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

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