The Sasaki join and admissible Kähler constructions

Huang

Legendre

et al. 2016

Int Math Res Notices

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Abstract. We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki manifold has vanishing Sasaki-Futaki invariant. We then apply this result to show that under the right conditions on the Sasaki join manifolds of [7] a polarized Sasaki manifold is K-semistable if only if it has constant scalar curvature.

Section: The Einstein-hilbert Functional On the W-cone Of A Sasaki Joinmentioning

confidence: 99%

“…w . These manifolds have recently been the object of study by the first and last authors [6,7,8,9]. They include an infinite number of homotopy types as well as an infinite number of contact structures of Sasaki type occurring on the same manifold [6,10].…”

Section: The Einstein-hilbert Functional On the W-cone Of A Sasaki Joinmentioning

confidence: 99%

The Einstein–Hilbert Functional and the Sasaki–Futaki Invariant

Huang

Legendre

et al. 2016

Int Math Res Notices

Self Cite

On positivity in Sasaki geometry

Tønnesen-Friedman

2019

Geom Dedicata

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It is well known that if the dimension of the Sasaki cone t + is greater than one, then all Sasakian structures in t + are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that dim t + > 1 there are three possibilities, either all elements of t + are positive, all are indefinite, or both positive and indefinite Sasakian structures occur in t + . We illustrate by examples how the type can change as we move in t + . If there exists a Sasakian structure in t + whose total transverse scalar curvature is non-positive, then all elements of t + are indefinite. Furthermore, we prove that if the first Chern class is a torsion class or represented by a positive definite (1, 1) form, then all elements of t + are positive.

The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature

Tønnesen-Friedman

2015

J Geom Anal

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We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki-Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension 2n + 1 and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds M l1,l2,w of dimension 2n + 3 which depend on four integer parameters l 1 , l 2 , w 1 , w 2 . Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two dimensional subcone of Sasaki Ricci solitons in the Sasaki cone, and a unique Sasaki-Einstein metric in each of the two dimensional sub cones.2000 Mathematics Subject Classification. Primary: 53D42; Secondary: 53C25.