Almost splitting maps, transformation theorems and smooth fibration theorems

2022/07/20 by Hongzhi Huang, Huang, Hongzhi, Xian-Tao Huang +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2207.10029

openalex publication_date 2022/07/20 · openalex created_date 2022/07/22 · openalex updated_date 2026/07/28

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 (\citeCN) and Cheeger-Jiang-Naber (\citeCJN21). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in \citeCJN21) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.

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