Existence of minimal-area metrics for all Riemann surfaces in the minimal-area string vertex construction

Prove the global existence (and, where appropriate, uniqueness) of minimal-area conformal metrics solving the length-constrained variational problem used to define closed-string vertices for arbitrary genus g and number of punctures n: namely, find on every punctured Riemann surface a metric of minimal reduced area under the constraint that the length of every nontrivial closed curve be at least 2π. Establish this for all cases beyond the genus-zero and partially covered higher-genus regions where existence is known empirically, thereby completing the foundations of the minimal-area vertex construction in closed bosonic string field theory.

Background

The minimal-area construction defines closed-string vertices by selecting on each punctured Riemann surface a conformal metric of least (reduced) area subject to a systolic-type constraint: every nontrivial closed curve must have length ≥ 2π. This gives a canonical section in the bundle of local coordinates, ensures compatibility with the BV master equation, and yields vertices without short nontrivial cycles.

While the construction works completely for genus-zero surfaces and large portions of higher-genus moduli spaces, a general mathematical existence proof of such minimizing metrics is missing. Convex optimization methods and numerical evidence (e.g., work of Headrick and Zwiebach) strongly support existence in many cases, but a full proof for arbitrary genus and punctures would settle the foundational status of the minimal-area approach.