Price Inequality and the Growth of Harmonic Functions on Non-Positively Curved Manifolds

TL;DR

This paper provides effective estimates for the growth of harmonic functions on non-positively curved manifolds, utilizing Price inequalities, and explores implications for the Singer conjecture.

Contribution

It introduces a double-sided Price inequality for harmonic functions and applies it to analyze potential counterexamples to the Singer conjecture.

Findings

Effective growth estimates for harmonic functions on curved manifolds

Development of a double-sided Price inequality

Insights into the structure of potential Singer conjecture counterexamples

Abstract

We obtain effective estimates for the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a double-sided Price inequality for harmonic functions. Finally, we apply this circle of ideas to study the analytical structure of a potential counterexample to the Singer conjecture in degree one.