Special Lagrangian 3-folds and integrable systems

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.Comment: 36 pages, LaTeX. (v2) substantially rewritten, references update

Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Free interpolation in Hardy spaces is caracterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces $H^p$, $p>0$. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to ``big'' Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and that are strictly bigger than $\bigcup_{p>0} H^p$. It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the weights of the majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz spaces will also be discussed in the general situation. We finish the paper with an example of a separated Blaschke sequence that is interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.Comment: 19 pages, 2 figure