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

TL;DR

This paper develops a unified framework extending Pohozaev and Kazdan-Warner identities to a broad class of conformally variational scalar invariants, revealing deep geometric and physical connections.

Contribution

It introduces a general extension of Pohozaev-Schoen and Kazdan-Warner identities applicable to conformally variational invariants, unifying previous results and exploring their physical significance.

Findings

Extended identities apply to a wide class of scalar invariants.

Identified gauge invariances underlying the identities.

Connected geometric identities to conservation laws in physics.

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.