The Weyl functional near the Yamabe invariant

Abstract

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant YC(M) and the Ln/2-norm WC(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant YC(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant WC(M) is arbitrarily large. We study the image of the mapYW:C→(YC(M), WC(M))∈R 2 near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.