Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary

TL;DR

This paper proves the existence of two free boundary embedded geodesics on a convex Riemannian 2-disk and analyzes their Morse indices, extending classical results to free boundary settings.

Contribution

It establishes the existence of free boundary embedded geodesics and determines their Morse indices, extending Grayson's theorem to free boundary scenarios.

Findings

Existence of two free boundary embedded geodesics on convex Riemannian 2-disks.

Existence of a simple closed geodesic with Morse index 1 and 2.

Flow convergence results for free boundary curve shortening flow.

Abstract

We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian $2$-disk with a strictly convex boundary. Moreover, we prove that there exists a simple closed geodesic with Morse Index $1$ and $2$. This settles the free boundary analog of Grayson's theorem.