Journal: Communications in Contemporary Mathematics Year: 2020

Abstract:

Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and
$J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie
group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for
hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent
one) is defined as a $G$-invariant hypersurface $\mathcal{E} \subset J^k$. We
describe a general method for constructing such invariant PDEs for $k\geq 2$.
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 PDEs for hypersurfaces in the Euclidean space
$\mathbb{E}^{n+1 }$ and in the conformal space $\mathbb{S}^{n+1}$. Our method
works under some mild assumptions on the action of $G$, namely:
A1) the group $G$ must have an open orbit in $J^{k-1}$, and
A2) the stabilizer $H^{(k-1)}\subset G$ of the fibre $J^k\to J^{k-1}$ must
factorize via the group of translations of the fibre itself.