A characterization of codimension one collapse under bounded curvature and diameter

TL;DR

This paper characterizes the conditions under which sequences of bounded curvature and diameter Riemannian manifolds collapse to spaces of at most codimension one, linking volume, injectivity radius, and curvature bounds.

Contribution

It establishes a bi-conditional relationship between codimension one collapse and uniform lower bounds on volume-to-injectivity radius ratios, extending understanding of manifold collapse behavior.

Findings

Limit spaces have at most codimension 1 under certain volume-injectivity bounds.

A lower bound on volume-to-injectivity ratio implies at most codimension 1 collapse.

Provides bounds on injectivity radius of fibers in Riemannian submersions.

Abstract

Let $\mathcal{M}(n,D)$ be the space of closed $n$-dimensional Riemannian manifolds $(M,g)$ with $diam(M) \leq D$ and $| \sec^M | \leq 1$. In this paper we consider sequences $(M_i,g_i)$ in $\mathcal{M}(n,D)$ 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 $1$ if there is a positive number $r$ such that the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ 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 $1$ then for all positive $r$ there is a positive constant $C(n,r,Y)$ bounding the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ uniformly from below for all $x \in M_i$. The proof uses results about the structure of collapse in $\mathcal{M}(n,D)$ by Cheeger, Fukaya and Gromov. In addition, we derive, for a submersion $M \rightarrow Y$ with uniformly bounded fundamental tensors, an upper bound on the injectivity radius of the fiber $F_p$, with $p \in Y$, which is proportional to the injectivity radius of $M$ at some $x \in F_p$, if the injectivity at $x$ is sufficiently small relative to the injectivity radius of $Y$. 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 $\mathcal{M}(n,D)$ with $C \leq \min_{x \in M}\frac{vol(B^{M}_r(x))}{inj^{M}(x)}$ for fixed positive numbers $r$ and $C$.