On the effect of the domain on the number of orthogonal geodesic chords
A celebrated theorem of Lyusternik and Schnirelmann states that a convex subset of \({\mathbb{R}}^n\) with smooth boundary has at least \(n\) orthogonal geodesic chords. The purpose of this paper is to investigate how the number of orthogonal geodesic chords changes according to the topology of the domain. More precisely, let \(M\) be an \(n\)-dimensional (\(n\geq 2\)) Riemannian manifold of class \(C^3\) and let \(\overline{\Omega}\subset M\) be the closure of an open subset \(\Omega\). Assume that \(\partial\Omega\) is a manifold of class \(C^2\) and \(\overline\Omega\) is convex. There are two possibilities, according to the case where \(\overline{\Omega}\) has one or at least two holes: (i) \(\overline{\Omega}\) is homeomorphic to an annulus in \({\mathbb{R}}^n\). Then \(\overline{\Omega}\) has at least two geometrically distinct orthogonal geodesic chords. (ii) There exist \(p\geq 2\), \(x_1,\dots, x_p\in{\mathbb{R}}^n\), and positive numbers \(\rho_1,\dots, \rho_p\) such that \(\overline{B(x_i,\rho_i)}\subset B(0,1)\) for any \(i\), \(\overline{B(x_i,\rho_i)}\cap\overline{B(x_j,\rho_j)}=\emptyset\) if \(i\not=j\) and \(\overline{\Omega}\) is homeomorphic to \(\overline{B(0,1)}\setminus \bigcup_{i=1}^p \overline{B(x_i,\rho_i)}\). Then there exists a sequence \(\{\gamma_m\}_m\) of orthogonal geodesic chords in \(\overline{\Omega}\) having energy going to \(+\infty\). The problem of finding orthogonal geodesic chords is related to the search of brake orbits in a potential well, which represent a special kind of periodic solutions of a Hamiltonian system. In this direction the main results of this paper and that of Lyusternik and Schnirelmann suggest the possibility to have at least two brake orbits, provided that the potential well satisfies the above condition (i), and infinitely many whenever it satisfies (ii). The paper is well written and the multiplicity results obtained by the authors show that they depend on the topology of the manifold in a strange way.
scientific article; zbMATH DE number 1145945
scientific article