Abstract

The minimal surface equation is the prototypic example of a nonlinear (quasilinear to be precise) second order elliptic PDE . It has been studied in depth and as such a lot is known about graphical minimal submanifolds in codimension one (the geometric objects which the equation describes). By contrast, relatively little is known about graphical minimal submanifolds in higher codimension. This lack of knowledge is in some sense ‘explained’ by the failure of standard, desirable PDE results for the minimal surface system, all of which is described in the wonderfully titled 1977 paper of Lawson and Osserman: “Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system”. The reality of course is that the failure of the standard results opens up many much more interesting questions, many of which are still open. I intend to sketch some less common proofs of well-known facts in the codimension one case, discuss whether or not they generalize to higher codimension and then possibly make some conjectures.