On Dynamical Systems and the Minimal Surface Equation

Abstract

The relationship between the theory of dynamical systems and differential geometry is a long-standing and profound one. It has largely been focused around the geodesic flow on the unit tangent bundle of a Riemannian manifold. One may recall early work by Poincaré and Birkhoff for the case of convex surfaces, Morse theory, contributions by Hadamard, and, more recently, by Anosov, in the case of negatively curved manifolds—to register the decisive influence that has been asserted by this particular example in the development of the general theory of dynamical systems. Conversely, analysis of the geodesic flow has been useful to study problems in Riemannian geometry; one may point to the recent successful structure theory for Hadamard manifolds by Ballmann, Eberlein, Gromov, Schroeder, and others. Here we will be concerned with a different, more unexpected role that dynamical systems have recently played in the very active field of partial differential equations that arise in Riemannian geometry, notably the minimal surface equation. Briefly, in the presence of a suitably chosen Lie group G of isometries, one studies the G-invariant solutions. A simple case of this idea goes back to Delaunay’s classification of rotationally invariant constant mean curvature surfaces [D]; more recently, the celebrated work of Bombieri, de Giorgi, and Giusti on the Bernstein problem may also be viewed in this context [BGG]. The ideas were developed in a more systematic way as a general program of “equivariant geometry” in a seminal paper by W.Y. Hsiang and B. Lawson [HL], and has since been developed, especially by Hsiang, to solve some long-standing open problems in Riemannian geometry [HI, H2, H3].