Length-minimizing level curves via calibrations

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 $\epsilon$ in the upper half plane endowed with the conformal metric $\left({\epsilon }^2+\frac{1}{y^2}\right)\left(d{x}^2+d{y}^2\right)$.

Typos corrected