# 20190417 Gonzalo Pérez, Jesús The Gibbons-Hawking ansatz and Blaschke products

Title: The Gibbons-Hawking ansatz and Blaschke products

Location: 425Mathematics Building

Date: April 17, 2019,15:00--16:00

Abstract: A hyperk\"akler structure in real dimension 4 consists of a Riemann metric $\, g\,$ together with three holomorphic structures $\, J_1,J_2,J_3\,$ which are parallel with respect to~$\, g\,$ and satisfy the quaternionic identities $\, J_1J_2=J_3$, etc. These are all local properties, and they correspond to Monge-Ampere equations.

Hyperk\"ahler structures preserved by the flow of some vector field~$\, X\,$ can be constructed, via the Gibbons-Hawking ansatz, from a harmonic function in~$\,{\mathbb R}^3$.

Structures with a vector field $\, Y$, whose flow is holomorphic for the~$\, J_i\,$ and expanding for the metric, correspond to families of contact forms parametrized by the $\, 2$-sphere (called contact spheres). A natural global property for these structures is {\it slice completeness:} that the metric be complete in directions transverse to the expanding flow.

Structures having both types of flows are constructed, via our version of the Gibbons-Hawking ansatz, from holomorphic data given on a Riemann surface. Blaschke products are special meromorphic functions; with them we construct slice-complete hyperk\"ahler metrics, thus solving a local and a global problem at once.