Periodic solutions with prescribed energy on non-complete Riemannian manifolds
The problem treated in this paper is to look for periodic solutions of the system \(D_t \dot x(t) = -\nabla V(x(t))\) that is defined on a Riemannian manifold for a \(C^1\) potential \(V\). The author considers the case where the Riemannian structure is not necessarily complete; instead, the author assumes the convexity of the boundary. The convexity is defined by an exhausting function whose gradient norm is finite and whose Hessian matrix is nonnegative. Then, by assuming that the fundamental group is finite or has infinite conjugacy classes, and with several additional conditions on the potential and convexity, the author shows the existence of non-trivial periodic solutions with energy estimate.
scientific article; zbMATH DE number 902562