Willmore-type variational problem for foliated hypersurfaces

We study new Willmore-type variational problem for a hypersurface M in Rⁿ⁺¹ equipped with an s-dimensional foliation F. Its general version is the Reilly-type functional WFn,s=∫M F(σF₁,…,σFₛ)\, dV, where σFᵢ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of F. The first and second variations of such functionals are calculated, conformal invariance of some of WFn,s is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation F is found and examples with s≤2 and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations.