  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
$2\pi$. 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 $2\pi$, 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.
