The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II)

TL;DR

This paper characterizes complete metrics with finite total Q-curvature as normal metrics across all dimensions and introduces a new volume entropy to analyze non-normal metrics, linking scalar curvature bounds to volume growth control.

Contribution

It provides a comprehensive characterization of finite total Q-curvature metrics and introduces a novel volume entropy to study non-normal metrics with geometric bounds.

Findings

Complete metrics with finite total Q-curvature are normal in all dimensions.

A new volume entropy is introduced to analyze non-normal metrics.

Scalar curvature bounds imply controlled volume growth.

Abstract

This is a continuation of our previous work (Advances in Mathematics 450 (2024), Paper No. 109768). In this paper, we characterize complete metrics with finite total Q-curvature as normal metrics for all dimensional cases. Secondly, we introduce another volume entropy to provide geometric information regarding complete non-normal metrics with finite total Q-curvature. In particular, we show that if the scalar curvature is bounded from below, the volume growth of such complete metrics is controlled.