Convexity criterion and radial-profile response for off-shell Kerr geometries: a fuzzy-dark-matter profile as an analytic benchmark

Abstract

We establish a sufficient one-minimum criterion for the off-shell Kerr family $\Delta(r) = r^2 - 2rm(r) + a^2$ with a positive, nondecreasing mass profile $m(r)$, showing that $1 - 2m'(r) - rm''(r) > 0$ ensures strict convexity and determines root counts for $\Delta$. Using a fuzzy-dark-matter-inspired benchmark satisfying this bound, we derive first-order responses for the outer horizon, extremal branch, photon sphere, and shadow functional under general deformations $m/M_{\text{ADM}} = 1 + \varepsilon h$. We demonstrate that static horizon and photon responses are profile-controlled, spin-odd shadow displacements are completion-dependent, and scale-consistent weak-field limits render local profile-gradient effects negligible ($\ll 10^{-20}$), confirming the strong-field box as a formal radial-profile benchmark rather than a self-consistent rotating scalar-field solution.