On the non-collapsed RCD spaces with local bounded covering geometry

Abstract

We consider a RCD$(-(N-1),N)$ space $(X,d,{\mathcal{H}}^N)$ with local bounded covering geometry. The first result is related to Gromov's almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover $\tilde{X}$, we have ${\mathcal{H}}^N(B_1(\tilde{p}))\ge v>0$ and the diameter of $X$ is sufficiently small, then $X$ is biHölder homeomorphic to an infranil-manifold. Moreover, if $X$ is a smooth Riemannian $N$-manifold with $\mathrm{Ric}\ge -(N-1)$, then $X$ is biHölder diffeomorphic to an infranil-manifold. An application of our argument is to confirm the conjecture that Gromov's almost flat manifold theorem holds in the $\mathrm{RCD}+\mathrm{CBA}$ setting. The second result concerns a regular fibration theorem. Let $(X_i,d_i,{\mathcal{H}}^N)$ be a sequence of RCD$(-(N-1),N)$ spaces converging to a compact smooth $k$-dimensional manifold $K$ in the Gromov-Hausdorff sense. Assume that for any $p_i\in X_i$, the local universal cover is non-collapsing, i.e., for any pre-image point ${\tilde{p}}_i$ of $p_i$ in the universal cover of the ball $B_3(p_i)$, we have ${\mathcal{H}}^N(B_1({\tilde{p}}_i))\ge v$ for some fixed $v>0$. Then for sufficiently large $i$, there exists a fibration map $f_i:X_i\to K$, where the fiber is an infra-nilmanifold and the structure group is affine.