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

Abstract

Let $M(n,D)$ be the space of closed n-dimensional Riemannian manifolds (M, g) with $\mathrm{diam}\left(M\right)\le D$ and $|{sec}^M|\le 1$ . 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 $\frac{\mathrm{vol}\left(B_r^{M_i}\left(x\right)\right)}{{\mathrm{inj}}^{M_i}\left(x\right)}$ can be uniformly bounded from below by a positive constant C(n, r, Y) for all points $x\in M_i$ . 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 $\frac{\mathrm{vol}\left(B_r^{M_i}\left(x\right)\right)}{{\mathrm{inj}}^{M_i}\left(x\right)}$ uniformly from below for all $x\in M_i$ . 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 $C\le \frac{\mathrm{vol}\left(M\right)}{\mathrm{inj}\left(M\right)}$ .