Partner symmetries of the complex Monge–Amp re equation yield hyper-K hler metrics without continuous symmetries

Abstract

We extend the Mason-Newman Lax pair for the elliptic complex MongeAmpère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. They imply the determining equation for symmetries of the complex Monge-Ampère equation as their differential compatibility condition. We shall identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kähler potential. This directly leads to a Legendre transformation. Studying the integrability conditions of the Legendre-transformed system we arrive at a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Ampère equation.Using these solutions we obtained explicit Legendre-transformed hyperKähler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors. They satisfy the Einstein field equations with Euclidean signature.We give the detailed derivation of the solution announced earlier and present a new solution with an added parameter. We compare our method of partner symmetries for finding non-invariant solutions to that of Dunajski and Mason who use 'hidden' symmetries for the same purpose.

Multi-Hamiltonian structure of Plebanski's second heavenly equation

“…A shorter proof is obtained by inverting (11) to arrive at the symplectic structure of the second heavenly system.…”

Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

“…We have called such a pair of mutually related symmetry characteristics partner symmetries [3]. Formulas (3.4) can be presented in the form…”

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

“…It is easy to see that if ϕ satisfies (2.3), then ψ also satisfies (2.3) and so the potential ψ of a symmetry ϕ is itself a symmetry of CMA [6,8]. These ϕ and ψ are called partner symmetries.…”

“…The Legendre transformation (3.6) maps the equations (2.7) with the choice of symmetries (3.5), their complex conjugates and the transformed CMA (3.7) to the following system of five independent equations (for more details see [6])…”

“…Recently, we have developed method of partner symmetries that yields non-invariant solutions to the elliptic and hyperbolic CMA and to the second heavenly equation. Using them as metric potentials, we have obtained some heavenly metrics with no Killing vectors [6,7,8] (nonexistence of conformal Killing vectors was not proved).…”