# Willmore-type variational problem for foliated hypersurfaces

Vladimir Rovenski 111Department of Mathematics, University of Haifa, Mount
Carmel, 3498838 Haifa, Israel
e-mail<EMAIL_ADDRESS>
###### 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 $WF_{n,s}=\int_{M}F(\sigma^{\cal
F}_{1},\ldots,\sigma^{\cal F}_{s})\,{\rm d}V$, where $\sigma^{\cal F}_{i}$ 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 $s\leq 2$ and $s=n$ are considered. Critical hypersurfaces of
revolution are found, and it is shown that they are a local minimum for
special variations.

Keywords: hypersurface, Euclidean space, foliation, second fundamental form,
mean curvature, Willmore’s type functional, Euler-Lagrange equations,
conformal invariant

Mathematics Subject Classifications (2010) 53C12; 53C15; 53C42

##  1 Introduction

Many authors, e.g., [3, 4, 8, 10, 15, 16], were looking for an immersion
$\phi:M^{n}\to\bar{M}^{n+1}$ of a smooth manifold $M^{n}\ (n\geq 2)$ into a
Riemannian manifold $(\bar{M},\bar{g})$, in particular, Euclidean space
$\mathbb{R}^{n+1}$, which is a critical point of the following functionals for
compactly supported variations of $\phi$:

$W_{n,p}=\int_{M}H^{p}\,{\rm d}V,\qquad J_{n,p}=\int_{M}\|{h}\,\|^{p}\,{\rm
d}V.$ (1)

Here, ${h}$ is the scalar second fundamental form of $\phi(M)$,
${H}=\frac{1}{n}\operatorname{trace}_{g}{h}$ is the mean curvature, ${\rm d}V$
is the volume form of the induced metric $g$ on $M$, and $p>0$. These
functionals measure how much $\phi(M)$ differs from a minimal hypersurface
($H=0$) or a totally geodesic hypersurface ($h=0$). The actions (1) are a
particular case of functionals $WF_{n}=\int_{M}F(H)\,{\rm d}V$ and
$JF_{n}=\int_{M}F(\|h\|)\,{\rm d}V$, where $F$ is a $C^{3}$-regular function
of one variable, e.g., [2, 6, 7, 9]. For a closed oriented smooth hypersurface
$M^{n}$ in ${\mathbb{R}}^{n+1}$, we get $W_{n,n}\geq C_{n}$, where
$C_{n}=\frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}$ is the area of the unit
$n$-sphere; the equality $W_{n,n}=C_{n}$ holds if and only if $M^{n}$ is
embedded as a hypersphere, see [3].

Variational problems for (1) were first posed by Thomas Willmore in [15] for
$W_{2,2}$, which belongs to conformal geometry. The Euler-Lagrange equation
for $W_{2,2}$ is the well known elliptic PDE

$\displaystyle\Delta H+2\,H(H^{2}-K)=0,$ (2)

where $\Delta$ is the Laplacian and $K$ – the gaussian curvature of $M^{2}$.
Solutions of (2) are called Willmore surfaces. An important class of Willmore
surfaces in $\mathbb{R}^{3}$ arise as the stereographic projection of minimal
surfaces in the 3-sphere. By Lawson’s theorem, any compact orientable surface
can be minimally embedded in the 3-sphere. For a closed orientable surface $M$
in $\mathbb{R}^{3}$, the inequality $W_{2,2}\geq C_{2}=4\pi$ holds with the
equality for round spheres. If $M^{2}$ is a torus in $\mathbb{R}^{3}$, then,
according the Willmore conjecture proven by F.C. Marques and A. Neves in [10],
we have $W_{2,2}\geq 2\,\pi^{2}$; the equality holds if and only if the
generating curve is a circle and the ratio of radii is $\frac{1}{\sqrt{2}}$.
Willmore surfaces have applications in biophysics, materials science,
architecture, etc., e.g., [14].

R. Reilly [12] and some mathematicians studied variations of more general
functionals than (1):

$\displaystyle WF_{n}=\int_{M}F(\sigma_{1},\ldots,\sigma_{n})\,{\rm d}V,\qquad
JF_{n}=\int_{M}F(\tau_{1},\ldots,\tau_{n})\,{\rm d}V,$ (3)

where $F\in C^{3}(\mathbb{R}^{n})$. The elementary symmetric functions
$\sigma_{r}=\sum_{i_{1}<\ldots<i_{r}}k_{i_{1}}\ldots k_{i_{r}}\ (0\leq r\leq
n)$ of the principal curvatures $k_{i}$ satisfy the equality
$\sum\nolimits_{\,r=0}^{\,n}\sigma_{r}t^{r}=\det({\rm id}_{\,TM}+tA)$, where
$A$ is the Weingarten operator, i.e., $g(AX,Y)=h(X,Y)$. The power sums of the
principal curvatures, $\tau_{i}=k_{1}^{i}+\ldots+k_{n}^{i}={\rm
trace}\,A^{i}$, can be expressed as polynomials of $\sigma_{r}$ using the
Newton formulas, e.g., [13]. For example, $\sigma_{0}=1$,
$\tau_{1}=\sigma_{1}=nH$, $\sigma_{n}=\det A$, $\tau_{2}=\|h\|^{2}$ and
$2\,\sigma_{2}=\tau_{1}^{2}-\tau_{2}$. The $r$-th ($r\leq n$) order Willmore
functional, introduced by Z. Guo in [6],

$\displaystyle W^{\rm conf}_{n,r}=\int_{M}Q_{\,r}^{\,n/r}\,{\rm d}V,\quad
Q_{r}=\sum\nolimits_{j=0}^{r}(-1)^{j+1}C^{j}_{r}S_{1}^{r-j}S_{j},$ (4)

is a special case of (3), invariant under conformal group of
$(\bar{M},\bar{g})$ and vanishing on totally umbilical hypersurfaces. Here,
$S_{r}=\sigma_{r}/C^{r}_{n}$ (where $C^{r}_{n}=\frac{n!}{r!(n-r)!}$ is a
binomial coefficient) is the $r$-th mean curvature function of a hypersurface.
In particular,
$Q_{2}=S_{1}^{2}-S_{2}=\frac{1}{n^{2}(n-1)}\,((n-1)\sigma_{1}^{2}-2\,n\,\sigma_{2})$.
Examples of hypersurfaces in $\mathbb{R}^{n+1}$ that are critical for (4) are
given in [7, 9].

An interesting problem is the generalization of the Willmore functional to
submanifolds with additional structures, such as foliations or almost
products. Let $M^{n}\ (n\geq 2)$ equipped with an $s$-dimensional $(1\leq
s\leq n)$ foliation ${\cal F}$ be immersed into a Riemannian manifold
$(\bar{M},\bar{g})$. Let $h_{\cal F}$ be the restriction of the second
fundamental form of $M$ on the leaves of ${\cal F}$. Denote by $\tau^{\cal
F}_{i}\ (1\leq i\leq s)$ the power sums, $\sigma^{\cal F}_{r}\ (1\leq r\leq
s)$ elementary symmetric functions of the eigenvalues $k^{\cal
F}_{1}\leq\ldots\leq k^{\cal F}_{s}$ of $h_{\cal F}$, and set $S^{\cal
F}_{r}=\sigma^{\cal F}_{r}/C^{r}_{s}$. We have $\tau^{\cal F}_{1}=\sigma^{\cal
F}_{1}=sH_{\cal F}=\operatorname{trace}_{g}h_{\cal F}$, $\tau^{\cal
F}_{2}=\|h_{\cal F}\|^{2}$, $(\tau^{\cal F}_{1})^{2}-\tau^{\cal
F}_{2}=2\,\sigma^{\cal F}_{2}$, etc. For foliation theory we refer to [5], the
extrinsic geometry of foliations was developed in [13]. We study Reilly-type
functionals for compactly supported variations of $(M^{n},{\cal F})$ immersed
in $\mathbb{R}^{n+1}$:

$\displaystyle WF_{n,s}=\int_{M}F(\sigma^{\cal F}_{1},\ldots,\sigma^{\cal
F}_{s})\,{\rm d}V,\qquad JF_{n,s}=\int_{M}F(\tau^{\cal
F}_{1},\ldots,\tau^{\cal F}_{s})\,{\rm d}V,$ (5)

which for $s=n$ reduce to (3). For $F=(\sigma_{1}^{\cal F}/s)^{p}$ and
$F=(\tau_{2}^{\cal F})^{p/2}$, the actions (5) read as

$W_{n,p,s}=\int_{M}{H}_{\cal F}^{p}\,{\rm d}V,\qquad
J_{n,p,s}=\int_{M}\|h_{\cal F}\|^{p}\,{\rm d}V,$ (6)

which reduce to (1) for $s=n$.

######  Remark 1.

A foliated hypersurface in $\mathbb{R}^{n+1}$, whose leaves $\\{L\\}$ are
minimal submanifolds in $\mathbb{R}^{n+1}$ is an example of a minimizer for
(6)1 with even $p$. A foliated hypersurface in $\mathbb{R}^{n+1}$ with an
asymptotic distribution $T{\cal F}$ (in particular, a ruled hypersurface) is a
minimizer for (6)2. It is interesting to find critical hypersurfaces for
actions (6) with ${H}_{\cal F}\neq 0$ or $h_{\cal F}\neq 0$ on an open dense
set of $M$.

The following special case of (5) is invariant under conformal group of
$(\bar{M},\bar{g})$, see Theorem 1:

$\displaystyle W^{\rm conf}_{n,s,r}=\int_{M}(Q^{\cal F}_{r})^{n/r}\,{\rm
d}V,\quad Q^{\cal F}_{r}=\sum\nolimits_{j=0}^{r}(-1)^{j+1}C^{j}_{r}(S^{\cal
F}_{1})^{r-j}S^{\cal F}_{j},\quad r\leq s,$ (7)

and reduces to (4) for $s=n$. Note that $Q^{\cal F}_{2}=(S^{\cal
F}_{1})^{2}-S^{\cal F}_{2}=\frac{1}{s^{2}(s-1)}\,((s-1)(\sigma^{\cal
F}_{1})^{2}-2s\,\sigma^{\cal F}_{2})$ and $Q^{\cal F}_{r}=0$ if $k^{\cal
F}_{1}=\ldots=k^{\cal F}_{s}$, e.g., for hypersurfaces of revolution in
$\mathbb{R}^{n+1}$ foliated by parallels.

We hope that foliated hypersurfaces, which are local minima for (5), will be
useful for natural sciences and technology related to layered (laminated) non-
isotropic materials.

The paper is organized as follows. Section 2 contains some lemmas (proved in
Sect. 5) that help us calculate variations of Reilly-type functionals on
foliated hypersurfaces in $\mathbb{R}^{n+1}$. In Section 3, conformal
invariance of (7) is shown, the first variations of the functionals (5)–(7)
are found, and the corresponding Euler-Lagrange equations for the case of
transversally harmonic (for example, Riemannian) foliation are obtained. Then
the second variations on critical hypersurfaces of some Willmore type
functionals are calculated. In Section 4, applications to hypersurfaces with
low-dimensional foliations are given, the critical hypersurfaces of revolution
for the actions (6) are presented, and it is shown that they are a local
minimum for special variations.

##  2 Auxiliary results

Let ${\bf r}:M^{n}\to\mathbb{R}^{n+1}$ be an immersion of a manifold $M$ into
$\mathbb{R}^{n+1}$ with Euclidean metric $\bar{g}$ and the Levi-Civita
connection $\bar{\nabla}$. We identify $M$ with its image ${\bf r}(M)$ and
restrict our calculations to a relatively compact neighborhood $U\subset M$
with induced metric $g=\langle\cdot\,,\cdot\rangle$ and normal coordinates
$(x^{1},\ldots,x^{n})$ centered at a point $x\in M$. Thus,
$g_{ij}=\delta_{ij}$ (the Kronecker symbol) and $\Gamma^{k}_{ij}=0$ at $x$.
Differentiation of a function $f$ (or a tensor) with respect to the variable
$x^{i}$ will be denoted by $f_{i}$.

Let $\partial_{i}=\partial/\partial x^{i}$ be the coordinate vector fields on
$U$. So, the vectors ${\bf r}_{i}=\bar{\nabla}_{\partial_{i}}{\bf r}$ form a
local coordinate basis for the tangent bundle $TM$ along $U$, and we get
$g=g_{ij}\,dx^{i}\,dx^{j}$, where $g_{ij}=\bar{g}({\bf r}_{i},{\bf
r}_{j})=\langle{\bf r}_{i},{\bf r}_{j}\rangle$ and Einstein summation rule is
used. Let ${\bf N}$ be a unit normal vector field to $M$ on $U$. The vectors
${\bf N}_{i}=\bar{\nabla}_{\partial_{i}}{\bf N}$ belong to the tangent space
$T_{x}M$, i.e., $\langle{\bf N}_{i},{\bf N}\rangle=0$.

Let ${h}$ be the scalar second fundamental form of $M$ with respect to unit
normal ${\bf N}$, $A=-\bar{\nabla}\,{\bf N}$ the Weingarten operator and
${H}=\frac{1}{n}\operatorname{trace}_{g}{h}$ the mean curvature. Denote by
$h^{j}$ the symmetric tensor dual to $A^{j}$, i.e., $h^{j}(X,Y)=\langle
A^{j}X,Y\rangle$. For example,
${h}^{2}=g^{kl}h_{li}h_{kj}\,dx^{i}dx^{j}={h}^{k}_{i}h_{kj}\,dx^{i}dx^{j}$.

Consider a one-parameter family of hypersurfaces ${\bf r}_{t}={\bf
r}+t\,u\,{\bf N}\ (|t|<1)$. We get a variation $\delta\,{\bf r}=u\,{\bf N}$,
where $\delta=(d/dt)|_{\,t=0}$ is the variational derivative operator, and
$u:U\to\mathbb{R}$ is a smooth function supported on relatively compact
neighborhood $U$. Obviously, $(\delta\,{\bf r})_{i}=u_{i}\,{\bf N}+u\,{\bf
N}_{i}$. The Hessian of a function $u$ is a (0,2)-tensor $({\rm
Hess}_{\,u})(X,Y)=X(Y(u))-(\nabla_{X}\,Y)u=(u_{ij}-\Gamma^{k}_{ij}u_{k})dx^{i}dx^{j}$,
see [11, pp. 48, 69]. The Laplacian is $\Delta
u=\operatorname{trace}_{g}\,{\rm Hess}_{\,u}=g^{ij}u_{ij}$. Note that $\langle
g,{\rm Hess}_{\,u}\rangle=\Delta\,u$. The divergence of a vector field
$X=X^{i}\partial_{i}$ on $M$ is ${\rm div}\,X=\nabla_{i}X^{i}$.

######  Lemma 1 (see [8] for $n=2$).

The following evolution equations are true:

$\displaystyle\delta\,g_{ij}=-2\,u\,h_{ij},$ (8)
$\displaystyle\delta\,g^{ij}=2\,u\,h^{ij},$ (9)
$\displaystyle\delta\,h_{ij}=u_{ij}-u{h}^{l}_{i}\,h_{jl}\ \Leftrightarrow\
\delta\,h={\rm Hess}_{\,u}-u\,h^{2},$ (10)
$\displaystyle\delta\,\|h\|^{2}=2\,\langle h,\ u{h}^{2}+{\rm
Hess}_{\,u}\rangle,$ (11) $\displaystyle\delta(nH)=\Delta\,u+u\,\|h\|^{2},$
(12) $\displaystyle\delta\,{\rm d}V=-n\,uH\,{\rm d}V.$ (13)

We carry out further calculations for a foliated hypersurface $(M,{\cal F})$
and a foliated neighborhood $U\subset M$ with normal coordinates
$(x^{1},\ldots,x^{n})$ adapted to ${\cal F}$, i.e., $(x^{1},\ldots,x^{s})$ are
variables along the leaves, see [5]. Let $\nabla^{\cal F}:TM\times T{\cal
F}\to T{\cal F}$ be the induced Levi-Civita connection on the subbundle
$T{\cal F}\subset TM$ and on the leaves of ${\cal F}$. The leafwise Laplacian
on functions is $\Delta_{\,\cal F}=\operatorname{trace}_{g}\,{\rm Hess}^{\cal
F}={\rm div}_{\cal F}\circ\nabla^{\,\cal F}$, where ${\rm Hess}^{\cal F}$ is
the Hessian on the leaves of $\cal F$. Let $P:TM\to T{\cal F}$ be the
orthoprojector, thus $P^{2}=P$ and $P$ is self-adjoint. For $h_{\cal F}$ and
its dual self-adjoint operator $A_{\cal F}$ we can write $h_{\cal
F}(X,Y)=h(PX,PY)\ (X,Y\in\mathfrak{X}_{M})$ and $A_{\cal F}=PAP$. Let $h_{\cal
F}^{j}$ be the symmetric tensor dual to $A_{\cal F}^{j}$, i.e., $h_{\cal
F}^{j}(X,Y)=\langle A_{\cal F}^{j}X,Y\rangle$. The symmetric tensor $h_{\rm
mix}$ is given by

$h_{\rm mix}(X,Y)=\frac{1}{2}\,\big{(}h(PX,Y)+h(X,PY)\big{)}-h(PX,PY),\quad
X,Y\in\mathfrak{X}_{M}.$

We have $\langle h_{\cal F},h_{\rm mix}\rangle=0$. The equality $h_{\rm
mix}=0$ means that $PA=AP$, i.e., $T{\cal F}$ is an invariant subbundle for
$A$. Let $h_{\cal F^{\perp}}$ be the restriction of $h$ on the normal
distribution to ${\cal F}$ in $M$, then $h_{\cal
F^{\perp}}^{2}=g^{\gamma\epsilon}h_{\alpha\gamma}h_{\alpha\epsilon}dx^{\alpha}dx^{\beta}$,
where $s<\alpha,\beta,\gamma,\epsilon\leq n$. Define symmetric tensors $h_{\rm
mix}^{2}=g^{\alpha\beta}h_{\alpha i}h_{\beta j}dx^{i}dx^{j}+g^{ij}h_{\alpha
i}h_{\beta j}\,dx^{\alpha}dx^{\beta}$ and ${\rm Hess}^{\,\rm
mix}_{\,u}=g^{ij}g^{\alpha\beta}u_{i\alpha}\,dx^{j}dx^{\beta}$, where $1\leq
i,j\leq s$ and $s<\alpha,\beta,\gamma,\epsilon\leq n$. Let $A_{\,\rm mix}$ be
the (1,1)-tensor dual to $h_{\rm mix}$, then $A^{2}_{\,\rm mix}$ is dual to
$h^{2}_{\,\rm mix}$.

The Newton transformations $T_{r}(A_{\cal F})$ of $A_{\cal F}$ are defined
inductively or explicitly by, e.g. [13],

$\displaystyle T_{0}(A_{\cal F})$ $\displaystyle=$ $\displaystyle{\rm
id}_{\,T\cal F},\quad T_{r}(A_{\cal F})=\sigma^{\cal F}_{r}\,{\rm id}_{\,T\cal
F}-A_{\cal F}\,T_{r-1}(A_{\cal F})\quad(0<r\leq s),$ $\displaystyle
T_{r}(A_{\cal F})$ $\displaystyle=$
$\displaystyle\sum\nolimits_{j=0}^{r}(-1)^{j}\sigma^{\cal F}_{r-j}\,A_{\cal
F}^{j}=\sigma^{\cal F}_{r}\,{\rm id}_{\,T\cal F}-\sigma^{\cal
F}_{r-1}\,A_{\cal F}+\ldots+(-1)^{r}A_{\cal F}^{\,r}.$

For example, $T_{1}(A_{\cal F})=\sigma^{\cal F}_{1}\,{\rm id}_{\,T\cal
F}-A_{\cal F}$ and $T_{s}(A_{\cal F})=0$ and the following equalities are
true:

$\displaystyle\operatorname{trace}T_{r}(A_{\cal F})=(s-r)\,\sigma^{\cal
F}_{r},\quad\operatorname{trace}(A_{\cal F}\,T_{r}(A_{\cal
F}))=(r+1)\,\sigma^{\cal F}_{r+1},$ $\displaystyle\operatorname{trace}(A_{\cal
F}^{2}\,T_{r}(A_{\cal F}))=\sigma^{\cal F}_{1}\sigma^{\cal
F}_{r+1}-(r+2)\,\sigma^{\cal F}_{r+2}.$ (14)

The “musical” isomorphism $\sharp:T^{*}M\to TM$ is used for tensors, e.g.,
$h^{\sharp}=A$, and for $(0,2)$-tensors $B$ and $C$ we have $\langle
B,C\rangle=\operatorname{trace}(B^{\sharp}C^{\sharp})=\langle
B^{\sharp},C^{\sharp}\rangle$.

######  Lemma 2.

The variations of $\tau^{\cal F}_{i}$ and $\sigma^{\cal F}_{r}$ are the
following:

$\displaystyle\frac{1}{i}\,\delta\,\tau^{\cal F}_{i}=\langle h_{\cal
F}^{i-1},{\rm Hess}^{\cal F}_{u}\rangle+u\,(\tau^{\cal F}_{i+1}+\langle
h_{\cal F}^{i-1},h^{2}_{\,\rm mix}\rangle),$
$\displaystyle\delta\,\sigma^{\cal F}_{r}=\langle T_{r-1}(A_{\cal F}),{\rm
Hess}^{\cal F\sharp}_{u}\rangle+u\big{(}\sigma^{\cal F}_{1}\sigma^{\cal
F}_{r-1}-(r+1)\,\sigma^{\cal F}_{r+1}+\langle T_{r-1}(A_{\cal F}),\,A_{\,\rm
mix}^{2}\rangle\big{)}.$ (15)

###### Proof.

By (10), we obtain $\delta A_{\cal F}={\rm Hess}^{\cal F\sharp}_{u}+u(A_{\cal
F}^{2}+A_{\,\rm mix}^{2})$. Using this, (2) and the the following variations
of $\tau^{\cal F}_{i}$ and $\sigma^{\cal F}_{r}$, see [13, Sect. 1.5.3]:

$\displaystyle\delta\,\tau^{\cal F}_{i}=i\operatorname{trace}(A_{\cal
F}^{i-1}\delta A_{\cal F}),\quad\delta\,\sigma^{\cal
F}_{r}=\operatorname{trace}(T_{r-1}(A_{\cal F})\,\delta A_{\cal F}),$

we get (2). ∎

######  Lemma 3.

The following evolution equations are true:

$\displaystyle\delta\,(s\,{H}_{\cal F})=\Delta_{\,\cal F}\,u+u\,(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2}),$ (16) $\displaystyle\delta\,\|h_{\cal
F}\|^{2}=2\,\langle h_{\cal F},\ u\,(h_{\cal F}^{2}+h_{\rm mix}^{2})+{\rm
Hess}^{\,\cal F}_{\,u}\rangle,$ (17) $\displaystyle\delta\,\|h_{\rm
mix}\|^{2}=u\,\langle h_{\cal F}+h_{\cal F^{\perp}},h_{\rm
mix}^{2}\rangle+2\,\langle h_{\rm mix},{\rm Hess}^{\,\rm mix}_{\,u}\rangle.$
(18)

For any smooth function $f:M\times\mathbb{R}\to\mathbb{R}$ the following
evolution is true:

$\displaystyle\delta\,(\Delta_{\,\cal F}f)=\Delta_{\,\cal
F}\dot{f}+2\,u\langle h_{\cal F},{\rm Hess}^{\,\cal
F}_{f}\rangle{+}su\langle\nabla^{\cal F}H_{\cal F},\nabla^{\cal
F}f\rangle{+}2\,h(\nabla^{\,\cal F}u,\nabla^{\cal F}f){-}sH_{\cal
F}\langle\nabla^{\cal F}u,\nabla^{\cal F}f\rangle.$ (19)

######  Remark 2.

The equations (16)–(17) can be deduced from Lemma 2, but we will prove them in
Section 5. To find the second variation of the functionals (5) we need also
the variation $\delta\,\langle h,{\rm Hess}^{\,\cal F}_{\,u}\rangle$, but we
omit this calculation and consider the second variation of the functionals (6)
only.

The following property helps to find the Euler-Lagrange equations using first
variation of (3)–(6):

$\displaystyle({\rm div}\,P)\circ P=0.$ (20)

Here, $({\rm
div}\,P)X=\sum\nolimits_{\,i=1}^{n}\langle(\nabla_{e_{i}}P)X,e_{i}\rangle$,
where $e_{1},\ldots,e_{n}$ is a local orthonormal basis on $M$. Note that
Riemannian foliations (and the leaves of twisted products, e.g., [13]) satisfy
(20).

######  Lemma 4.

A foliated Riemannian manifold $(M,g,{\cal F})$ satisfies (20) if and only if
${\cal F}$ is transversally harmonic, i.e., the normal distribution has zero
mean curvature.

###### Proof.

Using a local orthonormal frame on $M$ such that $e_{i}\in T{\cal F}\ (1\leq
i\leq s)$, we calculate:

$\displaystyle({\rm
div}\,P)(PX)=\sum\nolimits_{\,i=1}^{n}\langle(\nabla_{e_{i}}P)(PX),e_{i}\rangle=\sum\nolimits_{\,i=1}^{n}\big{\\{}\langle\nabla_{e_{i}}(P^{2}X),e_{i}\rangle-\langle
P\nabla_{e_{i}}(PX),e_{i}\rangle\big{\\}}$
$\displaystyle=\sum\nolimits_{\,i=1}^{n}\big{\\{}\langle\nabla_{e_{i}}(PX),e_{i}\rangle-\langle\nabla_{e_{i}}(PX),Pe_{i}\rangle\big{\\}}=\sum\nolimits_{\,i>s}\langle\nabla_{e_{i}}(PX),e_{i}\rangle=-\langle
X,(n-s)H^{\perp}\rangle,$

where $(n-s)H^{\perp}=P\sum_{\,i>s}\nabla_{e_{i}}e_{i}$ is the mean curvature
vector of $(T{\cal F})^{\perp}$ and $X\in\mathfrak{X}_{M}$. ∎

For any 2-tensor $B$ on $M$ define the adjoint of the covariant derivative
$\nabla^{*}B=-\sum_{\,i}(\nabla_{i}\,B)(e_{i},\cdot)$, see [11, p. 59]. We
have the formula $\int_{M}\langle B,\nabla B^{\prime}\rangle\,{\rm
d}V=\int_{M}\langle\nabla^{*}B,B^{\prime}\rangle\,{\rm d}V$, see [11, p. 60];
thus,

$\displaystyle\int_{M}\langle B,{\rm Hess}_{\,u}\rangle\,{\rm
d}V=\int_{M}\langle B,\nabla(\nabla u)\rangle\,{\rm
d}V=\int_{M}\langle\nabla^{*}B,\nabla u\rangle\,{\rm
d}V=\int_{M}u\,(\nabla^{*})^{2}(B)\,{\rm d}V.$ (21)

The next lemma generalizes (21) and the well known Green’s formula, e.g., [11,
p. 75].

######  Lemma 5.

If a foliated Riemannian manifold $(M,g,{\cal F})$ satisfies (20), then the
following formulas are valid for any compactly supported functions $u,f$ and
2-tensor $B$:

$\displaystyle\int_{M}f(\Delta_{\,\cal F}\,u)\,{\rm
d}V=\int_{M}u(\Delta_{\,\cal F}\,f)\,{\rm d}V,$ (22)
$\displaystyle\int_{M}\langle B,{\rm Hess}^{\cal F}_{\,u}\rangle\,{\rm
d}V=\int_{M}u\,(\nabla^{{\cal F}*})^{2}(B)\,{\rm d}V.$ (23)

##  3 Main results

In Sect.3.1, we find the Euler-Lagrange equations (and first variations) for
the functionals (5)–(7), and in Sect. 3.2 we find the second variations of (5)
and (6). First we check the conformity of (7).

######  Theorem 1.

The functional $W^{\rm conf}_{n,s,r}$ is a conformal invariant of a foliated
hypersurface $(M,{\cal F})$ in a Riemannian manifold $(\bar{M},\bar{g})$.

###### Proof.

Define a new Riemannian metric on $\bar{M}$ by $\bar{g}^{c}=\mu^{2}\bar{g}$
for some positive function $\mu\in C^{3}(\bar{M})$. Then $g^{c}=\mu^{2}g$ is
the new induced metric on $M$, thus, the new volume form of $M$ is ${\rm
d}V^{c}=\mu^{n}\,{\rm d}V$. If $X$ is a $\bar{g}$-unit vector, then
$X^{c}=X/\mu$ is a $g^{c}$-unit vector. By the well known formula for the
Levi-Civita connection, e.g., [13, p. 14], we get

$2\,\bar{\nabla}^{\,c}_{X}\,Y=2\,\bar{\nabla}_{X}\,Y+\mu^{-2}\big{(}X(\mu^{2})Y+Y(\mu^{2})X-\langle
X,Y\rangle\bar{\nabla}\mu^{2}\big{)}.$

By this with $X\in T{\cal F}$ and $Y={\bf N}^{\,c}$, the operators $A$ and
$A^{c}$ are related by
$A^{c}=\frac{1}{\mu}\big{(}A-\frac{1}{\mu}\langle\bar{\nabla}\mu,N\rangle\,{\rm
id}_{\,T\cal F}\big{)}$, see also [13, Sect. 2.1.2.2]. By the above and
$A_{\cal F}=PAP$, $A^{\,c}_{\cal F}=PA^{c}P$, we get

$A_{\cal F}^{c}=\frac{1}{\mu}\big{(}A_{\cal
F}-\frac{1}{\mu}\langle\bar{\nabla}\mu,N\rangle\,{\rm id}_{\,T\cal
F}\big{)},\quad H_{\cal F}^{c}=\frac{1}{s}\operatorname{trace}A_{\cal
F}^{c}=\frac{1}{\mu}\big{(}H_{\cal
F}-\frac{1}{\mu}\langle\bar{\nabla}\mu,N\rangle\big{)}.$

Set $B_{\cal F}=H_{\cal F}\,{\rm id}_{\,T\cal F}-A_{\cal F}$. Let
$\lambda^{B}_{i}$ be the eigenvalues of $B_{\cal F}$ on ${\cal F}$ and
$\sigma^{B}_{r}$ be elementary symmetric functions of $B_{\cal F}$. Obviously,
$B_{\cal F}^{\,c}=\frac{1}{\mu}\,B_{\cal F}$ holds; hence,
$\lambda^{B,c}_{i}=\frac{1}{\mu}\,\lambda^{B}_{i}$. One can show that $Q^{\cal
F}_{r}=-\sigma^{B}_{r}/C^{s}_{r}$ is true, see [6, Lemma 2.1]. By the above,
$\mu^{r}Q^{{\cal F},c}_{r}=Q^{\cal F}_{r}$ holds. Hence, $(Q^{\cal
F}_{r})^{n/r}{\rm d}V$ is a conformal invariant of $(M,{\cal F})$ in
$(\bar{M},\bar{g})$: $(Q^{{\cal F},c}_{r})^{n/r}{\rm d}V^{c}=(Q^{\cal
F}_{r})^{n/r}{\rm d}V$. Note that if $A_{\cal F}$ is a conformal operator on
$T{\cal F}$ (i.e., proportional to ${\rm id}_{\,T\cal F}$), then $B_{\cal
F}=0$, hence, $Q^{\cal F}_{r}=0$. ∎

###  3.1 The first variation

We can state our main theorem.

######  Theorem 2.

If (20) is valid, then Euler-Lagrange equations for the functionals (5) are

$\displaystyle\sum\nolimits_{\,r=1}^{s}\big{\\{}(\nabla^{{\cal
F}*})^{2}\big{(}F^{\prime}_{r}\cdot T_{r-1}(A_{\cal
F})\big{)}+F^{\prime}_{r}\big{(}\sigma^{\cal F}_{1}\sigma^{\cal
F}_{r-1}{-}(r+1)\sigma^{\cal F}_{r+1}{+}\langle T_{r-1}(A_{\cal F}),A_{\,\rm
mix}^{2}\rangle\big{)}\big{\\}}-nFH=0,$
$\displaystyle\sum\nolimits_{\,i=1}^{s}\frac{1}{i}\,\big{\\{}(\nabla^{{\cal
F}*})^{2}\big{(}F^{\prime}_{i}\cdot A_{\cal
F}^{i-1}\big{)}+F^{\prime}_{i}\big{(}\tau^{\cal F}_{i+1}+\langle h_{\cal
F}^{i-1},h^{2}_{\,\rm mix}\rangle\big{)}\big{\\}}-nFH=0.$ (24)

###### Proof.

Using (13), we get the following:

$\displaystyle\delta\,WF_{n,s}=\int_{M}\delta\,(F(\sigma^{\cal
F}_{1},\ldots,\sigma^{\cal F}_{s})\,{\rm
d}V)=\int_{M}\Big{\\{}\sum\nolimits_{r=1}^{s}F^{\prime}_{r}\cdot\delta\,\sigma^{\cal
F}_{r}-n\,uFH\Big{\\}}{\rm d}V,$
$\displaystyle\delta\,JF_{n,s}=\int_{M}\delta\,(F(\tau^{\cal
F}_{1},\ldots,\tau^{\cal F}_{s})\,{\rm
d}V)=\int_{M}\Big{\\{}\sum\nolimits_{i=1}^{s}F^{\prime}_{i}\cdot\delta\,\tau^{\cal
F}_{i}-n\,uFH\Big{\\}}{\rm d}V.$ (25)

From (3.1) and (2) we find the first variations of functionals (5):

$\displaystyle\delta\,WF_{n,s}=\int_{M}\Big{\\{}\sum\nolimits_{\,r=1}^{s}\langle
F^{\prime}_{r}\cdot T_{r-1}(A_{\cal F}),{\rm Hess}^{\cal F\sharp}_{u}\rangle$
$\displaystyle\quad+u\sum\nolimits_{\,r=1}^{s}F^{\prime}_{r}\big{(}\sigma^{\cal
F}_{1}\sigma^{\cal F}_{r-1}-(r+1)\sigma^{\cal F}_{r+1}+\langle T_{r-1}(A_{\cal
F}),A_{\,\rm mix}^{2}\rangle\big{)}-n\,uFH\Big{\\}}{\rm d}V,$
$\displaystyle\delta\,JF_{n,s}=\int_{M}\Big{\\{}\sum\nolimits_{\,i=1}^{s}\frac{1}{i}\,F^{\prime}_{i}\big{(}\langle
h_{\cal F}^{i-1},{\rm Hess}^{\cal F}_{u}\rangle+u\,(\tau^{\cal
F}_{i+1}+\langle h_{\cal F}^{i-1},h^{2}_{\,\rm
mix}\rangle)\big{)}-n\,uFH\Big{\\}}{\rm d}V.$ (26)

From (3.1), using (20) and (22), we obtain (2). ∎

Equations (27) and (28) of the next statement follow from Theorem 2, but we
will prove them.

######  Corollary 1.

If (20) is valid, then the Euler-Lagrange equations for the functionals
$W_{n,p,s},\,J_{n,p,s}$, see (6), and $W^{\rm conf}_{n,s,2}$, see (7), are,
respectively, the following:

$\displaystyle\Delta_{\,\cal F}\,({H}_{\cal F}^{p-1})+{H}_{\cal
F}^{p-1}\big{(}\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2}-\frac{n\,s}{p}\,H{H}_{\cal F}\big{)}=0,$ (27)
$\displaystyle(\nabla^{{\cal F}*})^{2}(\|h_{\cal F}\|^{p-2}h_{\cal
F})+\|h_{\cal F}\|^{p-2}\big{(}\langle h_{\cal F},\ h_{\cal F}^{2}+h_{\rm
mix}^{2}\rangle-\frac{n}{p}\,\|h_{\cal F}\|^{2}\,H\big{)}=0,$ (28)
$\displaystyle\Delta_{\,\cal F}\,\big{(}(Q^{\cal F}_{2})^{n/2-1}\sigma^{\cal
F}_{1}\big{)}-\frac{s}{s-1}\,(\nabla^{\cal F*})^{2}\big{(}(Q^{\cal
F}_{2})^{n/2-1}T_{1}(A_{\cal F})\big{)}+\big{\\{}\sigma^{\cal
F}_{1}(\sigma^{\cal F}_{1}-2\,\sigma^{\cal F}_{2}+\|A_{\,\rm mix}\|^{2})$
$\displaystyle\quad-\frac{s}{s-1}\,(\sigma^{\cal F}_{1}\sigma^{\cal
F}_{2}-3\,\sigma^{\cal F}_{3}+\langle T_{1}(A_{\cal F}),\,A_{\,\rm
mix}^{2}\rangle)-s^{2}Q^{\cal F}_{2}H\big{\\}}(Q^{\cal F}_{2})^{n/2-1}=0.$
(29)

###### Proof.

a) Using (13), (16) and (17) we calculate the variation

$\displaystyle\delta\,({H}_{\cal F}^{p}\,{\rm d}V)$ $\displaystyle={H}_{\cal
F}^{p-1}\Big{\\{}\frac{p}{s}\,\big{(}\Delta_{\,\cal F}\,u+u\,(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2})\big{)}-nu{H}{H}_{\cal F}\Big{\\}}{\rm d}V,$
$\displaystyle\delta\,(\|h_{\cal F}\|^{p}\,{\rm d}V)=\|h_{\cal
F}\|^{p-2}\big{\\{}p\,\langle h_{\cal F},\ u\,(h_{\cal F}^{2}+h_{\rm
mix}^{2})+{\rm Hess}^{\,\cal F}_{\,u}\rangle-n\,u\,\|h_{\cal
F}\|^{2}\,{H}\big{\\}}{\rm d}V.$

Hence, using (20), (22), (23) and (13), we find the first variation of the
actions (6):

$\displaystyle\delta\,W_{n,p,s}$ $\displaystyle=\int_{M}\delta\,({H}_{\cal
F}^{p}\,{\rm d}V)=\int_{M}{H}_{\cal
F}^{p-1}\Big{\\{}\frac{p}{s}\,\big{(}\Delta_{\,\cal F}\,u+u\,(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2})\big{)}-n\,u\,{H}_{\cal F}H\Big{\\}}{\rm d}V$
$\displaystyle=\frac{p}{s}\int_{M}u\Big{\\{}\Delta_{\,\cal F}({H}_{\cal
F}^{p-1})+{H}_{\cal F}^{p-1}\,\big{(}\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2}-\frac{n\,s}{p}\,{H}{H}_{\cal F}\big{)}\Big{\\}}{\rm d}V,$ (30)
$\displaystyle\delta\,J_{n,p,s}$ $\displaystyle=\int_{M}\|h_{\cal
F}\|^{p-2}\Big{\\{}p\,\langle h_{\cal F},\,u\,(h_{\cal F}^{2}+h_{\rm
mix}^{2})+{\rm Hess}^{\,\cal F}_{\,u}\rangle-n\,u\,\|h_{\cal
F}\|^{2}\,{H}\Big{\\}}{\rm d}V$
$\displaystyle=\int_{M}u\Big{\\{}p\,(\nabla^{{\cal F}*})^{2}(\|h_{\cal
F}\|^{p-2}h_{\cal F})+p\,\|h_{\cal F}\|^{p-2}\big{(}\langle h_{\cal F},h_{\cal
F}^{2}+h_{\rm mix}^{2}\rangle-n\,\|h_{\cal F}\|^{2}\,{H}\big{)}\Big{\\}}{\rm
d}V.$ (31)

From (3.1) and (3.1) the Euler-Lagrange equations (27) and (28) follow. By
(2)2 we get

$\displaystyle\delta\,\sigma^{\cal F}_{1}=\Delta_{\,\cal
F}\,u+u\big{(}\sigma^{\cal F}_{1}-2\,\sigma^{\cal F}_{2}+\|A_{\,\rm
mix}\|^{2},$ $\displaystyle\delta\,\sigma^{\cal F}_{2}=\langle T_{1}(A_{\cal
F}),{\rm Hess}^{\cal F\sharp}_{u}\rangle+u\big{(}\sigma^{\cal
F}_{1}\sigma^{\cal F}_{2}-3\,\sigma^{\cal F}_{3}+\langle T_{1}(A_{\cal
F}),\,A_{\,\rm mix}^{2}\rangle\big{)}.$

Using $Q^{\cal F}_{2}=\frac{1}{s^{2}(s-1)}\,((s-1)(\sigma^{\cal
F}_{1})^{2}-2\,s\,\sigma^{\cal F}_{2})$ and (2)2 for $r=1,2$, we get

$\displaystyle s^{2}(s-1)\delta\,Q^{\cal F}_{2}$
$\displaystyle=2(s-1)\sigma^{\cal F}_{1}\delta\,\sigma^{\cal
F}_{1}-2\,s\,\delta\,\sigma^{\cal F}_{2}=2(s-1)\sigma^{\cal
F}_{1}\big{(}\Delta_{\,\cal F}\,u+u(\sigma^{\cal F}_{1}-2\,\sigma^{\cal
F}_{2}+\|A_{\,\rm mix}\|^{2})\big{)}$ $\displaystyle-2\,s\big{(}\langle
T_{1}(A_{\cal F}),{\rm Hess}^{\cal F\sharp}_{u}\rangle+u\,(\sigma^{\cal
F}_{1}\sigma^{\cal F}_{2}-3\,\sigma^{\cal F}_{3}+\langle T_{1}(A_{\cal
F}),\,A_{\,\rm mix}^{2}\rangle)\big{)}.$

Hence

$\displaystyle\delta\,W^{\rm conf}_{n,s,2}$
$\displaystyle=\frac{n}{2}\int_{M}(Q^{\cal
F}_{2})^{n/2-1}\Big{\\{}\frac{1}{s^{2}(s-1)}\big{[}2(s-1)\sigma^{\cal
F}_{1}\big{(}\Delta_{\,\cal F}\,u+u(\sigma^{\cal F}_{1}-2\,\sigma^{\cal
F}_{2}+\|A_{\,\rm mix}\|^{2})\big{)}$ $\displaystyle\quad-2\,s\big{(}\langle
T_{1}(A_{\cal F}),{\rm Hess}^{\cal F\sharp}_{u}\rangle+u\,(\sigma^{\cal
F}_{1}\sigma^{\cal F}_{2}-3\,\sigma^{\cal F}_{3}+\langle T_{1}(A_{\cal
F}),A_{\,\rm mix}^{2}\rangle)\big{)}\big{]}-2\,Q^{\cal F}_{2}uH\Big{\\}}\,{\rm
d}V.$ (32)

Using (22) with $f=(Q^{\cal F}_{2})^{\frac{n}{2}-1}\sigma^{\cal F}_{1}$ and
(23) with $B=(Q^{\cal F}_{2})^{\frac{n}{2}-1}T_{1}(A_{\cal F})$ in (3.1), we
get (1). ∎

######  Remark 3.

(i) For a hypersurface $M\subset\mathbb{R}^{n+1}$ equipped with a line field
(i.e., $s=1$) of the normal curvature $\kappa$, the functionals (5) and (6)
coincide with $WF_{n,1}=\int_{M}F(\kappa)\,{\rm d}V$ and
$W_{n,p,1}=\int_{M}\kappa^{p}\,{\rm d}V$. For $W_{n,2,1}$ and $J_{n,2,1}$,
from (27) and (28) with $p=2$ and $s=1$, using $(\nabla^{{\cal
F}*})^{2}h_{\cal F}=\Delta_{\,\cal F}\,\kappa$, we get the following leaf-wise
elliptic PDE: $\Delta_{\,\cal F}\,\kappa+(\kappa^{2}-\|h_{\rm
mix}\|^{2}-\frac{n}{2}\,H\kappa)\,\kappa=0$.

(ii) The first variation of the functional $W^{\rm conf}_{n.s,r}$ and the
Euler-Lagrange equation can be obtained from (2)1 and (3.1)1, similarly to the
corresponding equations in [6] for $W^{\rm conf}_{n.r}$.

(iii) By (3.1)–(3.1) with $s=n$, the first variations of functionals (1) are
given by

$\displaystyle\delta\,W_{n,p}=\int_{M}{H}^{p-1}\Big{\\{}\frac{p}{n}\,\big{(}\Delta\,u+u\,\|{h}\|^{2}\big{)}-nu{H}^{2}\Big{\\}}{\rm
d}V,$
$\displaystyle\delta\,J_{n,p}=\int_{M}\|{h}\|^{p-2}\Big{\\{}p\,\langle{h},\
u\,{h}^{2}+{\rm Hess}_{\,u}\rangle-n\,u\,\|{h}\|^{2}{H}\Big{\\}}{\rm d}V.$

The corresponding Euler-Lagrange equations are well known:

$\displaystyle\Delta\,{H}^{p-1}+{H}^{p-1}\,\big{(}\|{h}\|^{2}-\frac{n^{2}}{p}\,{H}^{2}\big{)}=0,$
(33)
$\displaystyle(\nabla^{*})^{2}(\|{h}\,\|^{p-2}h)+\|{h}\,\|^{p-2}\big{(}\langle
h,h^{2}\rangle-\frac{n}{p}\,\|{h}\,\|^{2}H\big{)}=0,$ (34)

for example, [8, Corollary 1], where $n=2$ and $M^{2}\subset\mathbb{R}^{3}$.
For $p=n=2$, we can use the identity
$\|h\|^{2}-2\,H^{2}=\frac{1}{2}\,(k_{1}-k_{2})^{2}=2\,(H^{2}-K)$, where
$k_{1}$ and $k_{2}$ are the principal curvatures, $H=(k_{1}+k_{2})/2$ and
$K=k_{1}k_{2}$ is the gaussian curvature of a surface
$M^{2}\subset\mathbb{R}^{3}$. In this case, the Euler-Lagrange equation (33)
reduces to (2). Using the identity $\langle h,h^{2}\rangle=8\,H^{3}-6\,HK$,
the Euler-Lagrange equation (34) for $p=n=2$ reads as
$(\nabla^{*})^{2}h+4\,H(H^{2}-K)=0$.

###  3.2 The second variation

The following statement generalizes Corollary 1 in [8] when
$M^{2}\subset\mathbb{R}^{3}$.

######  Theorem 3.

If (20) is valid, then the Euler-Lagrange equation for (5)1 with $F=F(H_{\cal
F})$ is

$\displaystyle\Delta_{\,\cal F}\,F^{\prime}+F^{\prime}(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2})-s\,n\,FH=0.$ (35)

At a critical hypersurface satisfying (20), the second variation of (5)1 with
$F=F(H_{\cal F})$ is

$\displaystyle\delta^{2}WF_{n,s}=-\\!\int_{M}\frac{n}{s}\,\Big{\\{}F^{\prime}\Delta_{\,\cal
F}\,u-u\,\Delta_{\,\cal F}\,F^{\prime}\Big{\\}}uH\,{\rm
d}V+\\!\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\big{(}2\,u\langle h_{\cal
F},{\rm Hess}^{\,\cal F}_{u}\rangle+s\,u\langle\nabla^{\cal F}H_{\cal
F},\nabla^{\cal F}u\rangle$ $\displaystyle+2\,h(\nabla^{\cal F}u,\nabla^{\cal
F}u)-s\,H_{\cal F}\|\nabla^{\cal
F}u\|^{2}\big{)}+\frac{F^{\prime\prime}}{s^{2}}\,\Delta_{\,\cal
F}\,u\big{(}\Delta_{\,\cal F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}{\rm d}V$
$\displaystyle+\int_{M}u\Big{\\{}\big{(}\frac{F^{\prime\prime}}{s^{2}}\,(\|h_{\cal
F}\|^{2}-\|h_{\rm
mix}\|^{2})-\frac{n}{s}\,HF^{\prime}\big{)}\big{(}\Delta_{\,\cal
F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}-F(\Delta\,u+u\|h\|^{2})$
$\displaystyle+\frac{F^{\prime}}{s}\,\big{(}2\,\langle h_{\cal F},\
u\,(h_{\cal F}^{2}+h_{\rm mix}^{2})+{\rm Hess}^{\,\cal
F}_{\,u}\rangle-u\,\langle h_{\cal F}+h_{\cal F^{\perp}},h_{\rm
mix}^{2}\rangle-2\,\langle h_{\rm mix},{\rm Hess}^{\,\rm
mix}_{\,u}\rangle\big{)}\Big{\\}}{\rm d}V.$ (36)

###### Proof.

By (3.1)1 with $F=F(\sigma_{1}^{\cal F}/s)$, using $\langle{\rm
id}_{\,TM},h_{\rm mix}^{2}\rangle=\|h_{\rm mix}\|^{2}$ and $\langle{\rm
id}_{\,T\cal F},{\rm Hess}^{\cal F}_{u}\rangle=\Delta_{\,\cal F}\,u$, we find
the first variation of the functional (5)1 with $F=F(H_{\cal F})$:

$\displaystyle\delta\,WF_{n,s}=\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\Delta_{\,\cal
F}\,u+\big{(}\frac{F^{\prime}}{s}\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})-n\,FH\big{)}\,u\Big{\\}}{\rm d}V=0.$ (37)

If (20) is valid, then using (37) and (22) we obtain (35). Our next aim is to
calculate

$\displaystyle\delta^{2}WF_{n,s}=\delta\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\Delta_{\,\cal
F}\,u+\big{(}\frac{F^{\prime}}{s}\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})-n\,FH\big{)}\,u\Big{\\}}{\rm d}V$
$\displaystyle=-\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\Delta_{\,\cal
F}\,u+\big{(}\frac{F^{\prime}}{s}\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})-n\,FH\big{)}\,u\Big{\\}}n\,uH\,{\rm d}V$
$\displaystyle+\int_{M}\delta\Big{(}\frac{F^{\prime}}{s}\,\Delta_{\,\cal
F}\,u\Big{)}{\rm
d}V+\int_{M}\delta\,\Big{\\{}\Big{(}\frac{F^{\prime}}{s}\,(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2})-n\,FH\Big{)}\,u\Big{\\}}{\rm d}V.$ (38)

For the first integral in the last line of (3.2), using (16), (19) and $\delta
u=0$, we get

$\displaystyle\int_{M}\delta\Big{(}\frac{F^{\prime}}{s}\,\Delta_{\,\cal
F}\,u\Big{)}{\rm d}V=\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\big{(}2\,u\langle
h_{\cal F},{\rm Hess}^{\,\cal F}_{u}\rangle+s\,u\langle\nabla^{\cal F}H_{\cal
F},\nabla^{\cal F}u\rangle$ $\displaystyle+2\,h(\nabla^{\,\cal
F}u,\nabla^{\cal F}u)-sH_{\cal F}\langle\nabla^{\cal F}u,\nabla^{\cal
F}u\rangle\big{)}+\frac{F^{\prime\prime}}{s^{2}}\,\Delta_{\,\cal
F}\,u\big{(}\Delta_{\,\cal F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}{\rm d}V.$ (39)

For the second integral in the last line of (3.2), using (18), we get

$\displaystyle\int_{M}\delta\,\Big{\\{}\Big{(}\frac{F^{\prime}}{s}\,(\|h_{\cal
F}\|^{2}-\|h_{\rm mix}\|^{2})-n\,FH\Big{)}\,u\Big{\\}}{\rm d}V$
$\displaystyle=\int_{M}u\Big{\\{}\big{(}\frac{F^{\prime\prime}}{s^{2}}\,(\|h_{\cal
F}\|^{2}-\|h_{\rm
mix}\|^{2})-\frac{nF^{\prime}}{s}\,H\big{)}\big{(}\Delta_{\,\cal
F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}-F(\Delta\,u+u\|h\|^{2})$
$\displaystyle+\frac{F^{\prime}}{s}\,\big{(}2\,\langle h_{\cal F},\
u\,(h_{\cal F}^{2}+h_{\rm mix}^{2})+{\rm Hess}^{\,\cal
F}_{\,u}\rangle-u\,\langle h_{\cal F}+h_{\cal F^{\perp}},h_{\rm
mix}^{2}\rangle-2\,\langle h_{\rm mix},{\rm Hess}^{\,\rm
mix}_{\,u}\rangle\big{)}\Big{\\}}{\rm d}V.$ (40)

By (3.2), (3.2) and (3.2), noting that the first variation vanishes at a
critical immersion, we get

$\displaystyle\delta^{2}WF_{n,s}=-\int_{M}\frac{n}{s}\,\Big{\\{}F^{\prime}\Delta_{\,\cal
F}\,u+\big{(}F^{\prime}(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})-s\,n\,FH\big{)}\,u\Big{\\}}uH\,{\rm d}V$
$\displaystyle+\int_{M}\Big{\\{}\frac{F^{\prime}}{s}\,\big{(}2\,u\langle
h_{\cal F},{\rm Hess}^{\,\cal F}_{u}\rangle+s\,u\langle\nabla^{\cal F}H_{\cal
F},\nabla^{\cal F}u\rangle+2\,h(\nabla^{\cal F}u,\nabla^{\cal F}u)-s\,H_{\cal
F}\|\nabla^{\cal F}u\|^{2}\big{)}$
$\displaystyle+\frac{1}{s^{2}}\,F^{\prime\prime}\Delta_{\,\cal
F}\,u\big{(}\Delta_{\,\cal F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}\,{\rm d}V$
$\displaystyle+\int_{M}u\Big{\\{}\big{(}\frac{F^{\prime\prime}}{s^{2}}\,(\|h_{\cal
F}\|^{2}-\|h_{\rm
mix}\|^{2})-\frac{nF^{\prime}}{s}\,H\big{)}\big{(}\Delta_{\,\cal
F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm
mix}\|^{2})\big{)}-F(\Delta\,u+u\|h\|^{2})$
$\displaystyle+\frac{F^{\prime}}{s}\,\big{(}2\,\langle h_{\cal F},\
u\,(h_{\cal F}^{2}+h_{\rm mix}^{2})+{\rm Hess}^{\,\cal
F}_{\,u}\rangle-u\,\langle h_{\cal F}+h_{\cal F^{\perp}},h_{\rm
mix}^{2}\rangle-2\,\langle h_{\rm mix},{\rm Hess}^{\,\rm
mix}_{\,u}\rangle\big{)}\Big{\\}}{\rm d}V.$ (41)

From (3.2) and (35), at a critical hypersurface, we get (3). ∎

Similarly, one can obtain the Euler-Lagrange equation for the functional (5)2
with $F=F(\|h_{\cal F}\|)$, we do not present it here. From Theorem 3 with
$F=H_{\cal F}^{p}$ we obtain the following.

######  Corollary 2.

At a critical hypersurface satisfying (20), the second variation of the action
(6)1 is

$\displaystyle\delta^{2}W_{n,p,s}=-\int_{M}\frac{np}{s}\Big{\\{}H_{\cal
F}^{p-1}\Delta_{\,\cal F}\,u-u\,\Delta_{\,\cal F}\,(H_{\cal
F}^{p-1})\Big{\\}}uH\,{\rm d}V+\int_{M}\frac{p}{s}\,H_{\cal
F}^{p-2}\Big{\\{}H_{\cal F}\big{(}2\,u\langle h_{\cal F},{\rm Hess}^{\cal
F}_{u}\rangle$ $\displaystyle+s\,u\langle\nabla^{\cal F}H_{\cal
F},\nabla^{\cal F}u\rangle{+}2h(\nabla^{\cal F}u,\nabla^{\cal F}u){-}sH_{\cal
F}\|\nabla^{\cal F}u\|^{2}\big{)}{+}\frac{p{-}1}{s}\Delta_{\,\cal
F}u\big{(}\Delta_{\,\cal F}u{+}u\,(\|h_{\cal F}\|^{2}{-}\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}{\rm d}V$ $\displaystyle+\\!\int_{M}\\!H_{\cal
F}^{p-2}u\Big{\\{}\frac{p}{s}\Big{(}\frac{p{-}1}{s}(\|h_{\cal
F}\|^{2}{-}\|h_{\rm mix}\|^{2}){-}n\,HH_{\cal F}\Big{)}\big{(}\Delta_{\,\cal
F}\,u{+}u\,(\|h_{\cal F}\|^{2}{-}\|h_{\rm mix}\|^{2})\big{)}{-}H_{\cal
F}^{2}(\Delta\,u{+}u\|h\|^{2})$ $\displaystyle\quad+\frac{p}{s}\,H_{\cal
F}\big{(}2\,\langle h_{\cal F},u\,(h_{\cal F}^{2}+h_{\rm mix}^{2})+{\rm
Hess}^{\,\cal F}_{\,u}\rangle-u\,\langle h_{\cal F}+h_{\cal F^{\perp}},h_{\rm
mix}^{2}\rangle-2\,\langle h_{\rm mix},{\rm Hess}^{\,\rm
mix}_{\,u}\rangle\big{)}\Big{\\}}{\rm d}V.$ (42)

######  Remark 4.

By (3.2) with $s=n$, the second variation of the action
$WF_{n}=\int_{M}F(H)\,{\rm d}V$ is

$\displaystyle\delta^{2}WF_{n}=-\\!\int_{M}\Big{\\{}F^{\prime}\,\Delta\,u{+}\big{(}F^{\prime}\|h\|^{2}{-}n^{2}\,FH\big{)}\,u\Big{\\}}u\,H\,{\rm
d}V{+}\\!\int_{M}\Big{\\{}\frac{F^{\prime}}{n}\big{(}2\,u\langle{h},{\rm
Hess}_{u}\rangle{+}n\,u\langle\nabla\,{H},\nabla u\rangle$
$\displaystyle+2\,h(\nabla\,u,\nabla u){-}nH\,\|\nabla
u\|^{2}\big{)}+\frac{F^{\prime\prime}}{n^{2}}\Delta\,u\big{(}\Delta\,u+u\,\|h\|^{2}\big{)}\Big{\\}}{\rm
d}V$
$\displaystyle+\int_{M}u\Big{\\{}\big{(}\frac{F^{\prime\prime}}{n^{2}}\,\|h\|^{2}-HF^{\prime}-F\big{)}\big{(}\Delta\,u+u\,\|h\|^{2}\big{)}+\frac{2\,F^{\prime}}{n}\,\langle
h,\,u{h}^{2}+{\rm Hess}_{\,u}\rangle\Big{\\}}{\rm d}V.$ (43)

This is compatible with a special case of Eq. (7) in [8] for $n=2$. As a
special case of (4), the second variation of the functional (1)1 with $n=2$
has the following form compatible with [8, Eq. (45)]:

$\displaystyle\delta^{2}W_{2,p}=\\!\int_{M}H^{p-2}\Big{\\{}\frac{p(p-1)}{4}\,(\Delta\,u)^{2}+p\,H\big{(}h(\nabla\,u,\nabla\,u)+2\,u\langle
h,{\rm Hess}_{u}\rangle+u\langle\nabla\,H,\nabla u\rangle-H\|\nabla
u\|^{2}\big{)}$
$\displaystyle+\big{(}(2p^{2}{-}4p-1)H^{2}{-}p(p{-}1)K\big{)}u\Delta\,u+(4p(p-1)H^{4}{-}2(p-1)(2p+1)KH^{2}{+}p(p{-}1)K^{2})u^{2}\Big{\\}}{\rm
d}V.$

##  4 Applications

We consider critical hypersurfaces equipped with two-dimensional foliations
(i.e., $s=2$) in Sect. 4.1, and discuss critical hypersurfaces of revolution
and their stability in Sect. 4.2.

###  4.1 Hypersurfaces with two-dimensional foliations

For $s=2$, it is natural to present the functionals (5) in the following form:

$WF_{n,2}=\int_{M}F({H}_{\cal F},K_{\cal F})\,{\rm d}V.$ (44)

where ${K}_{\cal F}=\det A_{\cal F}=\sigma^{\cal F}_{2}$ is the Gaussian
curvature of the leaves. For $n=s=2$, (44) reduces to the functional
$WF_{2}=\int_{M}F(H,K)\,{\rm d}V$ seen in [8]. The following equalities are
true:

$\|h_{\cal F}\|^{2}=k_{{\cal F},1}^{2}+k_{{\cal F},2}^{2}=4\,{H}_{\cal
F}^{2}-2\,K_{\cal F},\quad\langle h_{\cal F},\ h_{\cal
F}^{2}\,\rangle=k_{{\cal F},1}^{3}+k_{{\cal F},2}^{3}=8\,{H}_{\cal
F}^{3}-6\,{H}_{\cal F}K_{\cal F}.$

From (16) and (17) with $s=2$, we obtain the following evolution equations:

$\displaystyle\delta\,(2\,{H}_{\cal F})=\Delta_{\,\cal F}\,u+u\,(4\,{H}_{\cal
F}^{2}-2\,K_{\cal F}-\|h_{\rm mix}\|^{2}),$ (45)
$\displaystyle\delta\,\|h_{\cal F}\|^{2}=4\,u\,(4\,{H}_{\cal
F}^{3}-3\,{H}_{\cal F}K_{\cal F})+2\,\langle h_{\cal F},\ u\,h_{\rm
mix}^{2}+{\rm Hess}^{\,\cal F}_{\,u}\rangle.$ (46)

Using (45) and (46) in $\delta\,K_{\cal F}=2\,{H}_{\cal
F}\,\delta\,(2\,{H}_{\cal F})-(1/2)\delta\,(\|h_{\cal F}\|^{2})$, we get the
evolution equation

$\displaystyle\delta\,K_{\cal F}=2\,{H}_{\cal F}\,\Delta_{\,\cal F}\,u-\langle
h_{\cal F},\ u\,h_{\rm mix}^{2}+{\rm Hess}^{{}_{\cal
F}}_{\,u}\rangle+2\,u\,{H}_{\cal F}\,(K_{\cal F}-\|h_{\rm mix}\|^{2}).$ (47)

For $n=2$, (45)–(47) reduce to the equations in [8, Lemma 1].

The next statement for $(M^{n},{\cal F}^{2})$ immersed in $\mathbb{R}^{n+1}$
generalizes Theorem 1 in [8] with $n=2$.

######  Theorem 4.

If (20) is valid, then the Euler-Lagrange equation for the action (44) with
$s=2$ is

$\displaystyle\Delta_{\,\cal F}\Big{(}\frac{1}{2}\,F^{\prime}_{H}+2\,{H}_{\cal
F}F^{\prime}_{K}\Big{)}-(\nabla^{{\cal F}*})^{2}(F^{\prime}_{K}h_{\cal
F})+F^{\prime}_{H}\big{(}2\,H_{\cal F}^{2}-K_{\cal F}-\frac{1}{2}\,\|h_{\rm
mix}\|^{2}\big{)}$ $\displaystyle\qquad+F^{\prime}_{K}\big{(}2\,{H}_{\cal
F}\,(K_{\cal F}-\|h_{\rm mix}\|^{2})-\langle h_{\cal F},\ h_{\rm
mix}^{2}\,\rangle\big{)}-n\,FH=0,$ (48)

where $F^{\prime}_{H},F^{\prime}_{K}$ denote partial derivatives of $F(H_{\cal
F},K_{\cal F})$ with respect to $H_{\cal F}$ and $K_{\cal F}$. At a critical
hypersurface foliated by surfaces $(s=2)$ and satisfying (20), the second
variation of the functional (44) with $F=F(H_{\cal F})$ is given by

$\displaystyle\delta^{2}WF_{n,2}=-\\!\int_{M}\frac{n}{2}\,\Big{\\{}F^{\prime}\Delta_{\,\cal
F}\,u-u\,\Delta_{\,\cal F}\,F^{\prime}\Big{\\}}uH\,{\rm
d}V+\\!\int_{M}\Big{\\{}\frac{F^{\prime}}{2}\big{(}2\,u\langle h_{\cal F},{\rm
Hess}^{\,\cal F}_{u}\rangle{+}2\,u\langle\nabla^{\cal F}H_{\cal
F},\nabla^{\cal F}u\rangle$ $\displaystyle\quad+2\,h(\nabla^{\cal
F}u,\nabla^{\cal F}u)-2\,H_{\cal F}\|\nabla^{\cal
F}u\|^{2}\big{)}+\frac{F^{\prime\prime}}{4}\,\Delta_{\,\cal
F}\,u\big{(}\Delta_{\,\cal F}\,u+u\,(4\,H_{\cal F}^{2}-2\,K_{\cal F}-\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}{\rm d}V$
$\displaystyle\quad+\int_{M}u\Big{\\{}\Big{(}\frac{F^{\prime\prime}}{4}\,(4\,H_{\cal
F}^{2}-2\,K_{\cal F}-\|h_{\rm
mix}\|^{2})-\frac{nF^{\prime}}{2}\,H\Big{)}\big{(}\Delta_{\,\cal
F}\,u+u\,(4\,H_{\cal F}^{2}-2\,K_{\cal F}-\|h_{\rm mix}\|^{2})\big{)}$
$\displaystyle\quad+\frac{F^{\prime}}{2}\big{(}4\,u\,{H}_{\cal F}(4{H}_{\cal
F}^{2}-3\,{K}_{\cal F})+2\,u\,\langle h_{\cal F},h_{\rm
mix}^{2}\rangle+2\,\langle h_{\cal F},{\rm Hess}^{\,\cal
F}_{\,u}\rangle-F(\Delta\,u{+}u\|h\|^{2})$ $\displaystyle\quad-u\langle
h_{\cal F}+h_{\cal F^{\perp}},h_{\rm mix}^{2}\rangle-2\langle h_{\rm mix},{\rm
Hess}^{\,\rm mix}_{\,u}\rangle\big{)}\Big{\\}}{\rm d}V.$ (49)

###### Proof.

Using (45) and (47) in $\delta\,F(H_{\cal F},K_{\cal
F})=\frac{1}{2}F^{\prime}_{H}\cdot\delta\,(2\,H_{\cal
F})+F^{\prime}_{K}\,\delta\,K_{\cal F}$, and applying (13), we calculate the
first variation of the functional (44) with $s=2$:

$\displaystyle\delta\,WF_{n,2}=\int_{M}\delta\big{(}F(H_{\cal F},K_{\cal
F})\,{\rm
d}V\big{)}=\int_{M}\Big{\\{}\frac{1}{2}\,F^{\prime}_{H}\big{(}\Delta_{\,\cal
F}\,u+u\,(4\,H_{\cal F}^{2}-2\,K_{\cal F}-\|h_{\rm mix}\|^{2})\big{)}$
$\displaystyle+F^{\prime}_{K}\big{(}2\,{H}_{\cal F}\,\Delta_{\,\cal
F}\,u-\langle h_{\cal F},\ u\,h_{\rm mix}^{2}+{\rm Hess}^{{}_{\cal
F}}_{\,u}\rangle+2\,u\,{H}_{\cal F}\,(K_{\cal F}-\|h_{\rm
mix}\|^{2})\big{)}-n\,u\,HF\Big{\\}}{\rm d}V$
$\displaystyle=\int_{M}\Big{\\{}\big{(}\frac{1}{2}\,F^{\prime}_{H}+2\,{H}_{\cal
F}F^{\prime}_{K}\big{)}\,\Delta_{\,\cal F}\,u-F^{\prime}_{K}\langle h_{\cal
F},\ {\rm Hess}^{{}_{\cal
F}}_{\,u}\rangle+\big{(}F^{\prime}_{H}\big{(}2\,H_{\cal F}^{2}-K_{\cal
F}-\frac{1}{2}\,\|h_{\rm mix}\|^{2}\big{)}$
$\displaystyle\quad+F^{\prime}_{K}\big{(}2\,{H}_{\cal F}\,(K_{\cal F}-\|h_{\rm
mix}\|^{2})-\langle h_{\cal F},\ h_{\rm
mix}^{2}\,\rangle\big{)}-n\,FH\big{)}\,u\Big{\\}}{\rm d}V.$ (50)

From (4.1), using (23), we get (4). From Theorem 3 with $s=2$ we get (4). ∎

######  Remark 5.

Let $F=F({H}_{\cal F})$, then from (4) we obtain

$\displaystyle\Delta_{\,\cal
F}\big{(}F^{\prime}_{H}\big{)}+F^{\prime}_{H}\big{(}4\,H_{\cal
F}^{2}-2\,K_{\cal F}-\|h_{\rm mix}\|^{2}\big{)}-2\,nFH=0.$

From this with $F={H}_{\cal F}^{2}$, or, from (27), we get the Euler-Lagrange
equation for $W_{n,2,2}$, see (6)1:

$\displaystyle\Delta_{\,\cal F}\,{H}_{\cal F}+\big{(}\,4{H}_{\cal
F}^{2}-2\,{K}_{\cal F}-\|h_{\rm mix}\|^{2}-n\,H{H}_{\cal F}\big{)}{H}_{\cal
F}=0.$

The following particular case of (4), or, Corollary 2 with $s=2$, is true.

######  Corollary 3.

At a critical hypersurface satisfying (20), the second variation of
$W_{n,p,2}$ is

$\displaystyle\delta^{2}W_{n,p,2}=-\int_{M}\frac{np}{2}\Big{\\{}H_{\cal
F}^{p-1}\Delta_{\,\cal F}\,u-u\,\Delta_{\,\cal F}\,(H_{\cal
F}^{p-1})\Big{\\}}uH\,{\rm d}V+\int_{M}\frac{p}{2}\,H_{\cal
F}^{p-2}\Big{\\{}2\,H_{\cal F}\big{(}u\langle h_{\cal F},{\rm Hess}^{\cal
F}_{u}\rangle$ $\displaystyle+u\langle\nabla^{\cal F}H_{\cal F},\nabla^{\cal
F}u\rangle{+}h(\nabla^{\cal F}u,\nabla^{\cal F}u){-}H_{\cal F}\|\nabla^{\cal
F}u\|^{2}\big{)}{+}\frac{p{-}1}{2}\Delta_{\,\cal F}u\big{(}\Delta_{\,\cal
F}u{+}u(4\,H_{\cal F}^{2}{-}2\,K_{\cal F}{-}\|h_{\rm
mix}\|^{2})\big{)}\Big{\\}}{\rm d}V$ $\displaystyle+\\!\int_{M}\\!H_{\cal
F}^{p-2}u\Big{\\{}\frac{p}{2}\Big{(}\frac{p-1}{2}\,(4\,H_{\cal
F}^{2}{-}2K_{\cal F}{-}\|h_{\rm mix}\|^{2})-n\,HH_{\cal
F}\Big{)}\big{(}\Delta_{\,\cal F}\,u+u\,(4\,H_{\cal F}^{2}{-}2K_{\cal
F}{-}\|h_{\rm mix}\|^{2})\big{)}$ $\displaystyle\quad-H_{\cal
F}^{2}(\Delta\,u+u\|h\|^{2})+\frac{p}{2}\,H_{\cal F}\Big{(}4\,u\,(4{H}_{\cal
F}^{3}-3\,{H}_{\cal F}{K}_{\cal F}^{2})+2\,\langle h_{\cal F},\ u\,h_{\rm
mix}^{2}+{\rm Hess}^{\,\cal F}_{\,u}\rangle$ $\displaystyle\quad-u\,\langle
h_{\cal F}+h_{\cal F^{\perp}},h_{\rm mix}^{2}\rangle-2\,\langle h_{\rm
mix},{\rm Hess}^{\,\rm mix}_{\,u}\rangle\Big{)}\Big{\\}}{\rm d}V.$

The following consequence of Theorem 4 was proven for $n=2$ in [8].

######  Corollary 4.

The Euler-Lagrange equation for the functional $WF_{2}$ with $F=F(H,K)$ is
given by

$\displaystyle\Delta\Big{(}\frac{1}{2}\,F^{\prime}_{H}+2\,HF^{\prime}_{K}\Big{)}-(\nabla^{*})^{2}(F^{\prime}_{K}h)+\big{(}2\,H^{2}-K\big{)}F^{\prime}_{H}+2\,HKF^{\prime}_{K}-n\,HF=0.$

###  4.2 Hypersurfaces of revolution

Hypersurfaces of revolution in Euclidean space $\mathbb{R}^{n+1}$ are
naturally foliated into $(n-1)$-spheres (parallels) and equipped with
rotationally symmetric metrics $g=d\rho^{2}+\rho^{2}ds^{2}_{n-1}$ – a special
case of a warped product metric, see [11, Sect. 4.2.3]. Such a hypersurface
can be represented as a graph $x_{n+1}=f(\rho)$, where the function $f\in
C^{2}$ is monotonic, $\rho=\sqrt{x_{1}^{2}+\ldots+x_{n}^{2}}$ and $(x_{i})$
are Cartesian coordinates in $\mathbb{R}^{n+1}$. We obtain the parametrization
${\bf r}={\bf r}(\phi_{1},\ldots,\phi_{n-1};f(\rho))$, of $M$, where
$(\phi_{1},\ldots,\phi_{n-1};\rho)$ are cylindrical coordinates in
$\mathbb{R}^{n+1}$. The principal curvatures of $M$ (functions of $\rho)$ are
$k_{1}=\ldots=k_{n-1}=\frac{f^{\prime}}{\rho\,(1+(f^{\prime})^{2})^{1/2}}\leq\frac{1}{\rho}$
for parallels and $k_{n}=\frac{f^{\prime\prime}}{(1+(f^{\prime})^{2})^{3/2}}$
for profile curves – geodesics on $M$. If the profile curve is a straight line
($f^{\prime\prime}\equiv 0$), then $k_{n}\equiv 0$ and $M$ is a cone, or a
cylinder, or a hyperplane. To exclude these cases, we will assume
$f^{\prime\prime}\not=0$. We get

$\displaystyle n\,H=(n-1)\,k_{1}+k_{n},\quad H_{\cal F}=k_{1},\quad\|h_{\cal
F}\|^{2}=(n-1)k_{1}^{2},\quad\|h\|^{2}=(n-1)k_{1}^{2}+k_{n}^{2},$
$\displaystyle\langle
h,{h}^{2}\rangle=(n{-}1)\,k_{1}^{3}+k_{n}^{3},\quad\langle h_{\cal F},h_{\cal
F}^{2}\rangle=(n{-}1)k_{1}^{3},\quad h_{\rm mix}=h_{\rm mix}^{2}=0.$ (51)

Recall that $\lambda_{j}=j(j+n-2)$ correspond to the solutions with
multiplicities $N_{j}=C^{n}_{n+j}=\frac{(n+j)!}{n!\,j!}$ of the eigenvalue
problem $\Delta\,u+\lambda\,u=0$ on a unit $(n-1)$-sphere. Any constant
function on the round sphere spans the space of $\lambda_{0}$-eigenfunctions
of the Laplacian. Let $\perp$ denote the orthogonality of functions with
respect to the $L^{2}$ inner product. The sphere $S^{2}(\rho)$ of radius
$\rho$ in $\mathbb{R}^{3}$ is not a local minimum of $W_{2,p}$ under volume-
preserving deformations for $p>2$. For $p\geq 1$, $S^{2}(\rho)$ is a local
minimum of $W_{2,p}$ under volume-preserving, nonconstant deformations $u$
provided $u\in\\{v:\Delta\,v=(2/\rho^{2})v\\}^{\perp}$, see Propositions 2 and
3 in [8]. According to (35), a hypersurface of revolution in
$\mathbb{R}^{n+1}$ foliated by $(n-1)$-spheres-parallels $\\{L_{\rho}\\}$ is a
critical point of the action (5)1 with $F=F(H_{\cal F})$ if and only if

$\displaystyle(F^{\prime}/F)\,((n-1)k_{1}^{2}+k_{n}^{2})-s\,n\,((n-1)k_{1}+k_{n})=0.$

In this case, $k_{n}$ and $k_{1}$ are functionally related; hence, $M$ is a
Weingarten hypersurface.

The following theorem studies the stability of hypersurfaces of revolution
critical for (6).

######  Theorem 5.

A hypersurface of revolution $M:x_{n+1}=f(\rho)\ (f^{\prime\prime}\not=0)$ in
$\mathbb{R}^{n+1}$ foliated by $(n-1)$-spheres-parallels $\\{L_{\rho}\\}$ is a
critical point of the action $W_{n,p,n-1}$ or $J_{n,p,n-1}$, see (6), if and
only if

$\displaystyle
f(\rho)=\int\frac{\sqrt{C_{1}\rho^{2p-2n+2}-\rho^{4p-4n+4}}}{C_{1}-\rho^{2p-2n+2}}\,{\rm
d}\rho+C_{2},\quad C_{1},C_{2}\in\mathbb{R}.$ (52)

A critical hypersurface is not a local minimum of $W_{n,p,n-1}$ for $p>n\geq
2$ with respect to general variations, but it is a local minimum for
variations $u=u(\phi_{1},\ldots,\phi_{n-1})$ satisfying
$u|_{L_{\rho}}\perp\ker\Delta_{\,\cal F}$.

###### Proof.

1\. Let $M$ be critical for the action $W_{n,p,n-1}$ under general
deformations. Since all principal curvatures $k_{i}$ are constant on
parallels, from (27) and (28) we get, respectively,

$\displaystyle p\,\|h_{\cal F}\|^{2}-n(n-1)\,H{H}_{\cal F}=0,\quad p\,\langle
h_{\cal F},\ h_{\cal F}^{2}\rangle-n\,\|h_{\cal F}\|^{2}\,H=0.$ (53)

Using (4.2) in (53), yields $k_{n}=(p-n+1)\,k_{1}\neq 0$, which is the
differential equation for $f=f(\rho)$,

$\displaystyle\rho\,f^{\prime\prime}=(p-n+1)\,f^{\prime}(1+(f^{\prime})^{2}).$
(54)

The solution of (54) is given by (52).

2\. Let $u=u(\phi_{1},\ldots,\phi_{n})$ be the eigenfunction of $\Delta$ on
$S^{n-1}(1)$ with the eigenvalue $\lambda_{j}$, then $\Delta_{\,\cal
F}\,u+\lambda_{j}\,\rho^{-2}\,u=0$. Since our hypersurface of revolution
$(M,g)$ is a warped product, its volume form is decomposed as ${\rm d}V={\rm
d}V_{\rho}\cdot d\,\rho$, see [1, Note 7.1.1.1]. For any function $a(\rho)$ we
have, see (61),

$\displaystyle\int_{M}a\|\nabla^{\,\cal F}u\|^{2}{\rm
d}V=\\!\int_{\rho_{1}}^{\rho_{2}}\\!a\big{(}\int_{L_{\rho}}\|\nabla^{\,\cal
F}u\|^{2}{\rm
d}V_{\rho}\big{)}d\,\rho=-\\!\int_{\rho_{1}}^{\rho_{2}}\\!a\big{(}\int_{L_{\rho}}u\,\Delta_{\,\cal
F}\,u\ {\rm d}V_{\rho}\big{)}d\,\rho=-\\!\int_{M}a\,u\,\Delta_{\,\cal
F}\,u\,{\rm d}V.$

Using (2) with $s=n-1$, Example 1, the equalities $\nabla u=\nabla^{\,\cal
F}u$ and

$\displaystyle\langle\nabla\,k_{1},\nabla u\rangle=0,\ h(\nabla^{\,\cal
F}u,\nabla u)=k_{1}\|\nabla^{\cal F}u\|^{2},\ \langle h,{\rm
Hess}_{\,u}\rangle=\langle h_{\cal F},{\rm Hess}^{\,\cal
F}_{u}\rangle=(n-1)\,{H}_{\cal F}\Delta_{\,\cal F}\,u,$

we find the second variation of $W_{n,p,n-1}$:

$\displaystyle\delta^{2}W_{n,p,n-1}=\frac{1}{(n-1)^{2}}\int_{M}k_{1}^{p-2}\big{\\{}p(p-1)(\Delta_{\,\cal
F}\,u)^{2}$ $\displaystyle+(n-1)(5\,np-n-9\,p+1)k_{1}^{2}\,u\,\Delta_{\,\cal
F}\,u-(n-1)^{2}(p-n)(p-n+1)k_{1}^{4}\,u^{2}\big{\\}}{\rm d}V.$ (55)

If the variation $u=u(\phi_{1},\ldots,\phi_{n-1})$ satisfies $\Delta_{\,\cal
F}\,u=0$, then by (4.2) we get

$\displaystyle\delta^{2}W_{n,p,n-1}=-\int_{M}(p-n)(p-n+1)k_{1}^{p+2}u^{2}\,{\rm
d}V,$

which is negative for $p>n$; hence, our critical hypersurface is not a local
minimum of $W_{n,p,n-1}$.

Let $u$ satisfy $u|_{L_{\rho}}\perp\ker\Delta_{\,\cal F}$. Using the
inequalities $\int_{S^{n-1}(1)}u\,\Delta\,u\,{\rm
d}V\geq\int_{S^{n-1}(1)}\lambda_{1}u^{2}\,{\rm d}V$ and
$\int_{S^{n-1}(1)}(\Delta\,u)^{2}\,{\rm
d}V\geq\int_{S^{n-1}(1)}\lambda_{1}u^{2}\,{\rm d}V$, see, for example, [8], we
get

$\displaystyle\int_{L_{\rho}}u\,\Delta_{\,\cal F}\,u\,{\rm
d}V_{L_{\rho}}\geq\int_{L_{\rho}}\lambda_{1}\rho^{-4}u^{2}\,{\rm
d}V_{L_{\rho}},\quad\int_{L_{\rho}}(\Delta_{\,\cal F}\,u)^{2}\,{\rm
d}V_{L_{\rho}}\geq\int_{L_{\rho}}\lambda_{1}\rho^{-4}u^{2}\,{\rm
d}V_{L_{\rho}}.$

By these estimates, (4.2) and the inequality $\rho^{-4}\geq k_{1}^{2}$, we
find

$\displaystyle\delta^{2}W_{n,p,n-1}\geq\frac{1}{(n-1)^{2}}\int_{M}\Big{\\{}p(p-1)(n-1)^{2}+(n-1)^{2}(5\,np-n-9\,p+1)$
$\displaystyle-(n-1)^{2}(p-n)(p-n+1)\Big{\\}}k_{1}^{p+2}u^{2}{\rm
d}V=\int_{M}\big{\\{}n(p-n)+p(6\,n-11)+1\big{\\}}k_{1}^{p+2}u^{2}{\rm d}V.$

Hence, $\delta^{2}W_{n,p,n-1}>0$ for all $p\geq n\geq 2$. ∎

######  Example 1.

(i) Let $M^{3}:x_{4}=f(\rho)\ (f^{\prime\prime}\not=0)$,
$\rho=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ be a hypersurface of revolution in
$\mathbb{R}^{4}$ foliated by parallels (2-spheres). We get
$H=\frac{1}{3}\,(2\,k_{1}+k_{3})$ and $H_{\cal F}=k_{1}$. Let $M^{3}$ be
critical for the functional $W_{3,p,2}$ or $J_{3,p,2}$ with $p\geq 3$, see
(6), under general deformations. From (53) we get $k_{3}=(p-2)k_{1}\neq 0$.
Solution of (54) is
$f(\rho)=\\!\int\\!\frac{\sqrt{C_{1}\rho^{2p+4}-\rho^{4p}}}{\rho^{2p}-C_{1}\,\rho^{4}}\,{\rm
d}\rho+C_{2}$, where $C_{1},C_{2}\in\mathbb{R}$.

(ii) Let $M^{2}:x_{3}=f(\rho)\ (f^{\prime\prime}\not=0)$,
$\rho=\sqrt{x_{1}^{2}+x_{2}^{2}}$ be a surface of revolution in
$\mathbb{R}^{3}$ foliated by parallels (circles). The principal curvatures are
$k_{1}=\frac{f^{\prime}}{\rho(1+(f^{\prime})^{2})^{1/2}}$ for parallels and
$k_{2}=\frac{f^{\prime\prime}}{(1+(f^{\prime})^{2})^{3/2}}$ for profile
curves. Let $M^{2}$ be critical for the action $W_{2,p,1}$ or $J_{2,p,1}$ with
$p\geq 2$ under general deformations. Then $k_{2}=(p-1)k_{1}\neq 0$; hence,
the equality $H^{2}/K=\frac{p^{2}}{4\,(p-1)}$ is true. Solution of (54) is
$f(\rho)=\\!\int\frac{\sqrt{C_{1}\rho^{2p+2}-\rho^{4p}}}{\rho^{2p}-C_{1}\,\rho^{2}}\,{\rm
d}\rho+C_{2}$, where $C_{1},C_{2}\in\mathbb{R}$, it is illustrated on Fig. 1
for $p=2,3,\ldots,8$.

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

##  5 Appendix

###### Proof.

(of Lemma 1). Using $\delta\,{\bf r}_{i}=u_{i}\,{\bf N}+u\,{\bf N}_{i}$, we
calculate

$\displaystyle\langle\delta\,{\bf r}_{i},{\bf r}_{j}\rangle=\langle
u_{i}\,{\bf N}+u\,{\bf N}_{i},{\bf r}_{j}\rangle=\,u\,\langle{\bf N}_{i},{\bf
r}_{j}\rangle=-u\,\langle{\bf N},{\bf r}_{ij}\rangle=-u\,h_{ij}.$

Thus, since the symmetry $h_{ij}=h_{ji}$ we get the equality (8):
$\delta\,g_{ij}=\langle\delta\,{\bf r}_{i},{\bf r}_{j}\rangle+\langle{\bf
r}_{i},\delta\,{\bf r}_{j}\rangle=-2\,u\,h_{ij}$. From
$g^{il}g_{lj}=\delta^{i}_{j}$ it follows that $(\delta
g^{il})g_{lj}=-g^{il}(\delta g_{lj})=2\,u\,g^{il}h_{lj}$; hence, (9) is true.

We will compute the variation of $h$. Using $\langle{\bf N},{\bf
N}_{i}\rangle=0$, we find

$\displaystyle\langle{\bf N},\ \delta\,{\bf r}_{ij}\rangle=\langle{\bf N},\
(u\,{\bf N})_{ij}\rangle=u_{ij}-u\,\langle{\bf N}_{i},\ {\bf
N}_{j}\rangle=u_{ij}-u\,\langle{h}^{l}_{i}\,{\bf r}_{l},\ {h}^{k}_{j}\,{\bf
r}_{k}\rangle=u_{ij}-u\,{h}^{l}_{i}\,h_{jl}.$

Note that $\delta\,{\bf N}=c^{i}{\bf r}_{i}$ for some functions $c^{i}$. Using
$\langle{\bf N},{\bf r}_{j}\rangle=0$, we get

$\displaystyle g_{ij}\,c^{i}=\langle\delta\,{\bf N},\ {\bf
r}_{j}\rangle=-\langle{\bf N},\ \delta\,{\bf r}_{j}\rangle=-\langle{\bf N},\
u_{j}\,{\bf N}+u\,{\bf N}_{j}\rangle=-u_{j}.$

It follows that $c^{i}=-g^{ij}u_{j}$ and $\delta\,{\bf N}=-g^{ij}u_{j}{\bf
r}_{i}=-u^{i}{\bf r}_{i}$. Using the Gauss equation for a hypersurface in
$\mathbb{R}^{n+1}$, we get at $x$: $\langle\delta\,{\bf N},\ {\bf
r}_{ij}\rangle=\langle-u^{i}{\bf r}_{i},\ h_{jl}{\bf N}+\Gamma^{k}_{jl}\,{\bf
r}_{k}\rangle=0$. Thus, (10) is true:

$\displaystyle\delta{h}_{ij}=\delta\langle{\bf r}_{ij},\ {\bf
N}\rangle=\langle\delta\,{\bf N},\ {\bf r}_{ij}\rangle+\langle{\bf N},\
\delta\,{\bf r}_{ij}\rangle=u_{ij}-u{h}^{l}_{i}\,h_{jl}.$

Calculating the variation of the mean curvature, we get (12):

$\displaystyle\delta\,(nH)=\delta\,(g^{ij}h_{ij})=2\,u{h}^{ij}\,h_{ij}+g^{ij}\big{(}u_{ij}-u{h}^{l}_{i}\,h_{jl}\big{)}$
$\displaystyle=u\,h_{ij}\,{h}^{ij}+g^{ij}u_{ij}=\Delta u+u\,\|{h}\|^{2}.$

The formula $\delta({\rm
d}V)=\frac{1}{2}\,(\operatorname{trace}_{g}\delta\,g)\,{\rm d}V$ for variation
of ${\rm d}V$ is valid for any variation $\delta g$ of a metric. for example,
[13]. Applying (8) to the above gives (13). Next, we calculate the variation
of $\|{h}\|^{2}$:

$\displaystyle\quad\delta\,\|{h}\|^{2}=\delta\,(g^{ik}g^{jl}h_{kl}h_{ij})$
$\displaystyle=2\,u\big{(}{h}^{ik}g^{jl}+{h}^{jl}g^{ik}\big{)}h_{kl}h_{ij}+g^{ik}g^{jl}\big{(}(u_{kl}-u{h}^{q}_{k}h_{lq})h_{ij}+(u_{ij}-u{h}^{q}_{i}h_{jq})h_{kl}\big{)}$
$\displaystyle=2\,(uh_{ij}{h}^{ik}{h}^{j}_{k}+u_{kl}\,g^{ik}g^{jl})=2\,u\,\langle{h},\
{h}^{2}\rangle+2\,\langle{h},{\rm Hess}_{\,u}\rangle,$

that proves (11). ∎

###### Proof.

(of Lemma 3) First, using (8) and (10), we get for $1\leq i,j\leq s$ and
$1\leq q\leq n$,

$\displaystyle\delta\,(s\,{H}_{\cal
F})=\delta\,(g^{ij}h_{ij})=2\,u{h}^{ij}\,h_{ij}+g^{ij}\big{(}u_{ij}-u{h}^{q}_{i}\,h_{jq}\big{)}=\Delta_{\,\cal
F}\,u+u\,(\|h_{\cal F}\|^{2}-\|h_{\rm mix}\|^{2}),$

that proves (16). Also for $1\leq i,j,k,l\leq s$ and $1\leq q\leq n$ we obtain
(17):

$\displaystyle\delta\,\|h_{\cal F}\|^{2}=\delta(g^{ik}g^{jl}h_{kl}h_{ij})$
$\displaystyle=2\,u\big{(}{h}^{ik}g^{jl}+{h}^{jl}g^{ik}\big{)}h_{kl}h_{ij}+g^{ik}g^{jl}\big{(}(u_{kl}-u{h}^{q}_{k}h_{lq})h_{ij}+(u_{ij}-u{h}^{q}_{i}h_{jq})h_{kl}\big{)}$
$\displaystyle=4\,u\,\langle h_{\cal F},h_{\cal F}^{2}\rangle+2\,\langle
h_{\cal F},{\rm Hess}^{\,\cal F}_{\,u}\rangle-2\,u\,\langle h_{\cal F},h_{\cal
F}^{2}\rangle-2\,u\,\langle h_{\cal F},h_{\rm mix}^{2}\rangle.$

For $1\leq i,j,k,l\leq s$, $s<\alpha,\beta,\gamma\leq n$ and $1\leq q\leq n$
we obtain (18):

$\displaystyle\delta\,\|h_{\rm
mix}\|^{2}=\delta(g^{ij}g^{\alpha\beta}h_{i\alpha}h_{j\beta})$
$\displaystyle=2\,u\big{(}{h}^{ij}g^{\alpha\beta}+{g}^{ij}h^{\alpha\beta}\big{)}h_{i\alpha}h_{j\beta}+g^{ij}g^{\alpha\beta}\big{(}u_{i\alpha}h_{j\beta}+u_{j\beta}h_{i\alpha}\big{)}-u\,g^{ij}g^{\alpha\beta}\big{(}h^{l}_{i}h_{\alpha
l}h_{j\beta}+h^{l}_{j}h_{\alpha i}h_{l\beta}\big{)}$ $\displaystyle=\langle
h_{\cal F}+h_{\cal F^{\perp}},u\,h_{\rm mix}^{2}\rangle+2\,\langle h_{\rm
mix},{\rm Hess}^{\,\rm mix}_{\,u}\rangle.$

The proof of (19) is similar to the proof of [8, Eq. (19)]: instead of $M^{2}$
we consider $s$-dimensional leaves of $\cal F$. The variation of the
Christoffel symbols is the following tensor, e.g., [8]:

$\displaystyle\delta\,\Gamma^{k}_{ij}=-u\,g^{kl}\big{(}h_{jl,i}+h_{il,j}-h_{ij,l}\big{)}-g^{kl}\big{(}u_{i}h_{jl}+u_{j}h_{il}-u_{l}h_{ij}\big{)}.$
(56)

For the Laplacian $\Delta_{\,\cal
F}\,f=g^{ij}\,(f_{ij}-\Gamma^{k}_{ij}\,f_{k})$ with $1\leq i,j,k\leq s$ it
follows that

$\displaystyle\delta\,(\Delta_{\,\cal
F}\,f)=\delta\,(g^{ij}f_{ij})-\delta\,(g^{ij}\Gamma^{k}_{ij}f_{k}).$ (57)

For the first term we get (for $1\leq i,j\leq s$)

$\displaystyle\delta\,(g^{ij}f_{ij})=2\,u\,h^{ij}f_{ij}+g^{ij}\dot{f}_{ij}=2\,u\,\langle
h_{\cal F},\,{\rm Hess}^{\,\cal F}_{f}\rangle+\Delta_{\,\cal F}\,\dot{f}.$
(58)

For the second term, using (56) and $\Gamma^{k}_{ij}=0$ at $x$, we get for
$1\leq i,j,k\leq s$ and $1\leq q\leq n$,

$\displaystyle\delta\,(g^{ij}\,\Gamma^{k}_{ij}f_{k})=g^{ij}\delta\,(\Gamma^{k}_{ij})f_{k}=-g^{ij}g^{kq}\big{\\{}u\,(h_{jq,i}+h_{iq,j}-h_{ij,q})f_{k}-(u_{i}h_{jq}+u_{j}h_{iq}-u_{q}h_{ij})f_{k}\big{\\}}$
$\displaystyle=-2\,u\,g^{ij}\,g^{kq}h_{jq,i}f_{k}+u\,g^{ij}\,g^{kq}h_{ij,q}f_{k}-2\,g^{ij}\,g^{kq}u_{i}h_{jq}f_{k}+g^{ij}\,g^{kq}\,u_{q}h_{ij}f_{k}.$
(59)

Using the Codazzi-Minardi equation $\nabla_{k}\,h_{ij}-\nabla_{j}\,h_{ik}=0$,
e.g., [11], we get for $1\leq i,j,k,l\leq s$,

$\displaystyle
g^{ij}\,g^{kl}(\nabla_{j}\,h_{il})f_{k}=g^{kl}(g^{ij}\nabla_{l}\,h_{ij})f_{k}=s\,(\nabla^{k}H_{\cal
F})f_{k}=s\,\langle\nabla^{\cal F}H_{\cal F},\,\nabla^{\cal F}f\rangle.$

Thus, using normal coordinates and
$-2\,u\,g^{ij}g^{kl}h_{jl,i}\,f_{k}+u\,g^{ij}g^{kl}h_{ij,l}f_{k}=-u\,g^{kl}(g^{ij}h_{ij,l})f_{k}$,
we get $s\,(H_{\cal F})_{l}=g^{ij}\,\nabla_{l}\,h_{ij}$ for $1\leq i,j,l\leq
s$. Therefore, (5) becomes

$\displaystyle\delta\,(g^{ij}\,\Gamma^{k}_{ij}f_{k})=-u\,g^{kl}(g^{ij}h_{ij,l})f_{k}-2\,g^{ij}g^{kl}u_{i}h_{jl}f_{k}+g^{kl}u_{l}(g^{ij}h_{ij})f_{k}$
$\displaystyle=-s\,u\,\langle\nabla^{\cal F}H_{\cal F},\nabla^{\cal
F}f\rangle-2\,h(\nabla^{\cal F}u,\nabla^{\cal F}f)+s\,H_{\cal
F}\langle\nabla^{\cal F}u,\nabla^{\cal F}f\rangle.$ (60)

Applying (58) and (5) to (57) completes the proof of (19). ∎

###### Proof.

(of Lemma 5) We have $\Delta_{\,\cal F}\,f_{2}={\rm div}_{\cal
F}(\nabla^{\,\cal F}f_{2})$. One can show that ${\rm div}_{\cal F}\,(PX)={\rm
div}\,(PX)-({\rm div}\,P)(PX)$ for all $X\in\mathfrak{X}_{M}$. Hence, using
$\nabla^{\,\cal F}f=P\nabla f$ and (20), we get

$\displaystyle f_{1}\Delta_{\,\cal F}\,f_{2}=f_{1}\,{\rm div}_{\,\cal
F}(\nabla^{\,\cal F}f_{2})=f_{1}\big{\\{}{\rm div}(P\nabla f_{2})-({\rm
div}\,P)(P\nabla f_{2})\big{\\}}$ $\displaystyle={\rm div}\,(f_{1}\,P\nabla
f_{2})-\langle P\nabla f_{1},P\nabla f_{2}\rangle={\rm div}\,(f_{1}\,P\nabla
f_{2})-\langle\nabla^{\,\cal F}f_{1},\nabla^{\,\cal F}f_{2}\rangle.$

Using the Divergence Theorem, gives

$\displaystyle\int_{M}f_{1}(\Delta_{\,\cal F}\,f_{2})\,{\rm
d}V=-\int_{M}\big{\langle}\nabla^{\,\cal F}f_{1},\nabla^{\,\cal
F}f_{2}\big{\rangle}\,{\rm d}V.$ (61)

By this, the formula (22) is true. Next, using (20) we will prove

$\displaystyle\int_{M}\langle\varphi_{1},\nabla^{\cal
F}\varphi_{2}\rangle\,{\rm d}V=\int_{M}\langle\nabla^{\cal
F*}\varphi_{1},\varphi_{2}\rangle\,{\rm d}V$ (62)

for any compactly supported $(s,t)$-tensor $\varphi_{1}$ and $(s,t+1)$-tensor
$\varphi_{2}$. Define a compactly supported 1-form $\omega$ by
$\omega(Y)=\langle\iota_{\,Y}\,\varphi_{2},\,\varphi_{1}\rangle$ for
$Y\in{\mathcal{X}}_{M}$. Take an orthonormal frame $(e_{i})$ such that
$\nabla_{Y}e_{i}=0$ for all $Y\in T_{x}M$ and some $x\in M$. To simplify
calculations, assume that $s=t=1$, then at $x\in M$,

$\displaystyle-\nabla^{*{\cal F}}\omega=\sum\nolimits_{j}(\nabla^{\cal
F}_{e_{j}}\,\omega)(e_{j})=\sum\nolimits_{i,j}\langle{P}e_{j},e_{i}\rangle\,(\nabla_{e_{i}}\,\omega)(e_{j})$
$\displaystyle=\sum\nolimits_{i,j,c}\langle{P}e_{j},e_{i}\rangle\big{(}\langle\nabla_{e_{i}}\varphi_{2}(e_{j},e_{c}),\varphi_{1}(e_{c})\rangle+\langle\varphi_{2}(e_{j},e_{c}),\nabla_{e_{i}}\varphi_{1}(e_{c})\rangle\big{)}$
$\displaystyle=\\!\sum\nolimits_{i,j,c}\big{[}\langle\,\langle{P}e_{j},e_{i}\rangle\nabla_{e_{i}}\varphi_{2}(e_{j},e_{c}),\varphi_{1}(e_{c})\rangle{+}\langle\varphi_{2}(e_{j},e_{c}),\langle{P}e_{j},e_{i}\rangle\nabla_{e_{i}}\varphi_{1}(e_{c})\rangle\big{]}$
$\displaystyle=\sum\nolimits_{j,c}\big{[}\langle\nabla^{\cal
F}_{e_{j}}\varphi_{2}(e_{j},e_{c}),\varphi_{1}(e_{c})\rangle+\langle\varphi_{2}(e_{j},e_{c}),\nabla^{\cal
F}_{e_{j}}\varphi_{1}(e_{c})\rangle\big{]}=\langle\varphi_{2},\nabla^{\cal
F}\varphi_{1}\rangle-\langle\nabla^{*{\cal F}}\varphi_{2},\varphi_{1}\rangle.$

The $\nabla^{{\cal F}*}$ is related to the ${\cal F}$-divergence of a vector
field $\omega^{\sharp}$ by ${\rm div}_{\cal F}\,\omega^{\sharp}=-\nabla^{{\cal
F}*}\,\omega$. By the above and $\int_{M}({\rm div}_{\cal F}\
\omega^{\sharp})\,{\rm d}V=\int_{M}{\rm div}(P\omega^{\sharp})\,{\rm d}V=0$,
we obtain (62). Applying this twice, we get (23). ∎

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