HOW COVARIANT CLOSED STRING THEORY SOLVES A MINIMAL AREA PROBLEM

For a given genus g Riemann surface with n greater-than-or-equal-to 0 punctures (n greater-than-or-equal-to 3 for g = 0) we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2-pi. The minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures. These metrics are induced by Jenkins-Strebel quadratic differentials. They arise from the string diagrams corresponding to restricted Feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra.