Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds
arXiv:2501.05119
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.
50 pages. v2: tidied up some parts, changed exposition. Main result and proof are the same