Volume renormalization of higher-codimension singular Yamabe spaces

Abstract

Given an embedded closed submanifold

in the closed Riemannian manifold

M n + k M^{n+k}

, where

k < n + 2 k<n+2

, we define extrinsic global conformal invariants of Σ by renormalizing the volume associated to the unique singular Yamabe metric with singular set Σ. 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 ≥ n + 2 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.