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arXiv · 2309.09896

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

Dongyeong Ko
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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.

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Dongyeong Ko. 2023-09-18. Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary. https://arxiv.org/abs/2309.09896

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