Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth

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--Nordstr\"om. In dimension four our result strengthens those of Chen--Chen, who proved exponential convergence to its asymptotic limit space for any ALH instanton.