Contact spheres and hyperkähler geometry (Q731336)

A differential \(1\)-form \(\alpha\) such that \(\alpha\wedge d\alpha\neq 0\) is called a contact form on a \(3\)-manifold \(M\). A triple of \(1\)-forms \((\alpha_1, \alpha_2,\alpha_3)\) on a \(3\)-manifold \(M\) such that any non-trivial linear combination with constant coefficients of these forms is a contact form is called a contact sphere. A contact sphere \((\alpha_1, \alpha_2,\alpha_3)\) on a \(3\)-manifold \(M\) is called taut if the contact form \(\lambda_1\alpha_1+\lambda_1\alpha_2+\lambda_3\alpha_3\) defines the same volume form for all \((\lambda_1,\lambda_2, \lambda_3)\in S^2\). In [Invent. Math. 121, No.1, 147-209 (1995; Zbl 1002.53501)], the authors gave a complete classification of the closed, orientable 3-manifolds that admit a taut contact circle or a taut contact sphere by showing that a closed \(3\)-manifold \(M\) admits a taut contact circle if and only if \(M\) is diffeomorphic to a quotient of the Lie group \({\mathcal G}\) under a discrete subgroup \(\Gamma\) acting by left multiplication, where \({\mathcal G}\) is one of the following: (i)\,\(S^3=\text{SU}(2)\), the universal cover of \(\text{SO}(3)\), (ii)\, \(\widetilde{\text{SL}}_2\), the universal cover of \,\(\text{PSL}_2\mathbb R\), (iii)\,\(\widetilde E_2\), or \(M\) admits a taut contact sphere if and only if it is diffeomorphic to a left-quotient of \(\text{SU}(2)\). In this paper, the authors completely determine the moduli of taut contact spheres on compact left-quotients of \(\text{SU}(2)\) and show that the moduli space of taut contact spheres is embedded into the moduli space of taut contact circles. The classification of taut contact spheres on closed \(3\)-manifolds includes the known classification of \(3\)-Sasakian \(3\)-manifolds. A taut contact sphere on a closed manifold \(M\) always gives rise to a flat hyper-Kähler metric on \(M\times\mathbb R\). Using a Monge-Ampere equation the authors construct a taut contact sphere on an open subset \(U\) of \(\mathbb R^3\) giving rise to a non-flat hyper-Kähler metric on \(U\times\mathbb R\). Finally, they discuss the Bernstein problem whether such examples can give rise to complete metrics.