Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth
Zetian Yan, Xingyu Zhu
TL;DR
The paper proves that complete noncompact Ricci-flat manifolds with linear volume growth have a unique asymptotic limit, which is always a cylinder $\mathbb{R}\times N$ under natural regularity assumptions; outside a compact set, the metric converges exponentially to the cylinder metric after an appropriate gauge. The authors adapt Cheeger–Tian’s framework to cylindrical ends, solving a divergence equation via Green functions, solving the infinitesimal Ricci deformation equation with a detailed radial-tangential decomposition, and eliminating non-decaying modes through a modified divergence constraint. In dimension four, the result is unconditional thanks to the codimension-4 regularity theorem, and the findings apply to asymptotically cylindrical Calabi–Yau manifolds, confirming exponential convergence to their asymptotic limits. The work extends the understanding of tangent-cone-like limits for linear growth and provides a robust method for proving uniqueness and exponential rates in cylindrical settings, with implications for ALH geometries and special holonomy manifolds.
Abstract
Under natural assumptions on curvature and cross section, we establish the uniqueness of asymptotic limits and the exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth, which are known to only admit cylindrical asymptotic limits. In dimension four, these assumptions hold automatically, yielding unconditional uniqueness and convergence. In particular, our results show that all asymptotically cylindrical Calabi--Yau manifolds converge exponentially to their asymptotic limits, thereby answering affirmatively a question by Haskins--Hein--Nordström. In dimension four our result strengthens those of Chen--Chen, who proved exponential convergence to its asymptotic limit space for any ALH instanton.