On the scalar curvature of self-dual manifolds (Q1318162)

We generalize LeBrun's explicit ``hyperbolic ansatz'' construction of self-dual metrics on connected sums of conformally flat manifolds and \(\mathbb{C} P^ 2\)'s through a systematic use of the theory of hyperbolic geometry and Kleinian groups. (This construction produces [\textit{C. LeBrun}, Self-dual manifolds and hyperbolic geometry, Lect. Notes Pure Appl. Math. 145, 99-131 (1993)], for example, all self-dual manifolds with semi-free \(S^ 1\)-action and with either nonnegative scalar curvature or positive-definite intersection form.) We then point out a simple criterion for determining the sign of the scalar curvature of these conformal metrics. Exploiting this, we then show that the sign of the scalar curvature can change on connected components of the moduli space of self-dual metrics, thereby answering a question raised by \textit{A. A. King} and \textit{D. Kotschick} [Math. Ann. 294, 591-609 (1992; Zbl 0765.58005)].