On the J-flow in Sasakian manifolds

Abstract

We study the space of Sasaki metrics on a compact manifold \(M\) by introducing an odd-dimensional analogue of the \(J\)-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold, there exists at most one normalised critical metric. The flow is a tool for texting the existence of such a metric. We show that some results proved by Chen (Commun. Anal. Geom. 12: 837–852, 2004) can be generalised to the Sasakian case. In particular, the Sasaki \(J\) -flow is a gradient flow which has always a long-time solution minimising the distance on the space of Sasakian potentials of a polarised Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form \(\chi \). When \(\chi \) has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.