Explicit self-similar solutions for a class of zero mean curvature equation and minimal surface equation

Abstract

In this paper, we consider the existence of self-similar solution for a class of zero mean curvature equations including the Born–Infeld equation, the membrane equation and maximal surface equation. By Calabi’s correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. Those models arise in string theory and geometric minimal surfaces theory. Moreover, 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 dimensional Euclidean space and gravitational instantons possessing two killing vectors.