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 $\hat{I}”(r)=r^2-2rm(r)+a^2$ with a positive, nondecreasing mass profile $m(r)$, showing that $1-2m^{\prime }(r)-r{m}^{\prime \prime }(r)>0$ ensures strict convexity and determines root counts for $\hat{I}”$. 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 $m/M_{\text{ADM}}=1+\epsilon h$. 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.