Abstract:In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -\kappa$ and $\mathrm{diam}(M)^2\mathrm{Ric}_M \geq -\delta$, where $\delta>0$ is sufficiently small depending only on $n$ and $\kappa$, fibers over a $b_1(M)$-torus. This removes the upper sectional curvature bound required in the earlier result of Yamaguchi \cite{Y88}. As a corollary, we obtain a refinement of Yamaguchi's smooth fibration theorem (\cite{Y91}), showing that the fiber itself (rather than a finite cover of it) fibers over a $b_1$-torus. Our results extend to manifolds satisfying a generalized Reifenberg condition introduced in \cite{HH24}, which encompasses both a lower bound on sectional curvature and the local rewinding Reifenberg condition. In the nonsmooth setting, a similar result also holds for a non-collapsed $\mathrm{RCD}(-\epsilon(D,r,n),n)$ space whose diameter is bounded by $D$ and which satisfies the $(r,\delta(n))$-local rewinding Reifenberg condition. The proofs rely on an equivariant regularity theorem for almost submetries under a lower Ricci curvature bound. In addition, we study the stability of rank of Abelian actions along equivariant Gromov-Hausdorff convergence in this paper.
| Comments: | 53pages. Comments are welcome |
| Subjects: | Differential Geometry (math.DG) |
| Cite as: | arXiv:2605.24380 [math.DG] |
| (or arXiv:2605.24380v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24380 arXiv-issued DOI via DataCite (pending registration) |
From: Xian-Tao Huang [view email]
[v1]
Sat, 23 May 2026 03:38:38 UTC (77 KB)
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