  A conformal geometry determines a distinguished, potentially singular,
variant of the usual Yamabe problem, where the conformal factor can change
sign. When a smooth solution does change sign, its zero locus is a smoothly
embedded separating hypersurface that, in dimension three, is necessarily a
Willmore energy minimiser or, in higher dimensions, satisfies a conformally
invariant analog of the Willmore equation. In any case the zero locus is
critical for a conformal functional that generalises the total Q-curvature by
including extrinsic data. These observations lead to some interesting global
problems that include natural singular variants of a classical problem solved
by Obata.
