A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.

Let M ( n , D ) be the space of closed n-dimensional Riemannian manifolds (M, g) with diam ( M ) ≤ D and | sec M | ≤ 1 . In this paper we consider sequences ( M i , g i ) in M ( n , D ) 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 vol ( B r M i ( x ) ) inj M i ( x ) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x ∈ M i . 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 vol ( B r M i ( x ) ) inj M i ( x ) uniformly from below for all x ∈ M i . 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 M ( n , D ) with C ≤ vol ( M ) inj ( M ) .