In the past 20 years, many techniques have been developed for proving the existence of complete minimal surfaces immersed in space. Few methods are known for classifying such surfaces. In order to study the structure of spaces of minimal surfaces, we introduce a variational method based in contemporary Teichmuller theory. We apply this method to demonstrate local uniqueness in a model case. We prove that the generalized Chen-Gackstatter surface of genus 2 is locally unique in the space of Weierstrass data of complete, orientable minimal surfaces immersed in space with exact height differential and smallest possible total Gaussian curvature.