Boundary exceptional sets for radial limits of superharmonic functions on non-positively curved Harmonic manifolds of purely exponential volume growth

TL;DR

This paper investigates the size of boundary sets where superharmonic functions on non-positively curved harmonic manifolds blow up faster than a certain rate, providing sharp bounds on their Hausdorff dimensions and revealing a gap due to curvature conditions.

Contribution

It establishes sharp Hausdorff dimension bounds for exceptional boundary sets of superharmonic functions on curved harmonic manifolds, extending classical Fatou theorems to this setting.

Findings

Sharp bounds on Hausdorff dimensions of exceptional sets for Poisson integrals

Construction of Green potentials with prescribed blow-up rates

Identification of a gap in Hausdorff dimensions due to curvature assumptions

Abstract

By classical Fatou type theorems in various setups, it is well-known that positive harmonic functions have non-tangential limit at almost every point on the boundary. In this paper, in the setting of non-positively curved Harmonic manifolds of purely exponential volume growth, we are interested in the size of the exceptional sets of points on the boundary at infinity, where a suitable function blows up faster than a prescribed growth rate, along radial geodesic rays. For Poisson integrals of complex measures, we obtain a sharp bound on the Hausdorff dimension of the exceptional sets, in terms of the mean curvature of horospheres and the parameter of the growth rate. In the case of the Green potentials, we obtain similar upper bounds and also construct Green potentials that blow up faster than a prescribed rate on lower Hausdorff dimensional realizable sets. So we get a gap in the corresponding Hausdorff dimensions due to the assumption of variable pinched non-positive sectional curvature. We also obtain a Riesz decomposition theorem for subharmonic functions. Combining the above results we get our main result concerning Hausdorff dimensions of the exceptional sets of positive superharmonic functions.