A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter

Let [Formula: see text] be the space of closed n-dimensional Riemannian manifolds (M, g) with [Formula: see text] and [Formula: see text] . In this paper we consider sequences [Formula: see text] in [Formula: see text] converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient [Formula: see text] can be uniformly bounded from below by a positive constant C(n, r, Y) for all points [Formula: see text] . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient [Formula: see text] uniformly from below for all [Formula: see text] . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in [Formula: see text] with [Formula: see text] .