Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

TL;DR

This paper constructs explicit anti-self-dual Riemannian metrics with no symmetries, expressed through a potential function, and suggests they can be realized on K3 surfaces or similar compact manifolds.

Contribution

It provides explicit examples of anti-self-dual metrics without Killing vectors, expressed via a potential function with exponential solutions, potentially on K3 surfaces.

Findings

Metrics are homogenous functions of degree zero in a potential.

Potential solutions are sums of exponential functions.

Metrics may be realized on K3 surfaces or their universal covers.

Abstract

Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on $K3$, or on surfaces whose universal covering is $K3$.