Convex programs for minimal-area problems

TL;DR

This paper formulates a convex optimization approach to solve the minimal-area problem on Riemann surfaces, providing new insights and numerical methods for conformal and systolic geometry.

Contribution

It introduces a convex programming formulation for the homology-based minimal-area problem, linking it to dual maximization and enabling numerical solutions.

Findings

Derived a convex program for the homology minimal-area problem.

Established an equivalent dual maximization program.

Provided a numerical approach for solving the minimal-area metric.

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. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem.