On the J-flow in Sasakian manifolds

TL;DR

This paper introduces an analogue of the J-flow for Sasakian manifolds, establishing existence, uniqueness, and convergence results for critical metrics, extending Kähler flow techniques to the Sasakian setting.

Contribution

It develops a Sasakian version of the J-flow, proving existence, uniqueness, and convergence of critical metrics, and generalizes Chen's results to Sasakian geometry.

Findings

The Sasaki J-flow is a gradient flow with long-time solutions.

The flow minimizes an energy functional related to the transverse Kähler form.

Under nonnegative curvature, the flow converges to a critical Sasakian structure.

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 K\"ahler 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 K\"ahler form $\chi$. When $\chi$ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.