On the multiplicity of brake orbits and homoclinics in Riemannian manifolds (Q2504816)

Summary: Let \((M,g)\) be a complete Riemannian manifold, \(\Omega \subset M\) an open subset whose closure is diffeomorphic to an annulus. If \(\partial \Omega\) is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in \(\overline{\Omega} = \Omega \bigcup \partial \Omega\) starting orthogonally to one connected component of \(\partial \Omega\) and arriving orthogonally onto the other one. The results given in [the authors, Adv. Differ. Equ. 10, No. 8, 931--960 (2005; Zbl 1118.37031)] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct \` brake orbits' for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least \(\text{dim} (M)\) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.