Stationary curves and complete integrability in the complex domain

Abstract

As is well known, the function theory and geometry of strictly pseudoconvex domains and real hypersurfaces in C ⁿ began around the turn of the century with the works of Poincaré, Hartogs, and Levi, and continued through the works of Oka, Cartan, Lelong, and many others. It began with the more qualitive and rigorous emphasis characteristic of a modern mathematical theory, with explicit computation playing less of a role. In fact, even at present it is essentially only for the unit ball, by reason of its symmetry, that the central objects of the theory can be computed. Examples of such “objects” include the Bergman and Szego kernels, the metrics of Caratheodory and Kobayashi, and the boundary invariants of Cartan, Chern, Moser, and Tanaka. In contrast to this, the classical theory of surfaces in real Euclidean space, for example, evolved in the light of many explicit examples and phenomena. The ellipsoidal surfaces clearly played a significant role in this, as may be seen in works of Jacobi, Lamé, Chasles, Liouville, Weierstrass, Klein, and many others.