Universal principles for Kazdan-Warner and Pohozaev-Schoen type identities

Abstract

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captured and extended by a single identity, due to Schoen in 1988. In each of the three cases the identity requires and involves an infinitesimal conformal symmetry. For structures with such a conformal vector field, we develop a very wide, and essentially complete, extension of this picture. Any conformally variational natural scalar invariant is shown to satisfy a Kazdan-Warner type identity, and a similar result holds for scalars that are the trace of a locally conserved 2-tensor. Scalars of the latter type are also seen to satisfy a Pohozaev-Schoen type identity on manifolds with boundary, and there are further extensions. These phenomena are explained and unified through the study of total and conformal variational theory, and in particular the gauge invariances of the functionals concerned. Our generalisation of the Pohozaev-Schoen identity is shown to be a complement to a standard conservation law from physics and general relativity.