP^2. We finish by asking whether our more general construction of special Lagrangian 3-folds can also be derived from a higher-dimensional integrable system, and whether the special Lagrangian equations themselves are in some sense integrable.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d7b378a8628bccce500115535daea2f3e8870838
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Citations in the corpus
- Lectures on special Lagrangian geometry (2001, methodology): “We now describe five methods of constructing special Lagrangian m-folds in C, drawn from papers by the author [15, 16, 18, 19, 20, 21, 22, 23], Bryant [1], Castro and Urbano [2], Goldstein [4, 5], Harvey [10, p.”
- Constructing special Lagrangian m-folds in C^m by evolving quadrics (2000, background): “D G ] 3 1 Ja n 20 01 Constructing special Lagrangian m-folds in C m by evolving quadrics Dominic Joyce, Lincoln College, Oxford”
- Special Lagrangian Cones in C3 and Primitive Harmonic Maps (2002, background): “grangian tori have been thoroughly discussed [6, 11, 13 ] and in particular the spectral data approach is unnecessary, since the equation (13) reduces to an o.d.e.”
- Special Lagrangian m-folds in ℂm with symmetries (2000, background): “D G ] 3 1 Ju l 2 00 1 Special Lagrangian m-folds in Cm with symmetries Dominic Joyce, Lincoln College, Oxford”
- The geometric complexity of special Lagrangian T2-cones (2003, methodology): “it construction of such tori in terms of elliptic functions and integrals will be reviewed in §5. Joyce has given some special examples of minimal Lagrangian tori with spectral curve genus equal to 4 [25]. For g > 4, no explicit examples are known of spectral curves which satisfy the periodicity ”