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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 adoi:10.2307/3597281 fatcat:rp3bsibiu5d53ceoodp4krggoe