Yang--Mills Monopoles in Extremal Reissner--Nordstr\"{o}m Black Hole Metric

We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang–Mills wave equation over the extremal Reissner–Nordstr"{o}m spacetime found in a recent work of Bizo'{n} and Kahl, is the unique solution to an associated domain-wall type energy minimization problem. We also derive the accompanying interior Yang–Mills wave equation within the same formalism. We obtain a few more closed-form solutions, regular and singular, among other solutions of oscillatory behavior, and discuss several interesting features of the solutions based on some energy consideration.