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arXiv · 2501.05119

Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

Michael B. Law
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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.

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Michael B. Law. 2025-01-09. Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds. https://arxiv.org/abs/2501.05119

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