E 3 with prescribed shape into a problem in Teichmüller theory: in a moduli space ∆ of compatible triples of flat structures, find a triple whose underlying Riemann surfaces coincide. This we solve nonconstructively by introducing a nonnegative height function H : ∆ → R + on ∆, which has the features of being proper, and whose only critical point is at a solution to our problem. The bulk of the paper is a description of this height function ( §4.3), a proof of its properness ( §4.6), and a

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... that its only critical points are at solutions ( §5, §6). An interesting feature of this proof is that it is inductive: the triples of flat structures for a slightly less complicated minimal surface lie on a boundary face of the compactified moduli space ∆ of compatible triples of flat structures for more complicated surfaces. We consider this solution (on ∆) to the less complicated problem as the point-at-infinity of a particularly good locus in the moduli space ∆ on which to restrict the height and look for a solution. 2. Background, notation, and a sketch of the argument 2.1. Minimal surfaces. 2.1.1. The Weierstrass representation. Any complete minimal surface M of finite total curvature in E 3 can be defined by z → Re z ·