On the decomposition of global conformal invariants II

Abstract

This paper is a continuation of [S. Alexakis, The decomposition of global conformal invariants I, submitted for publication, see also math.DG/0509571], where we complete our partial proof of the Deser–Schwimmer conjecture on the structure of “global conformal invariants.” Our theorem deals with such invariants P ( g n ) that locally depend only on the curvature tensor R i j k l (without covariant derivatives). In [S. Alexakis, The decomposition of global conformal invariants I, Ann. of Math., in press] we developed a powerful tool, the “super divergence formula” which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator I g n ( ϕ ) that measures the “non-conformally invariant part” of P ( g n ) . This paper resolves the problem of using this information we have obtained on the structure of I g n ( ϕ ) to understand the structure of P ( g n ) .