Existence of Typical Scales for Manifolds with Lower Ricci Curvature Bound

TL;DR

This paper demonstrates that for collapsing sequences of Riemannian manifolds with a uniform lower Ricci curvature bound, there exist specific scales at which the rescaled manifolds converge to a product space with consistent Euclidean and compact factors.

Contribution

It establishes the existence of typical scales for such manifolds where the rescaled limits have uniform Euclidean and compact factors, independent of base point choices.

Findings

Rescaled manifolds converge to a product of Euclidean and compact spaces.

All Euclidean factors have the same dimension across the sequence.

Compact factors satisfy uniform diameter bounds.

Abstract

For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors have the same dimension, all possible compact factors satisfy the same diameter bounds and their dimension does not depend on the choice of the base point (along a fixed subsequence).