PreprintCC BY 4.0arXiv
A Reilly approach to weighted curvature integral inequalities in hyperbolic space
Abstract
We establish a polarized weighted Reilly identity and an exact deficit decomposition for weighted curvature integrals Ij=∫ΣVEj dμ in hyperbolic space, where V=cosh r and Ej is the normalized jth mean curvature. For compact domains with smooth connected boundary, these yield Ik2≥ Ik−1Ik+1 under intrinsic sub-staticity at k=1 and static convexity with κ∈Γk∩Γ̅k+1 for 2≤ k≤ n−1, with equality precisely for geodesic balls. Under h≥0 and Vh−Vνg≥0, log-concavity and a Euclidean mixed-volume representation recover the centered-ball comparisons conjectured by Hu and Li \cite[Conjecture~1.1]{HuLi}, allowing the weight center to lie outside the domain. Equality in these comparisons holds precisely for balls centered at the weight center.
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BibTeX
@misc{bi2026reilly,
title = {{A Reilly approach to weighted curvature integral inequalities in hyperbolic space}},
author = {Shengguang Bi and Yingxiang Hu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02882},
eprint = {2611.02882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02882},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Reilly approach to weighted curvature integral inequalities in hyperbolic space
AU - Shengguang Bi
AU - Yingxiang Hu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02882
UR - https://arxiv.org/abs/2611.02882
AB - We establish a polarized weighted Reilly identity and an exact deficit decomposition for weighted curvature integrals I_j=∫_ΣVE_j dμ in hyperbolic space, where V=cosh r and E_j is the normalized jth mean curvature. For compact domains with smooth connected boundary, these yield I_k^2≥ I_k−1I_k+1 under intrinsic sub-staticity at k=1 and static convexity with κ∈Γ_k∩Γ̅_k+1 for 2≤ k≤ n−1, with equality precisely for geodesic balls. Under h≥0 and Vh−V_νg≥0, log-concavity and a Euclidean mixed-volume representation recover the centered-ball comparisons conjectured by Hu and Li \cite[Conjecture~1.1]{HuLi}, allowing the weight center to lie outside the domain. Equality in these comparisons holds precisely for balls centered at the weight center.
ER -