Yamabe invariants and the $\mathrm{Pin}^-(2)$-monopole equations

Abstract

Abstract. We compute the Yamabe invariants for a new infinite class of closed 4-dimensional manifolds by using a "twisted" version of the SeibergWitten equations, the Pin − (2)-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature. IntroductionThe Yamabe invariant is a diffeomorphism invariant of smooth manifolds, which arises from a variational problem for the total scalar curvature of Riemannian metrics. The Pin − (2)-monopole equations are a "twisted" version of the Seiberg-Witten equations. In this paper we will compute the Yamabe invariants for a new infinite class of closed 4-dimensional manifolds by using the Pin − (2)-monopole equations. We begin by recalling the Yamabe invariant. Let X be a closed, oriented, connected manifold of dim X = m ≥ 3, and M(X) the space of all smooth Riemannian metrics on X. For each metric g ∈ M(X), we denote by s g the scalar curvature and by dµ g the volume form. Then the normalised Einstein-Hilbert functional E X : M(X) → R is defined byThe classical Yamabe problem is to find a metricǧ in a given conformal class C such that the normalised Einstein-Hilbert functional attains its minimum on C: E X (ǧ) = inf g∈C E X (g). This minimising metricǧ is called a Yamabe metric, and a conformal invariant Y(X, C) := E X (ǧ) the Yamabe constant. We define a diffeomorphism invariant Y(X) by the supremum of Y(X, C) of all the conformal classes C on X:We call it the Yamabe invariant of X; it is also referred to as the σ-constant. See [16] and [28]. It is a natural problem to compute the Yamabe invariant. In dimension 4, Seiberg-Witten theory and LeBrun's curvature estimates have played a prominent role in this problem. LeBrun used the ordinary Seiberg-Witten equations to compute the Yamabe invariants of most algebraic surfaces [19,20]