Existence of variationally defined curves in complete Riemannian manifolds

Abstract

We present a method for proving the existence of solutions to a class of one dimensional variational problems. The method is demonstrated by two examples of optimal interpolation problems which are motivated by engineering applications. In each case we prove that the variational problem satisfies the Palais-Smale condition and the existence of a minimal solution and lower bounds for the number of stationary curves then follow.