The Weyl functional near the Yamabe invariant

For a compact manifold $M$ of $\dim M =n\geq 4$, we study two conformal invariants of a conformal class $C$ on $M$. These are the Yamabe constant $Y_C(M)$ and the $L^{\frac{n}{2}}$-norm $W_C(M)$ of the Weyl curvature. We prove that for any manifold $M$ there exists a conformal class $C$ such that the Yamabe constant $Y_C(M)$ is arbitrarily close to the Yamabe invariant $Y(M)$, and, at the same time, the constant $W_C(M)$ is arbitrarily large. We study the image of the map $\YW: C\mapsto (Y_C(M),W_C(M))\in \R^2$ near the line $\{(Y(M),w) | w\in \R\}$. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact K\"ahler surfaces of Kodaira dimension 0, 1 or 2.