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

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arxiv 2408.03640 v2 pith:CHT4B3O5 submitted 2024-08-07 math.DG math.AP

Mingxiang Li This is my paper · ORCID
classification math.DGmath.AP
keywords metricscompleteq-curvaturetotalvolumeentropyfinitegrowth
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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.

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  1. Positivity and non-positivity results for the sixth-order $Q$-curvature of conformal metrics in $\mathbb{R}^n$

    math.DG 2026-07 conditional novelty 8.0 of 10

    Under nonnegative top-order Q-curvature, Q^(6) is positive for 2m ≤ n ≤ 4m−6, but fails at some point for all n > N_m ≈ 10.55m, refuting the positivity conjecture.

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