A Partial Epsilon-Regularity Theorem for Integral Stationary Geodesic Networks in R^n

An integral stationary geodesic network is a locally finite embedded graph in a Riemannian manifold M whose edges are geodesics, and for which at each vertex, the sum of the unit tangent vectors leaving it are zero. These objects are critical points of the length functional on the space of 1-varifolds of M. It has been asked if locally the length of such networks can control the cardinality of their singular set. Earlier this year, this was proven when M is a surface. The general case of higher codimension has remained open. In this work, we show that such control holds when networks are sufficiently close to a multiplicity k line in R^n. The proof involves a novel potential function which models pairwise collision potentials of nearby geodesics.