On a mass functional for initial data in 4+1 dimensional spacetime

We consider a broad class of asymptotically flat, maximal initial data sets satisfying the vacuum constraint equations, admitting two commuting rotational symmetries. We construct a mass functional for `$t-{\varphi }^i$’ symmetric data which evaluates to the ADM mass. We then show that $$\mathbb{R}$ \times U(1)^2$-invariant solutions of the vacuum Einstein equations are critical points of this functional amongst this class of data. We demonstrate positivity of this functional for a class of rod structures which include the Myers-Perry initial data. The construction is a natural extension of Dain’s mass functional to $D=5$, although several new features arise.