General heavenly equation governs anti-self-dual gravity

We show that the general heavenly equation, suggested recently by Doubrov and Ferapontov \cite{fer}, governs anti-self-dual (ASD) gravity. We derive ASD Ricci-flat vacuum metric governed by the general heavenly equation, null tetrad and basis of 1-forms for this metric. We present algebraic exact solutions of the general heavenly equation as a set of zeros of homogeneous polynomials in independent and dependent variables. A real solution is obtained for the case of neutral signature.

💡 Research Summary

The paper establishes that the “general heavenly equation” (GHE), recently introduced by Doubrov and Ferapontov, serves as a universal governing equation for anti‑self‑dual (ASD) Ricci‑flat vacuum gravity. The authors begin by recalling the classical heavenly equations of Plebański, which generate ASD metrics in four dimensions, and then present the GHE in its most symmetric form: a second‑order nonlinear partial differential equation involving four independent variables (z^{1},z^{2},z^{3},z^{4}) and a single dependent function (u(z^{i})). The equation can be written as a linear combination of products of mixed second derivatives, \