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 χ. When χ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.

On the J-flow in Sasakian manifolds / Zedda, Michela; Vezzoni, Luigi. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 195:(2016), pp. 757-774. [10.1007/s10231-015-0488-9]

On the J-flow in Sasakian manifolds

ZEDDA, MICHELA;
2016

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 χ. When χ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.
On the J-flow in Sasakian manifolds / Zedda, Michela; Vezzoni, Luigi. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 195:(2016), pp. 757-774. [10.1007/s10231-015-0488-9]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2838545
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 0
social impact