Willmore-type variational problem for foliated hypersurfaces

Abstract

We study new Willmore-type variational problem for a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ equipped with an $s$-dimensional foliation $\mathcal{F}$. Its general version is the Reilly-type functional $W{F}_{n,s}={\int }_{M}F({\sigma }_1^{\mathcal{F}},\dots ,{\sigma }_s^{\mathcal{F}})\mathrm{d}V$, where ${\sigma }_i^{\mathcal{F}}$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\mathcal{F}$. The first and second variations of such functionals are calculated, conformal invariance of some of $W{F}_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\mathcal{F}$ is found and examples with $s\le 2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations.

14 pages, 1 figure