Almost splitting maps, transformation theorems and smooth fibration theorems

TL;DR

This paper introduces a generalized Reifenberg condition to prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounds, unifying previous results and utilizing transformation techniques for almost splitting maps.

Contribution

It develops a generalized Reifenberg condition and extends transformation theorems for almost splitting maps to collapsed manifolds, providing a unified approach to smooth fibration theorems.

Findings

Proves a smooth fibration theorem under the generalized Reifenberg condition.

Extends transformation theorems for almost splitting maps to collapsed manifolds.

Provides applications of transformation theorems in geometric analysis.

Abstract

In this paper, we introduce a notion, called generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounded below, which gives a unified proof of smooth fibration theorems in many previous works (including the ones proved by Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration theorem is the transformation technique for almost splitting maps, which originates from Cheeger-Naber (\cite{CN}) and Cheeger-Jiang-Naber (\cite{CJN21}). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in \cite{CJN21}) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.