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

TL;DR

This paper investigates the uniqueness of asymptotic limits for noncollapsed Ricci-flat manifolds with linear volume growth, establishing conditions under which such limits are unique and quantifying the convergence rate.

Contribution

It links the uniqueness of asymptotic limits to the existence of a harmonic function asymptotic to a Busemann function, providing new insights into Ricci-flat manifolds with linear volume growth.

Findings

Proves uniqueness of asymptotic limits under certain conditions.

Establishes a polynomial convergence rate for the asymptotic limit.

Shows that uniqueness implies the existence of a harmonic function without smoothness 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.