Differential invariants and group foliation for the complex Monge-Ampère equation

Abstract

We apply the method of group foliation to the complex Monge-Ampere equation ({CMA}2) with the goal of establishing a regular framework for finding its non-invariant solutions. We employ the infinite symmetry subgroup of the equation, the group of unimodular biholomorphisms, to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group. Accordingly, {CMA}2 is split into an automorphic system and a resolvent system which we derive in this paper. This is an intricate system and here we make no attempt to solve it in order to obtain non-invariant solutions. We obtain all differential invariants up to third order for the group of unimodular biholomorphisms and, in particular, all the basis differential invariants. We construct the operators of invariant differentiation from which all higher differential invariants can be obtained. Consequently, we are able to write down all independent partial differential equations with one real unknown and two complex independent variables which keep the same infinite symmetry subgroup as {CMA}2. We prove explicitly that applying operators of invariant differentiation to third-order invariants we obtain all fourth-order invariants. At this level we have all the information which is necessary and sufficient for group foliation. We propose a new approach in the method of group foliation which is based on the commutator algebra of operators of invariant differentiation. The resolving equations are obtained by applying this algebra to differential invariants with the status of independent variables. Furthermore, this algebra together with Jacobi identities provides the commutator representation of the resolvent system. This proves to be the simplest and most natural way of arriving at the resolving equations.