Elasticae in a Riemannian submanifold
The author considers compact Riemannian manifolds and closed regular curves \(\gamma(s)\) parametrized by arclength in the manifold. Let \(\tau\) be the curvature vector and \(\varphi\) a \(C^ \infty\) function defined on the unit tangent bundle of \(M\); \(\varphi\) therefore is bounded. The problem is to find a minimum for the functional \(F(\gamma)=\int_ \gamma\{|\tau|^ 2+\varphi(\gamma)\}ds\). By standard arguments, there exists a converging subsequence for which \(F(\gamma)\) tends to an infimum in \(C^ 1\) topology. Explicit inequalities show that the convergence is in fact in \(H^ 2\) and therefore there exists a minimum curve which is a classical solution of the Euler equation which can be shown to be \(C^ \infty\). The problem described by the title is a special case.
scientific article; zbMATH DE number 219892