Seifert conjecture in the even convex case (Q2922149)

Let \(J\) be the standard symplectic matrix on \(\mathbb{R}^{2n}\) and \(\omega_0\) the standard symplectic form \(\omega_0(x,y)=\left<Jx,y\right>\) and define an involution matrix \(N=\text{diag}(-I_n, I_n)\). Let \(H:\mathbb{R}^{2n}\setminus\{0\}\to\mathbb{R}\) be a \(C^2\) Hamiltonian which admits a \(C^1\)-extension over \(\mathbb{R}^{2n}\) and satisfies the reversible condition \(H\circ N=H\) and let \(\Sigma\) be a \(C^2\) compact hypersurface bounding a compact set \(C\) with nonempty interior. Moreover, assume that \(\Sigma\) has nonvanishing Gaussian curvature and that \(N(\Sigma)=\Sigma\). For \(x\in\Sigma\) denote the unit outward normal vector by \(n_\Sigma\). The task is to find a \(\tau>0\) and a \(C^1\)-curve \(x:\mathbb{R}\to\Sigma\) such that \(\dot{x}=Jn_\Sigma(x)\) satisfying \(x(-t)=Nx(t)\) and \(x(t+\tau)=x(t)\) for all \(t\in\mathbb{R}\). Such a solution \((\tau,x)\) is called a brake orbit on \(\Sigma\). Two brake orbits \((\tau_1,x_1)\) and \((\tau_2,x_2)\) are said to be equivalent if \(x_1(\mathbb{R})=x_2(\mathbb{R})\). If \(\Sigma\) is symmetric and strictly convex the authors prove that there are at least \(n\) equivalence classes of brake orbits. The main tool in the proof is Maslov index theory. To get Seifert's original setting (cf., [\textit{H. Seifert}, Math. Z. 51, 197--216 (1948; Zbl 0030.22103)]) consider the Hamiltonian \(H(p,q)=\frac12A(q)p\cdot p+V(q)\) where \(A\) is \(C^2\), \(A(q)\) is positive definite and the potential energy \(V\) is \(C^2\) as well. If \(x=(p,q)\) solves \(\dot{x}=JH'(x)\) with \(p(0)=p(\frac\tau2)=0\), then \((\tau,x)\) is a brake orbit on \(\Sigma:=H^{-1}(\{h\})\) where \(h\) is the total energy of the brake orbit, i.e., \(H(p(t),q(t))=h\) and \(V(q(0))=V(q(\tau))=h\). Let \(\Omega:=\{q|\;V(q)<h\}\). Seifert [op. cit.] proved that there must be a brake orbit on \(\Sigma\) if one assumes that \(\Bar{\Omega}\) is bounded and homeomorphic to the closed unit ball and that \(V'\not=0\) on \(\partial\Omega\). Moreover, he conjectured that there are at least \(n\) equivalence classes of brake orbits on \(\Sigma\) in this situation.