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We study the flow equations for BPS black holes in $\mathcal{N} = 2$five-dimensional gauged supergravity coupled to any number of vector multipletsvia FI couplings. We develop the Noether-Wald procedure in this context andexhibit the conserved charges as explicit integrals of motion, in the sensethat they can be computed at any radius on the rotating spacetime. The boundaryconditions needed to solve the first order differential equations are discussedin great detail. We extremize the entropy function that controls the nearhorizon geometry and give explicit formulae for all geometric variables attheir supersymmetric extrema. We have also considered a complexification of thenear-horizon variables that elucidates some features of the theory from thenear-horizon perspective.

The Attractor Flow for AdS$_5$ Black Holes in $\mathcal{N} = 2$ Gauged Supergravity