Closed geodesics in collapsing K3 surfaces

Abstract Motivated by some conjectures originating in the physics literature, we use Foscolo’s construction of Ricci-flat Kähler metrics on K3 surfaces to locate, with high precision, several closed geodesics and compute their index (their length is also approximately known). Interestingly, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873. Our construction is also of interest to modern physicists working on (supersymmetric) non-linear sigma models with target space such a K3 surface.