A homogeneous Gibbons–Hawking ansatz and Blaschke products

A homogeneous Gibbons–Hawking ansatz is described, leading to 4-dimensional hyperkähler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkähler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on 3-dimensional contact manifolds that induce complete Carnot–Carathéodory distances.