Convex programs for minimal-area problems
arXiv:1806.00449
The paper formulates the minimal‑area problem for Riemann surfaces—requiring all non‑contractible curves to have length ≥ 2π—as a local convex optimization problem using calibrations and the max‑flow min‑cut theorem, and provides a dual concave program suitable for numerical solution.
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 2Ï. 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.
70 pages, 14 figures. Clarifications, two new figures