  A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms,
all defining the same volume form. In the present paper we completely determine
the moduli of taut contact spheres on compact left-quotients of SU(2) (the only
closed manifolds admitting such structures). We also show that the moduli space
of taut contact spheres embeds into the moduli space of taut contact circles.
  This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in
hyperkahler geometry. The classification of taut contact spheres on closed
3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but
the local Riemannian geometry of contact spheres is much richer. We construct
two examples of taut contact spheres on open subsets of 3-space with nontrivial
local geometry; one from the Helmholtz equation on the 2-sphere, and one from
the Gibbons-Hawking ansatz. We address the Bernstein problem whether such
examples can give rise to complete metrics.
