On the free boundary min-max geodesics (Q2800175)

The main result of this paper is that, given a closed submanifold \(M\) of a Riemannian manifold, there exists a geodesic segment with free boundary on \(M\). In other words, there exists a geodesic segment that begins and ends in this submanifold, and does so orthogonally. This is proved through a min-max theory for free boundary variational problems, which is an important tool to produce critical points of positive Morse index. One of the key ingredients is an adaptation of Birkhoff's curve shortening process to curves whose endpoints lie on a given submanifold.