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On the J-flow in Sasakian manifolds

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 in [7] 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 polarized Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form $χ$. When $χ$ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.