Harmonic and Schr\"odinger functions of polynomial growth on gradient shrinking Ricci solitons

TL;DR

This paper investigates the properties of harmonic, caloric, and Schr"odinger functions with polynomial growth on gradient shrinking Ricci solitons, establishing finiteness and dimension estimates under various curvature conditions.

Contribution

It provides new finite dimensionality results for harmonic and caloric functions under scalar curvature decay and sharp estimates for Schr"odinger functions without curvature assumptions.

Findings

Finite dimensionality of harmonic functions with polynomial growth under quadratic scalar curvature decay.

Finite dimensionality of ancient caloric functions with polynomial growth under similar conditions.

Sharp finite dimensional estimates for Schr"odinger functions without curvature restrictions.

Abstract

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We also prove analogous results for ancient caloric functions. On the other hand, without any curvature condition, we prove sharp finite dimensional estimates for the space of Schr\"odinger functions with fixed polynomial growth degree.