A Brown-York Resolvent Functional: Curvature Moments and Asymptotic Scaling

This paper introduces a one-parameter Brown–York resolvent functional: the resolvent map A → A(I − tA)⁻¹ is applied to the physical and reference shape operators of a closed spacelike two-surface. The zero-scale member is exactly the Brown–York quasilocal energy, and the density is the logarithmic derivative of the flat-model normal-offset determinant, so the family is geometrically distinguished rather than an arbitrary analytic deformation. Its Taylor coefficients are the reference-subtracted power sums of the principal curvatures; the second coefficient carries a mean-curvature-square contribution related by the Gauss equation to Willmore and Hawking-type curvature data. The main theorem is a large-sphere asymptotic scale law: under standard asymptotically flat hypotheses, if t(r)/r → α with |α| < 1, the resolvent energy tends to M_ADM/(1 − α)², so the ADM normalization is recovered throughout the sublinear sector. Mean-curvature ordering alone does not determine monotonicity in t. By contrast, ordered principal curvatures with 0 ≤ κ_i ≤ κ̄_i make the family absolutely monotone: every t-derivative is nonnegative, strictly under strict ordering on a set of positive area. In a time-symmetric spherically symmetric slice, the family obeys an exact radial balance law at fixed fractional scale t = αr; on the Schwarzschild exterior it is globally decreasing outward exactly when α ≤ 1/3, and an exact scale-independent transform of E_t recovers the Misner–Sharp mass, as a calibration rather than a new mass. For nonzero t the functional is treated as a reference-calibrated geometric surface functional, not a Hamiltonian boundary charge; that analysis is carried out in the companion paper, "Brown–York Resolvent Energy: Dirichlet Obstruction and a Resolvent–Robin Boundary Condition" (doi:10.5281/zenodo.20650879). Version 3 retitles the paper (previously "A Scale-Dependent Brown–York Resolvent Functional for Quasilocal Energy: Curvature Moments and an Asymptotic Scale Law") and adds the log-determinant motivation, the absolute-monotonicity theorem, the spherical radial balance law with its sharp threshold, and the Misner–Sharp reconstruction.