  In this paper we study surfaces in R^3 that arise as limit shapes in a class
of random surface models arising from dimer models. The limit shapes are
minimizers of a surface tension functional, that is, they minimize, for fixed
boundary conditions, the integral of a quantity (the surface tension) depending
only on the slope of the surface. The surface tension as a function of the
slope has singularities and is not strictly convex, which leads to formation of
facets and edges in the limit shapes.
  We find a change of variables that reduces the Euler-Lagrange equation for
the variational problem to the complex inviscid Burgers equation (complex Hopf
equation). The equation can thus be solved in terms of an arbitrary holomorphic
function, which is somewhat similar in spirit to Weierstrass parametrization of
minimal surfaces. We further show that for a natural dense set of boundary
conditions, the holomorphic function in question is, in fact, algebraic. The
tools of algebraic geometry can thus be brought in to study the the minimizers
and, especially, the formation of their singularities. This is illustrated by
several explicitly computed examples.
