On the cut locus of closed sets

Abstract

Let C be a closed subset of a smooth manifold of dimension n, M, and let M be endowed with a Riemannian metric of class C2. We study the cut locus of C, cut(C). First, we show that cut(C) is a set of measure zero. Then, we assume that C is the boundary of an open bounded set, Ω⊂M (in particular, this assumption implies that cut(C)≠0̸.) We deduce that cut(C)∩Ω is invariant w.r.t. the (generalized) gradient flow associated with the distance function from the set C. As a consequence of the invariance, we have that cut(C)∩Ω has the same homotopy type as the set Ω. Furthermore, if M is a compact manifold, then cut(C) has the same homotopy type as M∖C. Finally, we show that the closure of the cut locus stays away from C if and only if C is a manifold of class C1,1.