Variational structure of the $v_{\frac{n}{2}}$-Yamabe problem

We define a new formal Riemannian metric on a conformal class in the context of the $v_{\frac{n}{2}}$-Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in a given conformal class.