Geodesics with unbounded speed on fluctuating surfaces (Q6580671)

The broad setting is a smooth closed hypersurface \(M\) in \(\mathbb{R}^d\) where \(d \geq 3\), and where \(M\) has the induced metric from \(\mathbb{R}^d\) with a \(2 \pi\)-time periodic embedding \(G_{\epsilon, t} : M \rightarrow \mathbb{R}^d\) such that \(G_{\epsilon, t}\) is \(O(\epsilon)\) close to the identity in \(C^{\infty}\). Then \(M_{\epsilon, t} = G_{\epsilon,t} (M)\) is a fluctuating hypersurface in \(\mathbb{R}^d\) that is \(O(\epsilon)\)-close to \(M\) in \(C^{\infty}\). A geodesic flow on \(M_{\epsilon, t}\) is defined by using the coordinates from \(M\) via the embedding \(G_{\epsilon, t}\) and the pullback metric \(g_{\epsilon}(\cdot, t) = (G_{\epsilon, t} )^*\). The geodesic flow on the fluctuating surface \(M_{\epsilon, t}\) is the flow \(\phi_{\epsilon,t}\) corresponding to the Hamiltonian \(H_{\epsilon} (q,p,t) = \frac{1}{2} g_{\epsilon} (q, t)(p,p)\).\N\NThe author's main result is as follows: If \(M\) is a \(C^{\infty}\) closed surface in \(\mathbb{R}^3\) with Euclidean metric such that the corresponding geodesic flow has a hyperbolic periodic orbit and a transverse homoclinic, then there are \(C^{\infty}\) \(2 \pi\)-time periodic embeddings \(G_{\epsilon, t}: M \rightarrow \mathbb{R}^3\) and points \((q_0, p_0)\) and \((q_1, p_1)\) in \(TM\) such that \(H_{\epsilon}(\phi_{\epsilon, t}(q_0, p_0), t) \rightarrow \infty\) as \(t \rightarrow \infty\), lim inf \(H_{\epsilon}(\phi_{\epsilon, t}(q_1, p_1, t) = c\) and lim sup \(H_{\epsilon}(\phi_{\epsilon, t}(q_1, p_1, t) = \infty\) as \(t \rightarrow \infty\) where \(c\) is a finite constant, and the initial speed \(||p_j||\) is of order 1.\N\NBecause the Hamiltonian -- thus the energy -- is unbounded, the geodesic flow on these fluctuating surfaces has orbits whose speed goes to infinity, hence the title of the paper.