Length-minimizing level curves via calibrations

TL;DR

This paper introduces an elementary criterion to establish the length-minimizing property of certain curves, including harmonic function level curves and specific conic sections, within conformal metrics.

Contribution

It provides a new, simple criterion to prove length-minimization of curves in conformal metrics, expanding understanding of geodesic properties in these settings.

Findings

Level curves of harmonic functions are length-minimizing.

Certain conic sections are length-minimizing in specified conformal metrics.

The criterion applies broadly to a large class of conformal metrics.

Abstract

We present an elementary criterion to show the length-minimizing property of geodesics for a large class of conformal metrics. In particular, we prove the length-minimizing property of level curves of harmonic functions and the length-minimizing property of a family of the conic sections with the eccentricity $\varepsilon$ in the upper half plane endowed with the conformal metric $ \left( {\varepsilon}^{2} + \frac{1}{\;{y^2} \;} \right) \left(dx^{2} + dy^{2} \right)$.