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 $\mathscr F_\xi$ induced by the Reeb vector field ξ. 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 $(\mathscr F_\xi,\bar J_t)_{t\in\mathcal B}$ of the transversely holomorphic foliation preserving $\mathscr F_\xi$ 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 $t\in \mathcal B$ and of the Reeb vector $\xi \in \mathfrak z$ in the center of $\mathfrak g$. If G=T⊆Aut(g,ξ) 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⊆Aut(g,ξ) is a maximal torus a slice of the above family of Sasaki-extremal metrics is Sasaki–Einstein. Thus, for each $t\in \mathcal B$ there is a $\xi_t\in \mathfrak z$ so that the Sasaki-extremal metric with Reeb vector field ξt 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.