A Reilly approach to weighted curvature integral inequalities in hyperbolic space

We establish a polarized weighted Reilly identity and an exact deficit decomposition for weighted curvature integrals Iⱼ=∫_ΣVEⱼ~dμ in hyperbolic space, where V= 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∩Γₖ₊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 [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.