Canonical identification at infinity for Ricci-flat manifolds

TL;DR

This paper introduces a method to identify scales in non-compact Ricci-flat manifolds with Euclidean volume growth using gradient flows, especially when the tangent cone at infinity has a smooth cross section.

Contribution

It provides a natural scale identification technique for Ricci-flat manifolds based on gradient flows of elliptic equations, applicable at arbitrarily large scales.

Findings

Establishes a canonical way to compare distant scales in Ricci-flat manifolds.

Uses gradient flow of elliptic solutions for scale identification.

Applicable when tangent cone at infinity has smooth cross section.

Abstract

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.