Steiner Trees on Curved Surfaces

Abstract. The Steiner tree problem on surfaces is more complicated than the corresponding one in the Euclidean plane. There are not many results on it to date. In this paper we first make a comparison of Steiner minimal trees on general curved surfaces with Steiner minimal trees in the Euclidean plane. Then, we focus our study on the Steiner trees on spheres. In particular, we detail the properties of locally minimal Steiner points, and the Steiner points for spherical triangles.