Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

TL;DR

This paper proves that in any Riemannian 2-sphere, there exist immersed closed curves with constant geodesic curvature for nearly all prescribed curvatures, using a novel min-max approach for a weighted length functional.

Contribution

It introduces a new min-max scheme for a weighted length functional to establish the existence of such curves in Riemannian 2-spheres.

Findings

Existence of immersed closed curves with constant geodesic curvature in Riemannian 2-spheres.

Applicability to almost every prescribed curvature.

Development of a min-max scheme for a weighted length functional.

Abstract

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.