Stable cut loci on surfaces

Abstract

Let M be a 2-dimensional compact connected smooth manifold without boundary. Let p ∈ M be fixed. Take a geodesic g(t), 0 ≤ t ≤ ∞, starting at p. Then the first point on this geodesic where the geodesic ceases to minimize distance from p is called the cut point of p along the geodesic g(t). The cut locus C(p) is the set of all cut points of p. Since M is compact, C(p) = ∅. The graph G is said to be smoothly embedded in M if for every point q ∈ G, there exists a smooth coordinate chart ρ : V → R2 where V is an open neighborhood of q in M , such that, for every edge e of G with q ∈ e, ρ(e ∩ V ) is contained in a 1-dimensional affine subspace of R2. Suppose G is a connected finite graph which is smoothly embedded in M , and whose vertices have degree 1 or 3 only. Furthermore, suppose that for every vertex v of G of degree 3, the tangent vectors to M at v in the directions of the three edges of G incident to v are not contained in a closed half-space of TvM . Also, suppose that the inclusion map ι : G → M induces an isomorphism ι∗ : H1(G;Z/2) → H1(M ;Z/2). In §1 3, with the preceding hypothesis, we construct a smooth Riemannian metric α on M and find a point p ∈ M so that the cut locus C(p, α) of p with respect to α is G, and in §4, we show that the cut locus C(p, α) is stable for α.