Minimal-area metrics on the Swiss cross and punctured torus

TL;DR

This paper numerically determines the minimal-area conformal metric on a punctured torus and Swiss cross shape using convex programming, revealing curvature properties and geodesic band structures relevant to string theory.

Contribution

It introduces a numerical convex programming approach to find minimal-area metrics on complex surfaces with crossing geodesic bands, advancing understanding of string field theory geometries.

Findings

Positively curved in two-band regions

Flat in single-band regions

Develops a third geodesic band for small boundaries

Abstract

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2\pi. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the "Swiss cross" challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.