A pinching theorem for homotopy spheres

Abstract

The (length) excess of a triangle measures how much the triangle inequality fails to be an equality. This notion was first studied seriously in [AG]. We say that a (bounded) metric space X = (X, d) has excess :5 e if there are points p, q E X such that d(p, x)+d(x, q) ~ d(p, q)+e for all x EX. The excess, e(X) , of X is the infimum of all e ~ 0 where X has excess ~ e. Notice that the standard spheres have excess = O. In general, a closed Riemannian manifold M has excess = 0 if and only if there is a point p E M whoSe cutlocus C(P) is a single point q EM. Such a manifold is clearly a twisted sphere, i.e., it is the union of two discs glued together along their common boundary. Conversely ·according to a result of Weinstein [B; Appendix C.4], any twisted sphere carries a Riemannian metric for which there is a point whose cut-locus is another point. With this in mind, we are interested in closed Riemannian manifolds with small excess. However; without any further geometric restrictions, small excess has no topological significance (see Problem 8, though). In fact, any closed n-manifold M can be given a Riemannian metric withdiam M = 1C and arbitrarily small excess. Simply make sure that all topological complexities of M (i.e~, the complement of a topological n-disc) are contained in a sufficiently small metric ball and what remains is metrically close to the complement of a small ball in the unit sphere Sn (1). Our main result is Theorem A. Given k E R, D, v > 0, and an integer n ~ 2, there is an' t = e(n, k, D, v) such that any closed Riemannian n-mani/old M with sectional curvature KM ~ k, diamM ~ D, VolM ~ v, and e(M) ~ e is a homotopy sphere.