Explicit singular minimal surface solutions for gravitational instantons

TL;DR

This paper introduces explicit singular minimal surface solutions that generate gravitational instantons with two symmetries, expanding the set of known solutions through a novel geometric correspondence.

Contribution

It presents new explicit solutions for minimal surfaces and their application to constructing gravitational instantons with specific symmetry properties.

Findings

Derived a family of explicit singular minimal surface solutions.

Established a correspondence between minimal surfaces and gravitational instantons.

Produced new gravitational instanton metrics from these solutions.

Abstract

We construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimesional Euclidean space and gravitational instantons possessing two killing vectors. By Calabi's correspondence, we derive a family of explicit maximal surface solution for spacelike surface with zero mean curvature equation.