From cmc surfaces to Hamiltonian stationary Lagrangian surfaces

Abstract

Introduction Minimal surfaces and surfaces with constant mean curvature (cmc) have fascinated di#erential geometers for over two centuries. Indeed these surfaces are solutions to variational problems whose formulation is elegant, modelling physical situations quite simple to experiment using soap films and bubbles; however their richness has not been exhausted yet. Advances in the comprehension of these surfaces draw on complex analysis, theory of Riemann surfaces, topology, nonlinear elliptic PDE theory and geometric measure theory. Furthermore, one of the most spectacular development in the past twenty years has been the discovery that many problems in differential geometry -- including the minimal and cmc surfaces -- are actually integrable systems. The theory of integrable systems developed in the 1960's, beginning essentially with the study of a now famous example: the Korteweg-de Vries equation, u t + 6uu x + u xxx = 0, modelling the waves in a shallow flat channel . In the