Proper \(r\)-harmonic submanifolds into ellipsoids and rotation hypersurfaces (Q1748239)

The \(r\)-harmonic maps generalize harmonic maps to a variational problem of higher order. An \(r\)-harmonic map is a mapping between manifolds which is a critical point of the \(r\)-energy functional \(E_r(\varphi):=\int|(d+d^*)^r\varphi|^2\,dx\). An \(r\)-harmonic submanifold is a submanifold of some ambient manifold such that its inclusion map is \(r\)-harmonic. If the ambient manifold is a Euclidean space, we should expect \(r\)-harmonic submanifolds to be 1-harmonic (i.e. minimal) by Chen's or Maeta's conjecture. In the current paper, the authors give simple and explicit examples of proper \(r\)-harmonic submanifolds of ellipsoids and paraboloids of revolution. Here proper means that they are not \(s\)-harmonic for any \(s<r\).