Cut Loci and Distance Spheres on Alexandrov Surfaces

The purpose of the present paper is to investigate the structure of distance spheres and cut locus C(K) to a compact set K of a complete Alexandrov surface X with curvature bounded below. The structure of distance spheres around K is almost the same as that of the smooth case. However C(K) carries different structure from the smooth case. As is seen in examples of Alexandrov surfaces, it is proved that the set of all end points Ce(K) of C(K) is not necessarily countable and may possibly be a fractal set and have an infinite length. It is proved that all the critical values of the distance function to K is closed and of Lebesgue measure zero. This is obtained by proving a generalized Sard theorem for one-valuable continuous functions. Our method applies to the cut locus to a point at infinity of a noncompact X and to Busemann functions on it. Here the structure of all co-points of asymptotic rays in the sense of Busemann is investigated. This has not been studied in the smooth case. Résumé. L’objet de cet article est d’étudier la structure des sphères de distance et du cut locus C(K) d’un ensemble compact. M.S.C. Subject Classification Index (1991) : 53C20. Acknowledgements. Research of the authors was partially supported by Grant-in-Aid for Cooperative Research, Grant No. 05302004 c © Séminaires & Congrès 1, SMF 1996