Stability of Sasaki-Extremal Metrics under Complex Deformations

We consider the stability of Sasaki-extremal metrics under deformations of the transversal complex structure of the foliation F-xi induced by the Reeb vector field xi. Let g be a Sasaki-extremal metric on M, G a compact connected subgroup of the automorphism group of the Sasaki structure, and suppose the reduced scalar curvature satisfies s(g)(G) = 0. And consider a G-equivariant deformation (F-xi, (J) over bar (t))(t is an element of B) of the transversely holomorphic foliation preserving F-xi as a smooth foliation. Provided the Futaki invariant relative to G of g is nondegenerate, the existence of Sasaki-extremal metrics is preserved under small variations of t is an element of B and of the Reeb vector xi is an element of z in the center of g. If G = T subset of Aut(g, xi) is a maximal torus, the nondegeneracy of the Futaki invariant is automatic. So such deformations provide the easiest examples. When the initial metric g is Sasaki-Einstein and G = T subset of Aut(g, xi) is a maximal torus a slice of the above family of Sasaki-extremal metrics is Sasaki-Einstein. Thus, for each t is an element of B, there is a xi(t) is an element of xi(z) so that the Sasaki-extremal metric with Reeb vector field xi(t) is Sasaki-Einstein. We apply this to deformations of toric 3-Sasaki 7-manifolds to obtain new families of Sasaki-Einstein metrics on these manifolds, which are deformations of 3-Sasaki metrics.