Volume renormalization of higher-codimension singular Yamabe spaces

Given an embedded closed submanifold $\Sigma^n$ in the closed Riemannian manifold $M^{n + k}$, where $k < n + 2$, we define extrinsic global conformal invariants of $\Sigma$ by renormalizing the volume associated to the unique singular Yamabe metric with singular set $\Sigma$. In case $n$ is odd, the renormalized volume is an absolute conformal invariant, while if $n$ is even, there is a conformally invariant energy term given by the integral of a local Riemannian submanifold invariant. In particular, the renormalized volume gives a global conformal invariant of a knot embedding in the three-sphere. We compute the variations of these quantities with respect to variations of the submanifold. We extend the construction of energies for even $n$ to general codimension by considering formal solutions to the singular Yamabe problem; except that, for each fixed $n$, there are finitely many $k \geq n + 2$, which we identify, for which the smoothness of the formal solution is obstructed and we obtain instead a pointwise conformal invariant. We compute the new quantities in several cases.