On Einstein four-manifolds with \(S^1\)-actions (Q1762704)

Let \((N,h)\) be a closed, simply connected, \(4\)-dimensional Einstein manifold. Normalize the metric such that \(\text{ Ric}_h = 3h\), and assume that there exists a semi-free isometric \(S^1\)-action on \((N,h)\) such that \((N\setminus F, h)\) is equivariantly isometric to \((U \times S^1, g + u^2d\theta^2)\) with its standard \(S^1\)-action, where \(F\) is the fixed point set of the action, \((U,g)\) is a \(3\)-dimensional Riemannian manifold, \((S^1,d\theta^2)\) is the circle of radius one, and \(u\) is a smooth positive function on \(U\). The author proves that \(F\) consists of totally geodesic \(2\)-dimensional spheres with Gaussian curvature \(K \geq 1\). Moreover, if \(K = 1\) on \(F\), then \((N,h)\) is isometric to the \(4\)-dimensional sphere with constant sectional curvature one. If \(K \leq 3\) on \(F\) and \(F\) has more than one component, then \((N,h)\) is isometric to the Riemannian product of two \(2\)-dimensional spheres with constant Gaussian curvature \(3\). The proof uses the Bochner-Weitzenböck formula for one-forms and the theory of minimal surfaces in \(3\)-manifolds.