Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

TL;DR

This paper classifies all polynomial growth solutions to drift-harmonic equations on manifolds with paraboloidal asymptotics, revealing their asymptotic behavior and dimensions, especially on steady gradient Ricci solitons.

Contribution

It provides a complete classification and dimension count of polynomial growth drift-harmonic functions on asymptotically paraboloidal manifolds, including steady gradient Ricci solitons.

Findings

All polynomial growth drift-harmonic functions asymptotically separate variables.

Dimensions of spaces of such functions are explicitly computed.

Constructive inductive method used for classification.

Abstract

We construct and classify all polynomial growth solutions to certain drift-harmonic equations on complete manifolds with paraboloidal asymptotics. These encompass the natural drift-harmonic equations on certain steady gradient Ricci solitons. Specifically, we show that all drift-harmonic functions with polynomial growth asymptotically separate variables, and compute the dimensions of spaces of drift-harmonic functions with a given polynomial growth rate. The proof uses an inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.