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 ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(σ^{\cal F}_1,\ldots,σ^{\cal F}_s)\,{\rm d}V$, where $σ^{\cal F}_i$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\cal F$. The first and second variations of such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations.

Figures (1)

• Figure 1: Graphs of $f(\rho)$ for $f(\frac{2}{5})=1$, $f'(\frac{2}{5})=\frac{2}{5}$, $n=2$ and $p=2,3,\ldots,8$.

Theorems & Definitions (31)

• Remark 1
• Lemma 1: see GTT-2019 for $n=2$
• Lemma 2
• proof
• Lemma 3
• Remark 2
• Lemma 4
• proof
• Lemma 5
• Theorem 1