[Paper Review] Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth II

Zetian Yan, Xingyu Zhu|arXiv (Cornell University)|Jan 4, 2026

Geometry and complex manifolds0 citations

TL;DR

Relates the uniqueness of asymptotic limits for noncollapsed Ricci-flat manifolds with linear volume growth to the existence of a harmonic function asymptotic to a Busemann function, establishing uniqueness and a polynomial convergence rate, and proving existence under uniqueness assumptions.

ABSTRACT

We relate the uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth to the existence of a harmonic function asymptotic to a Busemann function. Parallel to the work of Colding--Minicozzi in the Euclidean volume growth setting, we prove uniqueness of the asymptotic limit and establish a quantitative polynomial convergence rate via a monotone quantity associated with this harmonic function, assuming such harmonic function exists and one asymptotic limit is smooth. Conversely, for an open manifold with nonnegative Ricci curvature, we show that uniqueness of the asymptotic limit implies the existence of the desired harmonic function, without assuming smoothness of the cross section.

Motivation & Objective

• Motivate the study of asymptotic limits for noncollapsed Ricci-flat manifolds with linear volume growth and their potential uniqueness.
• Develop a monotone framework to prove uniqueness of asymptotic limits and quantify convergence rates.
• Relate the uniqueness result to the existence of a harmonic function asymptotic to a Busemann function on the manifold’s end.
• Provide an existence result for such harmonic functions under a uniqueness assumption for the asymptotic limit.

Proposed method

• Define and analyze a monotone quantity derived from a harmonic replacement and a Busemann function.
• Smooth a non-smooth monotone quantity via a constructed harmonic function to obtain a tractable, approximate functional.
• Establish a Łojasiewicz–Simon type inequality for the approximate functional to derive decay and convergence rates.
• Translate geometric convergence into estimates on Gromov–Hausdorff distances to cylinders, using Hessian bounds and partial differential inequalities.
• Prove an existence result for a harmonic function asymptotic to a Busemann function by transplanting from the limit cylinder and applying elliptic regularity.

Experimental results

Research questions

• RQ1 Does a noncollapsed Ricci-flat manifold with linear volume growth have a unique asymptotic limit for divergent translation sequences?
• RQ2 Can a harmonic function asymptotic to a Busemann function exist on the manifold’s end, given uniqueness of the asymptotic limit?
• RQ3 If a unique asymptotic limit exists, what is the rate at which the manifold converges to a cylinder along translation sequences?
• RQ4 How can monotone quantities and their smooth approximations yield quantitative convergence rates to the asymptotic limit?

Key findings

• A monotone quantity associated with a harmonic function (and its smooth approximation) yields a polynomial convergence rate to the unique cylindrical asymptotic limit under suitable smoothness assumptions.
• If one asymptotic limit is smooth and a harmonic function asymptotic to the Busemann function exists, the asymptotic limit is unique and the convergence to a cylinder is polynomial in t.
• There exists an effective uniqueness statement: small discrepancies in a monotone quantity and near-cylindrical geometry over a time interval imply controlled closeness to a cylinder across the interval.
• Under a uniqueness assumption for the asymptotic limit, there is an existence result for a harmonic function on the end which is asymptotic to the Busemann function.
• The approach connects a geometric monotonicity framework with an analytic Łojasiewicz–Simon inequality to obtain decay estimates for the derivative of the monotone quantity.
• The work removes several previous technical assumptions from earlier papers and clarifies the role of a harmonic function asymptotic to a Busemann function in the linear volume growth setting.