Projectively and affinely invariant PDEs on hypersurfaces (Q6622899)

Let us say that a hypersurface equation on a manifold \(M\) is a partial differential equation whose solutions are hypersurfaces in \(M\), so that in any local coordinates\N\[\N(u,x^1,\dots,x^n)\N\]\Non \(M\), the hypersurface equation is a system of partial differential equations for \(u\) as a function of \(x^1,\dots,x^n\). In previous work, the authors have developed a method to compute all \(G\)-invariant hypersurface equations on any homogeneous space \(M=G/H\). In this paper, they consider the special case of \(M\) being affine space or projective space, and \(G\) the affine group or projective linear group. The striking result: to each of these invariant hypersurface equations is associated a hypersurface in the space of trace free cubic forms on a finite dimensional pseudo-Euclidean vector space, and each invariant hypersurface equation arises in this way uniquely up to isomorphism.