Splitting and Slow Volume Growth for Open Manifolds with Nonnegative Ricci Curvature
Hongzhi Huang, Xian-Tao Huang
TL;DR
This work proves that open manifolds with nonnegative Ricci curvature and linear volume growth have universal covers that split isometrically as a product R^k × N, with k determined by the finite-index Z^k-subgroup of the fundamental group. It develops a robust framework using stability of Euclidean factors in equivariant asymptotic cones, harmonic functions with almost linear growth, and slicing arguments to obtain both a fibration structure and higher codimension splittings under weaker volume growth hypotheses. Notably, the results imply finiteness of the fundamental group when Ricci is positive somewhere and, under an infimum volume growth bound below 3 with Euclidean growth on the universal cover, an R^{n-2} splitting, connecting to questions of PY24 and PY24. The work extends the compact-case structure theorem to open manifolds, offering precise topological and geometric descriptions such as fiber bundle structures over tori and constraints on the soul of M in the low growth regime. Overall, it provides a comprehensive splitting and rigidity picture for nonnegatively curved open manifolds under linear and sublinear volume growth conditions, with implications for the topology of fundamental groups and the geometry of universal covers.
Abstract
In \cite{NPZ24}, Navarro-Pan-Zhu proved that the fundamental group of an open manifold with nonnegative Ricci curvature and linear volume growth contains a subgroup isomorphic to $\mathbb{Z}^k$ with finite index. They further asked whether the existence of a torsion-free element in the fundamental group forces the universal cover to split off an isometric $\mathbb{R}$-factor (Question 1.3 of \cite{NPZ24}). In this article, we provide an affirmative answer to this question. Specifically, we prove that if an open manifold with nonnegative Ricci curvature has linear volume growth, then its universal cover is isometric to a metric product $\mathbb{R}^k \times N$, where $N$ is an open manifold with linear volume growth and $k$ is the integer such that $π_1(M)$ contains a $\mathbb{Z}^k$-subgroup of finite index. As a direct consequence, if the Ricci curvature is positive at some point, then the fundamental group is finite. We also establish that for an open manifold $M$ with nonnegative Ricci curvature, if the infimum of its volume growth order is strictly less than $3$ and $\tilde{M}$ has Euclidean volume growth, then the universal cover $\tilde{M}$ splits off an $\mathbb{R}^{n-2}$-factor. As an application, if $M$ has first Betti number $b_1 = n-2$ and $\tilde{M}$ has Euclidean volume growth, then its universal cover admits such a splitting. This result provides a partial answer to \cite[Question 1.6]{PY24}.