Some existence results for dynamical systems on non-complete Riemannian manifolds (Q1809489)

Let \(({\mathcal M}^*, \langle \cdot,\cdot\rangle_\mathbb{R})\) be a finite dimensional Riemannian manifold, \({\mathcal M}\subseteq{\mathcal M}^*\) be an open unbounded connected subset such that \(({\mathcal M},\langle \cdot,\cdot\rangle_\mathbb{R})\) is a Riemannian manifold with bounded topological boundary \(\partial{\mathcal M}\), and \(V:{\mathcal M}\rightarrow\mathbb{R}\) a \(C^2\) potential function. The authors look for curves \(x:[0,T]\rightarrow{\mathcal M}\) having prescribed period \(T\) or joining two fixed points of \({\mathcal M}\), satisfying the system \(D_t(\dot x(t))=-\nabla_\mathbb{R} V(x(t))\), where \(D_t(\dot x(t))\) is the covariant derivative of \(\dot x\) along the direction of \(\dot x\) and \(\nabla_\mathbb{R} V\) the Riemannian gradient of \(V\). The potential \(V\) is supposed to be subquadratic at infinity, \(V(x)\rightarrow-\infty\) if \(d(x,\partial{\mathcal M}) \rightarrow 0\), and \(\mathcal M^*\) is non-complete. Some suitable hypotheses on the sectional curvature of \(\mathcal M\) at infinity are required for the periodic case. Variational methods with a penalization technique and Morse index estimates are used.