A criterion for flatness in minimal area metrics that define string diagrams

It has been proposed that the string diagrams of closed string field theory be defined by a minimal area problem that requires that all nontrivial homotopy curves have length greater than or equal to 2. Consistency requires that the minimal area metric be flat in a neighbourhood of the punctures. The theorem proven in this paper, yieds a criterion which if satisfied, will ensure this requirement. The theorem states roughly that the metric is flat in an open set,U if there is a unique closed curve of length 2 through every point inU and all of these closed curves are in the same free homotopy class.Supported in part by funds provided by the US Department of Energy (DOE) under contract # DE-AC02-76ER03069