A general method to construct invariant PDEs on homogeneous manifolds

Let M = G/H be an (n + 1)-dimensional homogeneous manifold and Jk(n,M) =: Jk be the manifold of k-jets of hypersurfaces of M. The Lie group G acts naturally on each Jk. A G-invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G-invariant hypersurface E of Jk. We describe a general method for constructing such invariant partial differential equations for k>1. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup H(k-1) of the (k-1)-prolonged action of G. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space n+1 and in the conformal space n+1. Our method works under some mild assumptions on the action of G, namely: A1) the group G must have an open orbit in Jk-1, and A2) the stabilizer H(k-1) in G of the fiber Jk → Jk-1 must factorize via the group of translations of the fiber itself.