Partner Symmetries and Non-Invariant Solutions of Complex Monge-Ampere Equations
Partner symmetries of the elliptic and hyperbolic complex Monge-Ampere equations (CMA and HCMA) provide non-invariant solutions of these equations. By using these solutions as metric potentials, we construct four-dimensional Ricci-.at metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors. Forthe elliptic CMA some solutions describe gravitational instan-tons. The most important gravitational instanton is the Kummer surface K3, an explicit construction of which is still an unsolved challenging problem. The metric on K3 should have no Killing vectors and hence the correspond-ing solution of CMA should have no symmetries, i.e. be a non-invariant solution.
April 8, 2009, 16:15, FENS 2019