Min-max theory for networks of constant geodesic curvature

TL;DR

This paper develops a min-max theory for finding curves of constant geodesic curvature on closed surfaces, revealing new classifications of blowups and solutions with specific junction properties.

Contribution

It introduces a novel min-max framework for prescribing constant geodesic curvature and classifies blowups, advancing understanding of geometric variational problems.

Findings

Solutions are almost embedded curves with finitely many junction points.

Each smooth segment of the solution has multiplicity one.

A new classification of blowups is established, even for the case c=0.

Abstract

We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smooth segment has multiplicity one. The key is a classification of blowups which is new even for $c=0$.