The Plumbing of Minimal Area Metrics

Abstract

We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than or equal to $2π$. By constructing deformations of admissible metrics we establish necessary conditions on minimal area metrics and a partial converse to Beurling's criterion for extremal metrics. We explicitly construct new minimal area metrics that do not arise from quadratic differentials. Under the physically motivated assumption of existence of the minimal area metrics, we show there exist neighborhoods of the punctures isometric to a flat semiinfinite cylinder of circumference $2π$, allowing the definition of canonical complex coordinates around the punctures. The plumbing of surfaces with minimal area metrics is shown to induce a metric of minimal area on the resulting surface. This implies that minimal area string diagrams define a consistent quantum closed string field theory.