The dual Thurston norm and the geometry of closed 3-manifolds (Q2504127)

In this nice paper, the authors characterize Riemannian three manifolds satisfying an extremal condition for norms of monopole classes. Thurston defined a (pseudo-)norm in the first cohomology of a three manifold. It is a norm when the manifold is closed and does not contain tori nor spheres which are non-separating (homologically nontrivial). By Poincaré duality, it induces a dual norm on the second cohomology group with real coefficients. Monopole classes in the second real cohomology group are (complex) Chern classes of \(\text{Spin}^c\) structures which support solutions to the Seiberg-Witten equations. \textit{P. B. Kronheimer} and \textit{T. S. Mrowka} have shown in [Math. Res. Lett. 4, No. 6, 931--937 (1997; Zbl 0892.57011)] that the convex hull of monopole classes is precisely the unit ball of the dual of the Thurston norm. In addition, they have shown that the dual Thurston norm equals the supremum over all metrics of \(4\pi\) times the \(L^2\) norm of the harmonic representative divided by the \(L^2\)-norm of the scalar curvature. In particular, for a monopole class, the \(L^ 2\) norm of the harmonic representative is less than or equal to the \(L^ 2\)-norm of the scalar curvature over \(4\pi\). The main theorem of the paper characterizes when there is equality: when the metric is locally modeled in the product of the hyperbolic plane and \(\mathbb R\). In particular such a manifold is Seifert fibered with trivial rational Euler number.