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

A. A. Malykh·Y. Nutku·M. B. Sheftel
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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 $K3$.

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A. A. Malykh, Y. Nutku, M. B. Sheftel. 2003-07-31. Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?. https://doi.org/10.1088/0264-9381%2F20%2F22%2Fl01

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