Multiple Brake Orbits and Homoclinics in Riemannian Manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Omega\subset M$ an open subset whose closure is homeomorphic to an annulus. We prove that if $\partial \Omega$ is smooth and it satisfies a strong concavity assumption, then there are at least two 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. Using the results given in Giambò et al. (Adv Differ Equ 10:931–960, 2005), we then 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, the result is improved by showing the existence of at least $\Dim(M)$ pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giambò et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290–337, 2010).