#  $\lambda$-Biharmonic hypersurfaces in the product space
$L^{m}\times\mathbb{R}$

Chao Yang∗, Zhen Zhao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou,
730070, China

> Abstract: In this paper, we study $\lambda$-biharmonic hypersurfaces in the
> product space $L^{m}\times\mathbb{R}$, where $L^{m}$ is an Einstein space
> and $\mathbb{R}$ is a real line. We prove that $\lambda$-biharmonic
> hypersurfaces with constant mean curvature in $L^{m}\times\mathbb{R}$ are
> either minimal or vertical cylinders, and obtain some classification results
> for $\lambda$-biharmonic hypersurfaces under various constraints.
> Furthermore, we investigate $\lambda$-biharmonic hypersurfaces in the
> product space $L^{m}(c)\times\mathbb{R}$, where $L^{m}(c)$ is a space form
> with constant sectional curvature $c$, and categorize hypersurfaces that are
> either totally umbilical or semi-parallel.
>
> Keywords: product space; $\lambda$-biharmonic hypersurfaces; constant mean
> curvature; totally umbilical hypersurfaces; semi-parallel hypersurfaces
>
> Mathematics Subject Classification (2020) 53C50

000This work is supported by National Natural Science Foundation of China (No.
12161078), Science and Technology Project of Gansu Province (No.
20JR5RA515).000E-mail address<EMAIL_ADDRESS>
##  1 Introduction

Let $\varphi:M^{m}\rightarrow N^{m}\times\mathbb{R}$ be an isometric
immersion. Denote by $E(\varphi)$ and $E_{2}(\varphi)$ the energy and bienergy
functionals of $\varphi$, respectively. The functional

$E_{2,\lambda}(\varphi)=E_{2}(\varphi)+\lambda E(\varphi),$

is called the $\lambda$-bienergy functional of $\varphi$. We say that a map
$\varphi$ is $\lambda$-biharmonic if it is a critical point of the functional
$E_{2,\lambda}(\varphi)$. The Euler-lagrange equation of
$E_{2,\lambda}(\varphi)$ yields

$\tau_{2}(\varphi)-\lambda\tau(\varphi)=0,$ (1.1)

where

$\displaystyle\quad\quad\quad\quad\ \ \tau(\varphi)=tr(\nabla d\varphi),$
$\displaystyle\tau_{2}(\varphi):=$
$\displaystyle\Delta\tau(\varphi)-tr_{g}\widetilde{R}(d\varphi,\tau(\varphi)d\varphi)$
$\displaystyle=$ $\displaystyle
tr_{g}(\nabla^{\varphi}\nabla^{\varphi}\tau(\varphi)-\nabla^{\varphi}_{\nabla^{M}}\tau(\varphi))-tr_{g}\widetilde{R}(d\varphi,\tau(\varphi)d\varphi),$

$\nabla^{\varphi}$ and $\nabla$ are induced connections of vector bundle
$\varphi^{-1}TN^{m+1}$ and Levi-Civita connection on $M^{m}$ respectively, and
$\widetilde{R}$ is the curvature operator on $N^{m+1}$.

Special cases are discussed, such as when $\lambda=0$ leading to automatically
biharmonic hypersurfaces. In the case of the ambient space $N^{m+1}$ having
constant curvature $c$, the equation (1.1) simplifies to
$\Delta\overrightarrow{H}=(mc-\lambda)\overrightarrow{H}$, where
$\overrightarrow{H}$ is the mean curvature vector field, defining
hypersurfaces with proper mean curvature vector fields.

The first result on $\lambda$-biharmonic hypersurfaces was obtained in [5] by
B. Y. Chen in 1988. It was proved that every $\lambda$-biharmonic hypersurface
in $\mathbb{R}^{3}$ is either minimal or an open portion of a circular
cylinder. Furtherly, A. Ferrández and P. Lucas showed in [9] that
$\lambda$-biharmonic hypersurfaces in $\mathbb{R}^{m+1}$ with at most two
distinct principal curvatures are minimal or locally isometric to
$\mathbb{R}^{k}\times S^{m-k}(a)$. For $\lambda$-biharmonic hypersurfaces with
three or more distinct principal curvatures, it is difficult to obtain
classification results. But geometers conjectured that their mean curvatures
are constants. In 1995, T. Hasanis and T. Vlachos showed that
$\lambda$-biharmonic hypersurfaces in $\mathbb{R}^{4}$ have constant mean
curvatures in [13]. In 2015, Y. Fu derived in [11] that $\lambda$-biharmonic
hypersurface in $\mathbb{R}^{m+1}$ with at most three distinct principal
curvatures have constant mean curvatures. Afterwards, Y. Fu and X. Zhan proved
in [12] that $\lambda$-biharmonic hypersurfaces in $\mathbb{R}^{5}$ have
constant mean curvatures.

For $\lambda$-biharmonic hypersurfaces in non-flat space forms, C. Yang, J.-C.
Liu and L. Du gave in [20] classification results under the condition that
hypersurfaces have at most two distinct principal curvatures. Scholars have
also discussed $\lambda$-biharmonic hypersurfaces in pseudo-Riemannian space
forms (see [1, 2, 3, 7, 8, 10, 14, 15, 16, 19, 20]), aiming to prove that
their mean curvatures are constants.

Naturally, it is meaningfull to study $\lambda$-biharmonic hypersurfaces in
the product space $L^{m}\times\mathbb{R}$ of an Einstein space $L^{m}$ and a
real line $\mathbb{R}$. The focus of this paper is on $\lambda$-biharmonic
hypersurfaces in the product space $L^{m}\times\mathbb{R}$. Significant
results are presented, showing that hypersurfaces with constant mean curvature
in $L^{m}\times\mathbb{R}$ are either minimal or vertical cylinders. Moreover,
classifications for hypersurfaces in $L^{m}(c)\times\mathbb{R}$ as totally
umbilical or semi-parallel are discussed.

##  2 Preliminaries

Let $\varphi:M^{m}\rightarrow(N^{m}\times\mathbb{R},g^{N}+\text{d}t^{2})$ be a
hypersurface with unit normal vector $\xi$. $T$ denotes the tangential
component of $\partial_{t}$ along the tangent space to $M^{m}$ and $\alpha$ be
the angle made by $\partial_{t}$ and $\xi$. We can decompose $\partial_{t}$ in
the following form

$\partial_{t}=T+\cos\alpha\xi,$

where $\cos\alpha=\langle\partial_{t},\xi\rangle$. Let $\widetilde{Ric}$ be
the Ricci curvature tensor of $N^{m}\times\mathbb{R}$, $A$ and $H$ be the
shape operator and the mean curvature of the hypersurface $M^{m}$, then the
Laplacian of $\theta:=\cos\alpha$ is given by

$\Delta\theta=-m\langle\nabla
H,\partial_{t}\rangle-\theta(|A|^{2}+\widetilde{Ric}(\xi,\xi)).$ (2.1)

Since $\partial_{t}$ is parallel in $N^{m}\times\mathbb{R}$, we have

$\left\\{\begin{array}[]{l}\nabla_{X}T=\cos\alpha AX\\\ X(\cos\alpha)=-\langle
AX,T\rangle,\end{array}\right.$ (2.2)

for every tangent vector field $X$ on $M^{m}$.

Let $\widetilde{Ric}$, $\widetilde{R}$ and $\widetilde{S}$ be the Ricci
curvature tensor, Riemannian curvature tensor, and scalar curvature of
$N^{m+1}\times\mathbb{R}$, respectively. Denote by $Ric$, $R$ and $S$
corresponding ones of $M^{m}$. Then the relationships of Ricci curvature
tensors and scalar curvatures between the hypersurface and the ambient space
are given by

$\widetilde{Ric}(X,Y)=Ric(X,Y)+\langle AX,AY\rangle-mH\langle
AX,Y\rangle+\widetilde{R}(X,\xi,Y,\xi),$

and

$\widetilde{S}=S+|A|^{2}-m^{2}H^{2}+2\widetilde{Ric}(\xi,\xi).$ (2.3)

The second fundamental form $B$ and the shape operator $A$ satisfy

$\langle B(X,Y),\xi\rangle=\langle A(X),Y\rangle.$

Moreover, the Codazzi equation is given by

$(\widetilde{R}(X,Y)Z)^{\bot}=(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z).$

If the hypersurface $M^{m}$ is totally umbilical if and only if

$\langle B(X,Y),\xi\rangle=\langle X,Y\rangle H.$

We say the hypersurface $M^{m}$ is semi-parallel if and only if

$B(R(X,Y)U,V)+B(U,R(X,Y)V)=0,$

for any $X,Y,U,V\in\Gamma(M)$.

In this paper, we denote by $L^{m}$ and $L^{m}(c)$ the $m$-dimensional
Einstein manifold and the space with constant curvature $c=1,-1$ or $0$,
respectively. For simplicity, we call $L^{m}\times\mathbb{R}$ Einstein product
space, and $L^{m}(c)\times\mathbb{R}$ constant curvature product space. The
Riemannian curvature tensor of $L^{m}(c)\times\mathbb{R}$ is given by ([13])

$\displaystyle\widetilde{R}(X,Y)Z=$ $\displaystyle c\\{\langle Y,Z\rangle
X-\langle X,Z\rangle Y-\langle Y,\partial_{t}\rangle\langle
Z,\partial_{t}\rangle X+\langle X,\partial_{t}\rangle\langle
Z,\partial_{t}\rangle Y$ $\displaystyle+\langle X,Z\rangle\langle
Y,\partial_{t}\rangle\partial_{t}-\langle Y,Z\rangle\langle
X,\partial_{t}\rangle\partial_{t}\\},$

where $X,Y,Z$ are vector fields on $L^{m}(c)\times\mathbb{R}$.

Recall in [6], that a rotation hypersurface $M^{m}$ in the product space
$S^{m}\times\mathbb{R}$, parametrizing the profile curve as

$\gamma(s)=(\cos s,0,\cdots,0,\sin s,h(s)),$

for some smooth function $h$, then the rotation hypersurface $M^{m}$ can be
parametrized as

$f(s,v_{1},\cdots,v_{m-1})=(\cos s,\varphi_{1}(v_{1},\cdots,v_{m-1})\sin
s,\cdots,\varphi_{m}(v_{1},\cdots,v_{m-1})\sin s,h(s)),$

where $\varphi=(\varphi_{1},\cdots\varphi_{m})$ is an orthogonal
parametrization of the unit sphere $S^{m-1}$ in $\mathbb{R}^{m}$, we compute
the principal curvatures as follows

$\lambda_{1}=-\frac{h^{\prime\prime}(s)}{(1+h^{\prime}(s)^{2})^{\frac{3}{2}}},\
\ \lambda_{2}=-\frac{h^{\prime}(s)\cot
s}{(1+h^{\prime}(s)^{2})^{\frac{1}{2}}}.$

The similar argument for a rotation hypersurface $M^{m}$ in a product space
$H^{m}\times\mathbb{R}$, the principal curvatures as follows

$\lambda_{1}=-\frac{h^{\prime\prime}(s)}{(1+h^{\prime}(s)^{2})^{\frac{3}{2}}},\
\ \lambda_{2}=-\frac{h^{\prime}(s)\tan
a(s)}{(1+h^{\prime}(s)^{2})^{\frac{1}{2}}}.$

To obtain main results, the following lemmas are required.

##  3 Some Lemmas

Lemma 3.1  _Let
$\varphi:(M^{m},g)\rightarrow(N^{m}\times\mathbb{R},g^{N}+\text{d}t^{2})$ be
an isometric immersion. Then $\varphi$ is $\lambda$-biharmonic if and only if
both its component maps $\pi_{1}\circ\varphi:(M,g)\rightarrow(N,g^{N})$ and
$\pi_{2}\circ\varphi:(M,g)\rightarrow(\mathbb{R},\text{d}t^{2})$ are
$\lambda$-biharmonic maps. In particular, the height function
$h=\pi_{2}\circ\varphi$ of a $\lambda$-biharmonic hypersurface is a
$\lambda$-biharmonic function on the hypersurface._

Proof Since

$\text{d}\varphi(X)=\text{d}(\pi_{1}\circ\varphi)(X)+\text{d}(\pi_{2}\circ\varphi)(X),\forall
X\in\Gamma M,$

we know

$\nabla^{\varphi}_{X}(\text{d}\varphi(Y))=\nabla^{\varphi}_{X}(\text{d}(\pi_{1}\circ\varphi))(Y)+\nabla^{\varphi}_{X}(\text{
d}(\pi_{2}\circ\varphi))(Y),\forall X,Y\in\Gamma M.$

It follows that

$\displaystyle\tau(\varphi)=$
$\displaystyle\sum^{m}_{i=1}\\{\nabla^{\varphi}_{e_{i}}(\text{d}\varphi(e_{i}))-\text{
d}\varphi(\nabla_{e_{i}}{e_{i}})\\}$ $\displaystyle=$
$\displaystyle\sum^{m}_{i=1}\\{\nabla_{e_{i}}^{\pi_{1}\circ\varphi}(\text{d}(\pi_{1}\circ\varphi)(e_{i}))-\text{d}(\pi_{1}\circ\varphi)(\nabla^{M}_{e_{i}}e_{i})\\}$
$\displaystyle+\sum^{m}_{i=1}\\{\nabla_{e_{i}}^{\pi_{2}\circ\varphi}\text{d}(\pi_{2}\circ\varphi)(e_{i})-\text{
d}(\pi_{2}\circ\varphi)(\nabla^{M}_{e_{i}}e_{i})\\}$ $\displaystyle=$
$\displaystyle\tau(\pi_{1}\circ\varphi)+\tau(\pi_{2}\circ\varphi).$

Since

$\tau_{2}(\varphi)=-J^{\varphi}(\tau(\varphi)),$
$J^{\varphi}(X):=-\\{tr(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})X-tr\widetilde{R}(\text{d}\varphi,X)\text{d}\varphi\\},$

we get

$\displaystyle\tau_{2}(\varphi)$ $\displaystyle=-J^{\varphi}(\tau(\varphi))$
$\displaystyle=-J^{\pi_{1}\circ\varphi}(\tau(\pi_{1}\circ\varphi))-J^{\pi_{2}\circ\varphi}(\tau(\pi_{2}\circ\varphi))$
$\displaystyle=\tau_{2}(\pi_{1}\circ\varphi)+\tau_{2}(\pi_{2}\circ\varphi).$

That $\varphi$ is $\lambda$-biharmonic means that
$\tau_{2}(\varphi)-\lambda\tau(\varphi)=0$, i.e.

$\left\\{\begin{array}[]{l}\tau_{2}(\pi_{1}\circ\varphi)-\lambda\tau(\pi_{1}\circ\varphi)=0,\\\
\tau_{2}(\pi_{2}\circ\varphi)-\lambda\tau(\pi_{2}\circ\varphi)=0.\end{array}\right.$

Then we obtain the lemma. $\square$

Lemma 3.2  _Let
$\varphi:(M^{m},g)\rightarrow(N^{m}\times\mathbb{R},g^{N}+\text{d}t^{2})$ be
an isometric immersion. If the height function $h=\pi_{2}\circ\varphi$ is
$\lambda$-biharmonic, then we have_

$\Delta^{2}h=\lambda\Delta h.$

Proof Calculate $\tau(h)$, we derive

$\displaystyle\tau(h)$ $\displaystyle=tr(\nabla\text{d}h)$
$\displaystyle=\nabla_{e_{i}}(\text{d}h(e_{i}))-\text{d}h(\nabla_{e_{i}}e_{i})$
$\displaystyle=e_{i}e_{i}h-(\nabla_{e_{i}}e_{i})h$ $\displaystyle=\Delta h.$

Considering the curvature operator $\bar{R}$ on Euclidean space is zero, we
know that

$\tau_{2}(h)=\Delta\tau(h)-tr\bar{R}(\text{d}h,\tau(h)dh)=\Delta^{2}h.$

So, $\tau_{2}(h)=\lambda\tau(h)$ is equivalent to $\Delta^{2}h=\lambda\Delta
h$. $\square$

Lemma 3.3  _Let $M^{m}$ is a $\lambda$-biharmonic hypersurface in
$N^{m}\times\mathbb{R}$. Then we have the following identity_

$\Delta(H\theta)=\lambda H\theta.$ (3.1)

Proof Let $h$ be the height function of the hypersurface, we know from [3]
that $\Delta h=m\theta H$. Furthermore

$\Delta^{2}h=\Delta(\Delta h)=m\Delta(H\theta),$

which together with $\Delta^{2}h=\lambda\Delta h$ (cf. Lemma 3.2) deduced the
lemma. $\square$

Lemma 3.4([21])  _(Yau’s maximum principle)_

  * •

_Let_ $M^{m}$ is a complete Riemannian manifold with nonnegative Ricci
curvature and $u$ is a positive smooth harmonic function. Then $u$ is a
constant function;

  * •

_Let_ $M^{m}$ is a complete Riemannian manifold and $u$ is a nonnegative
smooth subharmonic function. Then $\int_{M}u^{p}=+\infty$ for $p>1$, unless
$u$ is a constant function.

Lemma 3.5([17])  _Let $u\in(0,C](C>0)$ be a superharmonic function on complete
noncompact manifold $M^{m}$. If_

$\int_{M}(\log^{(k)}\frac{Ce^{(k)}}{u})^{p}\text{d}v_{g}<+\infty,\text{f}or\text{s}omep>0,k\in\mathbb{N},$

_where
$\log^{(1)}=\log,e^{(1)}=e,\log^{(k)}=\log(\log^{(k-1)}),e^{(k)}=e^{e^{(k-1)}}$,
then $u$ is a constant._

Lemma 3.6  _A hypersurface $\varphi:M^{m}\rightarrow N^{m+1}$ in a Riemannian
manifold is $\lambda$-biharmonic if and only if_

$\left\\{\begin{array}[]{l}\Delta H-H|A|^{2}+H\widetilde{Ric}(\xi,\xi)-\lambda
H=0,\\\ 2A(\nabla H)+\frac{m}{2}\nabla
H^{2}-2H(\widetilde{Ric}(\xi))^{T}=0,\end{array}\right.$ (3.2)

_where $\widetilde{Ric}$ denotes Ricci curvature operator on $N^{m+1}$, and
$\langle\widetilde{Ric}(Z),W\rangle:=\widetilde{Ric}(Z,W)$ for any
$Z,W\in\Gamma M$. _

Proof Choose a local orthonormal frame $\\{e_{i}\\},i=1,\cdots,m$ on $M^{m}$,
such that $\\{d\varphi(e_{1}),\cdots,d\varphi(e_{m}),\xi\\}$ is an local
orthonormal frame of $N^{m+1}$. Notice that $\tau(\varphi)=mH\xi$, we compute
the bitension field of $\varphi$

$\displaystyle\tau_{2}(\varphi)$
$\displaystyle=\sum_{i=1}^{m}\\{\nabla^{\varphi}_{e_{i}}\nabla^{\varphi}_{e_{i}}(mH\xi)-\nabla^{\varphi}_{\nabla_{e_{i}}{e_{i}}}(mH\xi)-\widetilde{R}(\text{d}\varphi(e_{i}),mH\xi)\text{d}\varphi(e_{i})\\}$
$\displaystyle=m(\Delta H)\xi-2mA(\nabla H)-mH\Delta^{\varphi}\xi-
mH\sum_{i=1}^{m}\widetilde{R}(\text{d}\varphi(e_{i}),\xi)\text{d}\varphi(e_{i}),$

where $\tilde{R}$ denotes the Riemannian curvature operator of the ambient
space $N^{m+1}$.

As

$\sum^{m}_{i=1}\langle\widetilde{R}(\text{d}\varphi(e_{i}),\tau(\varphi))\text{d}\varphi(e_{i}),\xi\rangle=-mH\widetilde{Ric}(\xi,\xi),$

and

$\displaystyle\langle\Delta^{\varphi}\xi,\xi\rangle$
$\displaystyle=\sum^{m}_{i=1}\langle\widetilde{\nabla}_{e_{i}}\widetilde{\nabla}_{e_{i}}\xi-\widetilde{\nabla}_{\nabla_{e_{i}}{e_{i}}}\xi,\xi\rangle$
$\displaystyle=\sum^{m}_{i=1}\langle\widetilde{\nabla}_{e_{i}}\xi,\widetilde{\nabla}_{e_{i}}\xi\rangle=|A|^{2},$

the normal part of $\tau_{2}(\varphi)-\lambda\tau(\varphi)$ can be expressed
as

$\displaystyle(\tau_{2}(\varphi)-\lambda\tau(\varphi))^{\bot}$
$\displaystyle=\langle\tau_{2}(\varphi)-\lambda\tau(\varphi),\xi\rangle\xi$
$\displaystyle=\Delta H-H|A|^{2}+H\widetilde{Ric}(\xi,\xi)-\lambda H.$

Because of

$\displaystyle\langle\Delta^{\varphi}\xi,e_{k}\rangle e_{k}$
$\displaystyle=\sum^{m}_{i,k=1}\langle\widetilde{\nabla}_{e_{i}}\widetilde{\nabla}_{e_{i}}\xi-\widetilde{\nabla}_{\nabla_{e_{i}}{e_{i}}}\xi,e_{k}\rangle
e_{k}$ $\displaystyle=m(\nabla H)-(\widetilde{Ric}(\xi))^{\top},$

and

$\sum^{m}_{i,k=1}\langle\widetilde{R}(d\varphi(e_{i}),\xi)d\varphi(e_{i}),e_{k}\rangle
e_{k}=-[\widetilde{Ric}(\xi,e_{k})]e_{k}=-(\widetilde{Ric}(\xi))^{\top},$

the tangent part of $\tau_{2}(\varphi)-\lambda\tau(\varphi)$ is

$\displaystyle(\tau_{2}(\varphi)-\lambda\tau(\varphi))^{\top}$
$\displaystyle=\langle\tau_{2}(\varphi)-\lambda\tau(\varphi),e_{k}\rangle
e_{k}$ $\displaystyle=-2A(\nabla H)-\frac{m}{2}(\nabla
H^{2})+2H(\widetilde{Ric}(\xi))^{\top}).$

The hypersurface $M^{m}$ is $\lambda$-biharmonic if and only if the tangent
and normal parts of $\tau_{2}(\varphi)-\lambda\tau(\varphi)$ are equal to
zero, i.e. (3.2) formula holds. $\square$

##  4 $\lambda$-biharmonic hypersurface in $L^{m}\times\mathbb{R}$

Using Lemmas in Section 3, we can give some classification results for
$\lambda$-biharmonic hypersurface in a product space $L^{m}\times\mathbb{R}$
of an Einstein space $L^{m}$ and a real line $\mathbb{R}$. In the following,
we classify the $\lambda$-biharmonic hypersurface with constant mean curvature
in such a product space.

Theorem 4.1  _Let $L^{m}$ be an Einstein manifold and $M^{m}$ is a
$\lambda$-biharmonic $(\lambda\geq 0)$ hypersurface in $L^{m}\times\mathbb{R}$
with constant mean curvature. Then $M^{m}$ is minimal, or a vertical cylinder
over a $\lambda$-biharmonic hypersurface in $L^{m}$._

Proof If $H=0$, then $M^{m}$ is minimal.

If $H\neq 0$, then (3.1) can be reduced to

$\Delta\theta=\lambda\theta.$ (4.1)

It is easy to find that

$\widetilde{Ric}(\xi,\xi)=\mu(1-\theta^{2}),$ (4.2)

where $\mu$ is a constant.

Substitute (4.2) into (2.1) and the first equation of (3.2), and considering
$H$ is a constant, we have

$\Delta\theta=-\theta(2|A|^{2}+\lambda).$ (4.3)

Combine (4.1) and (4.3), we have $\theta(|A|^{2}+\lambda)=0$. As $|A|^{2}>0$,
$\lambda\geq 0$, the equation $\theta(|A|^{2}+\lambda)=0$ implies
$\theta\equiv 0$, i.e. $\partial_{t}$ is tangent to the hypersurface. So, we
conclude that the hypersurface $M^{m}$ is a vertical cylinder over a
$\lambda$-biharmonic hypersurface in $L^{m}$. $\square$

Theorem 4.2  _Let $L^{m}$ be an Einstein manifold and $M^{m}$ is a complete
$\lambda$-biharmonic $(\lambda\geq 0)$ hypersurface in $L^{m}\times\mathbb{R}$
with constant angle function. If the mean curvature $H$ is nonnegative and
$H\in L^{p}(M),1<p<\infty$, then $M^{m}$ is minimal, or a vertical cylinder
over a $\lambda$-biharmonic hypersurface in $L^{m}$._

Proof If $\theta=\langle\xi,\partial_{t}\rangle\equiv 0$, then the
hypersurface $M^{m}$ is a vertical cylinder over a $\lambda$-biharmonic
hypersurface in $L^{m}$.

If the constant $\theta\neq 0$, then, we have $\Delta H=\lambda H$ by (3.1).
As $\lambda\geq 0$ and $H$ is nonnegative, it follows that $\Delta H\geq 0$.
Applying $\text{Y}au^{,}s$ maximum principle, we find that $H$ is a constant.
We complete the proof by using Theorem 4.1. $\square$

Theorem 4.3  _Let $L^{m}$ be an Einstein manifold and $M^{m}$ is a totally
umbilical $\lambda$-biharmonic $(\lambda\geq 0)$ hypersurface in
$L^{m}\times\mathbb{R}$ with constant angle function. Then $M^{m}$ is minimal,
or a vertical cylinder over a $\lambda$-biharmonic hypersurface in $L^{m}$._

Proof Choose $\\{e_{1},e_{2},\cdots e_{m}\\}$ be a local orthonormal frame on
$M^{m}$, then $A(e_{i})=He_{i},i=1,2,\cdots m$ and $|A|^{2}=mH^{2}$.

Substitute $\widetilde{Ric}(\xi,\xi)=\mu(1-\theta^{2})$ into the first
equation of (3.2), we have

$\Delta H-mH^{3}+H\mu(1-\theta^{2})-\lambda H=0.$

If $\theta\neq 0$, we know $\Delta H=\lambda H$ from (3.1). So the above
equation can be reduced to

$mH^{3}=H\mu(1-\theta^{2}),$

which means that $H$ is a constant. Applying Theorem 4.1, we finish the proof.
$\square$

Proposition 4.4  _Let $M^{m}$ is a complete $\lambda$-biharmonic $(\lambda\geq
0)$ hypersurface with non-negative Ricci curvature in $L^{m}\times\mathbb{R}$
. Assume that_

$\int_{M}H^{2p}dv_{g}<+\infty,$

_and_

$\int_{M}(\log^{(k)}\frac{e^{(k)}}{\theta^{2}+\varepsilon})^{q}dv_{g}<+\infty,$

_for some $p>1,q>0,k\in\mathbb{N}$ and $\varepsilon>0$. Then $M^{m}$ is
minimal, or a vertical cylinder over a $\lambda$-biharmonic hypersurface in
$L^{m}$._

Proof It follows from Lemma 3.3 that

$\Delta(H\theta)^{2}=2|\nabla(H\theta)|^{2}+2\lambda(H\theta)^{2}\geq 0.$

As $-1\leq\theta\leq 1$, we have

$\int_{M}{(H\theta)^{2}}^{p}dv_{g}\leq\int_{M}H^{2p}dv_{g}<+\infty,p>1,$

Then by Yau’s maximum principle, we conclude that $H\theta$ is a constant.

As $\Delta h=mH\theta$ and $H\theta$ is a constant, the Ricci identity

$\Delta\nabla_{i}h=\nabla_{i}\Delta h+Ric_{ij}\nabla_{j}h.$

can be simplify as

$\Delta\nabla_{i}h=Ric_{ij}\nabla_{j}h.$

From $h=\pi_{2}\circ\varphi$, we have

$\langle\Delta T,T\rangle=Ric(T,T).$

Due to

$\frac{1}{2}\Delta|T|^{2}=|\nabla T|^{2}+\langle\Delta T,T\rangle=|\nabla
T|^{2}+Ric(T,T)\geq 0.$ (4.4)

Also because of $\partial_{t}=T+\theta\xi$, we have $\langle
T,T\rangle=1-\theta^{2}$. So

$\frac{1}{2}\Delta|T|^{2}=\frac{1}{2}\Delta(1-\theta^{2})=-\frac{1}{2}\Delta(1+\theta^{2}).$
(4.5)

Combining (4.4) and (4.5), we have

$-\frac{1}{2}\Delta(\theta^{2}+1)\geq 0.$

Therefore $\Delta(\theta^{2}+1)\leq 0$, that is $\theta^{2}+\varepsilon$ is a
superharmonic function. From Lemma 3.5, we obtain that $\theta^{2}+1$ is a
constant, i.e. $\theta$ is a constant. It together with that $H\theta$ is a
constant imply that $H$ is also a constant. By Theorem 4.1, the proof is
completed. $\square$

Proposition 4.5  _let $M^{m}$ be a complete $\lambda$-biharmonic hypersurface
with non-negative Ricci curvature in a product space $L^{m}\times\mathbb{R}$.
Assume that_

  * •

$H$_is harmonic and bounded from below, or_

  * •

$\theta$_is harmonic and scalar curvature of_ $M^{m}$ is a constant.

_Then $M^{m}$ is either minimal, or a vertical cylinder over a
$\lambda$-biharmonic hypersurface in $L^{m}$._

Proof (i) Assume that $\Delta H=0$ and $H\geq-C$ for some positive $C$, take
$u=H+C+\varepsilon>0$, where $\varepsilon$ is some positive constant. Then
$\Delta u=\Delta H=0$. According to Yau’s maximum principle, we know $u$ is a
constant. Hence, $H=u-C-\varepsilon$ is a constant. By Theorem 4.1, the result
follows.

(ii)We suppose that $\Delta\theta=0$ and the scalar curvature $S$ is a
constant. Set $u=\theta+2$, then $u>0$ and $\Delta u=\Delta\theta=0$. It
follows from Yau’s maximum principle that $u$ and $\theta$ are constant.

If $\theta\equiv 0$, then the $\lambda$-biharmonic hypersurface is a vertical
cylinder over a $\lambda$-biharmonic hypersurface in $L^{m}$.

If $\theta\neq 0$, we have $\Delta H=\lambda H$. Then using the equation
$\widetilde{Ric}(\xi,\xi)=\mu(1-\theta^{2})$, the first equation of (3.2) use
can be rewritten as

$H[|A|^{2}-\mu(1-\theta^{2})]=0.$

When $H\equiv 0$, then $M^{m}$ is minimal. When $H\neq 0$ at some point $p\in
M$, then the above formula tells us that

$|A|^{2}=\mu(1-\theta^{2}).$

By (2.3), one get $S=\widetilde{S}-3\mu(1-\theta^{2})+m^{2}H^{2}$. Noting that
$\widetilde{S}=\mu m$. It follows that $\nabla S=2m^{2}H\nabla H$. We find
from $S$ is a constant $\nabla S=0$, which together with $2m^{2}H\nabla
H=\nabla S$ deduced that $H$ is a constant.

Next we can use Theorem 4.1 to complete the proof.

In particular, when the product space is $L^{m}(c)\times\mathbb{R}$, we have
the following Theorem. $\square$

##  5 $\lambda$-biharmonic hypersurface in $L^{m}(c)\times\mathbb{R}$

Lemma 5.1  _A hypersurface $M^{m}$ in a product space
$L^{m}(c)\times\mathbb{R}$ is $\lambda$-biharmonic if and only if_

$\left\\{\begin{array}[]{l}\Delta
H-H[|A|^{2}-c(m-1)\sin^{2}\alpha+\lambda]=0,\\\ A(\nabla H)+\frac{m}{2}H\nabla
H+c(m-1)\cos\alpha HT=0.\end{array}\right.$ (5.1)

Proof Set $\\{e_{i}\\},i=1,\cdots,m$ be a local orthonormal frame on $M^{m}$.
Using (2.3), we get

$\widetilde{Ric}(\xi,\xi)=\sum^{m}_{i=1}\langle\widetilde{R}(e_{i},\xi)\xi,e_{i}\rangle=c(m-1)\sin^{2}\alpha$

and

$(\widetilde{Ric}(\xi))^{T}=\sum^{m}_{i=1}\langle\widetilde{R}(e_{i},\xi)e_{k},e_{i}\rangle
e_{k}=-c(m-1)\cos\alpha T.$

Substituting the above two equations into $\lambda$-biharmonic equations
(3.2), we obtain the lemma. $\square$

Theorem 5.2  _A rotation hypersurface $M^{m}$ in $L^{m}(c)\times\mathbb{R}$ is
$\lambda$-biharmonic, then we have_

$(\frac{m}{2}H-\alpha^{\prime}(s)\cos\alpha)H^{\prime}-c(m-1)\sin\alpha H=0.$
(5.2)

Proof As in [6], we choose a local orthonormal frame
$\\{e_{1},e_{2},\cdots,e_{m}\\}$, so that

$\nabla H=e_{1}(H)e_{1}=-\cos\alpha H^{\prime},$
$A(e_{1})=\lambda_{1}e_{1},A(e_{i})=\lambda_{i}e_{i},2\leq i\leq m.$

Combining $\langle T,T\rangle=\sin^{2}\alpha$, the second equation of (5.1)
can be rewriten as

$(\frac{m}{2}H+\lambda_{1})H^{\prime}-c(m-1)\sin\alpha H=0.$

$\square$

Theorem 5.3  _Any totally umbilical $\lambda$-biharmonic ($\lambda\geq 0$)
hypersurface $M^{m}$ in $L^{m}(c)\times\mathbb{R}$ is minimal._

Proof Since $M^{m}$ is a totally umbilical $\lambda$-biharmonic hypersurface,
we have $|A|^{2}=mH^{2}$ and $A(\nabla H)=H\nabla H$. Then the equation (5.1)
can be rewritten as

$\left\\{\begin{array}[]{l}\Delta
H-H[mH^{2}-c(m-1)\sin^{2}\alpha+\lambda]=0\\\ \frac{m+2}{2}\nabla
H+c(m-1)\cos\alpha T=0.\end{array}\right.$ (5.3)

If $H\equiv 0$, then $M^{m}$ is minimal.

Now, we assume that $H\neq 0$ on an open set $\Omega$.

(i) When $\sin\alpha\equiv 0$, we know $\cos\alpha=\pm 1$. Take $X=\nabla H$
in the second equation of (2.2), and combine $A(\nabla H)=H\nabla H$ and the
second equation of (5.3), we have $H|\nabla H|^{2}=0$, which implies that
$|\nabla H|=0$, i.e. $H$ is non-zero constant. Thus, we derive from the first
equation of (5.3) that $\lambda=-mH^{2}$, which contradicts with $\lambda\geq
0$.

(ii) When $\sin\alpha\neq 0$ at some point $p\in M^{m}$. Considering
$|T|^{2}=\sin^{2}\alpha$, we can choose a local orthonormal frame
$\\{e_{1},e_{2},\cdots,e_{m}\\}$ on $M^{m}$, such that

$T=\sin\alpha e_{1}.$ (5.4)

Combining $\nabla H=\sum_{i}e_{i}(H)e_{i}$ and (5.4), we obtain from the
second equation of (5.3) that

$e_{2}(H)=e_{3}(H)=\cdots=e_{m}(H)=0,$

and

$e_{1}(H)=-\frac{2c(m-1)}{m+2}\sin\alpha\cos\alpha.$ (5.5)

Using the first equation of (2.2), we have

$\langle\cos
Ae_{j},e_{j}\rangle=H\cos\alpha=\langle\nabla_{e_{j}}T,e_{j}\rangle=e_{j}\langle
T,e_{j}\rangle-\langle T,\nabla_{e_{j}}e_{j}\rangle=-\sin\alpha\langle
e_{1},\nabla_{e_{j}}e_{j}\rangle,$

which deduce that

$\langle e_{1},\nabla_{e_{j}}e_{j}\rangle=-H\cot\alpha.$

Then we have

$\displaystyle\Delta H$
$\displaystyle=e_{1}e_{1}(H)+\sum_{i}(\nabla_{e_{i}}e_{i})(H)$ (5.6)
$\displaystyle=e_{1}e_{1}(H)+(m-1)\cot\alpha He_{1}(H).$

Put $X=e_{1}$ in the second equation of (2.2), and combine $T=\sin\alpha
e_{1}$, we obtain

$H=e_{1}(\alpha).$ (5.7)

Differentiating (5.5) along $e_{1}$, we derive

$e_{1}e_{1}(H)=-\frac{2c(m-1)}{m+2}\cos(2\alpha)H.$ (5.8)

Substitute (5.6) into the first equation of (5.3), and combine (5.7) and
(5.8), we have

$-e_{1}(\alpha)(\frac{2c(m-1)}{m+2}\cos(2\alpha)+\frac{2c(m-1)^{2}}{m+2}\cos^{2}\alpha-c(m-1)\sin^{2}\alpha+m(e_{1}(\alpha))^{2}+\lambda)=0.$
(5.9)

If $e_{1}(\alpha)\equiv 0$, then $H=e_{1}(\alpha)\equiv 0$, a contradiction.

If $e_{1}(\alpha)\neq 0$ on some neighborhood $\Omega$ on $M^{m}$, then the
equation (5.9) reads

$\frac{2c(m-1)}{m+2}\cos(2\alpha)+\frac{2c(m-1)^{2}}{m+2}\cos^{2}\alpha-c(m-1)\sin^{2}\alpha+m(e_{1}(\alpha))^{2}+\lambda=0.$
(5.10)

It is shown ([4]) that the Sine-Gordon equation

$e_{1}e_{1}(2\alpha)+c\sin(2\alpha)=0.$ (5.11)

Differentiating (5.10) along $e_{1}$ and combining (5.11), we have

$2c\sin\alpha\cos\alpha[\frac{4(m-1)+2(m-1)^{2}+(m-1)(m+2)+m(m+2)}{m+2}]=0,$
(5.12)

If $\sin\alpha=1$, then $\cos\alpha=0$, we have $\alpha$ is a constant, so
$e_{1}(\alpha)=0$, a contradiction. Hence, (5.12) is equivalent to

$4m^{2}+3m-4=0,$

which implies that $m$ is a non-positive integer, a contradiction. $\square$

Theorem 5.4  _Any semi-parallel $\lambda$-biharmonic ($\lambda\geq 0$)
hypersurface in $S^{m}\times\mathbb{R}(m\geq 3)$ is minimal or a vertical
cylinder over a $\lambda$-biharmonic hypersurface in $S^{m}$._

Proof Let $M^{m}$ is a semi-parallel hypersurface in
$S^{m}\times\mathbb{R}(m\geq 3)$, then according to [18], one of the following
is true,

(I) $M^{m}$ is totally umbilical;

(II) $M^{m}$ is an open part of rotation hypersurface with
$\lambda_{1}\lambda_{2}=-\cos^{2}\alpha$, which $\lambda_{1}$ and
$\lambda_{2}$
are principal curvatures in hypersurface $M^{m}$;

(III) $M^{m}\subset\tilde{M}^{m-1}\times\mathbb{R}$, where $\tilde{M}$ is a
semi-parallel hypersurface of $S^{m}$.

Suppose $M^{m}$ satisfies $\lambda_{1}\lambda_{2}=-\cos^{2}\alpha$, then we
have

$\lambda_{1}=-\alpha^{\prime}(s)\cos\alpha,\ \ \lambda_{2}=-\sin\alpha\cot s.$

If $u=-\sin\alpha$, then

$uu^{\prime}\cot s=u^{2}-1.$

Solving this equation, we obtain $u=\pm\sqrt{1+C\sec^{2}(s)}$. However, it
does not satisfy the equation (5.2), which is a contradiction. $\square$

Similarly, we can give the following result.

Theorem 5.5  _Any semi-parallel $\lambda$-biharmonic ($\lambda\geq 0$)
hypersurface in $H^{m}\times\mathbb{R}(m\geq 3)$ is minimal or a vertical
cylinder over a $\lambda$-biharmonic hypersurface in $H^{m}$._

## References

  * [1] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, Hypersurfaces of $E^{4}_{s}$ with proper mean curvature vector, J. Math. Soc. Japan 59:3 (2007), 797–809.
  * [2] A. Arvanitoyeorgos, G. Kaimakamis, M. Magid, Lorentz hypersurfaces in $E^{4}_{1}$ satisfying $\Delta\vec{H}=\alpha\vec{H}$, Illinois J. Math. 53:2 (2009), 581–590.
  * [3] A. Arvanitoyeorgos, G. Kaimakamis, Hypersurfaces of type $M^{3}_{2}$ in $E^{4}_{2}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106.
  * [4] G. Calvaruso, D. Kowalczyk, J. Van der Veken, On extrinsically symmetric hypersurfaces of $\mathbb{H}^{n}\times\mathbb{R}$, Bull. Aust. Math. Soc. 82 (2010), 390–400.
  * [5] B.-Y. Chen, Null 2-type surfaces in $\mathbb{E}^{3}$ are circular cylinders, Kodai Math. J. 11:2 (1988), 295–299.
  * [6] F. Dillen, J. Fastenakels, J. Van der Veken, Rotation hypersurfaces in $\mathbb{S}^{n}\times\mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, Note Mat. 29 (2009), 41–54.
  * [7] L. Du, J. Zhang, X. Xie, Hypersurfaces satisfying $\tau_{2}(\phi)=\eta\tau(\phi)$ in pseudo-Riemannian space forms, Math. Phys. Anal. Geom. 20:2 (2017), 17, 23 pp.
  * [8] L. Du, On $\eta$-bihanmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms, J. Math. Anal. Appl. 518 (2023), 126670, 20 pp.
  * [9] A. Ferrández, P. Lucas, Null finite type hypersurfaces in space forms, Kodai Math. J. 14 (1991), 406–419.
  * [10] A. Ferrández, P. Lucas, On surfaces in the 3-dimensional Lorentz-Minkowski space, Pacific J. Math. 152:1 (1992), 93–100.
  * [11] Y. Fu, Null 2-type hypersurfaces with at most three distinct principal curvatures in Euclidean space, Taiwanese J. Math. 19 (2015), 519–533.
  * [12] Y. Fu, M. C. Hong, X. Zhan, On Chen’s biharmonic conjecture for hypersurfaces in $\mathbb{E}^{5}$, Adv. Math. 383 (2021), 107697, 28 pp.
  * [13] T. Hasanis, T. Vlachos, Hypersurfaces with constant scalar curvature and constant mean curvature, Ann. Glob. Anal. Geom. 13 (1995), 69–77.
  * [14] J.-C. Liu, C. Yang, Hypersurfaces in $E^{n+1}_{s}$ satisfying $\Delta\overrightarrow{H}=\lambda\overrightarrow{H}$ with at most three distinct principal curvatures, J. Math. Anal. Appl. 419 (2014), 562–573.
  * [15] J.-C. Liu, C. Yang, Lorentz hypersufaces in $E^{n+1}_{s}$ satisfying $\Delta\overrightarrow{H}=\lambda\overrightarrow{H}$ with at most three distinct principal curvatures, J. Math. Anal. Appl. 434 (2016), 222–240.
  * [16] J.-C. Liu, C. Yang, Hypersurfaces in $\mathbb{E}^{n+1}_{s}$ satisfying $\Delta\vec{H}=\lambda\vec{H}$ with at most two distinct principal curvatures, J. Math. Anal. Appl. 451 (2017), 14–33.
  * [17] Y. Luo, S. Maeta, Biharmonic hypersurfaces in a sphere, Proc. Amer. Math. Soc. 145 (2017), 3109–3116.
  * [18] J. Van der Veken, L. Vrancken, Parallel and semi-parallel hypersurfaces of $\mathbb{S}^{n}\times\mathbb{R}$, Bull. Braz. Math. Soc. (N. S.) 39 (2008), 355–370.
  * [19] C. Yang, J.-C. Liu, L. Du, Four dimensional hypersufaces with proper mean curvature vector field in pseudo-Riemannian space forms, arxiv: 2303. 02362v1, 2023, 1–47.
  * [20] C. Yang, J.-C. Liu, L. Du, PMCV hypersurfaces in non-flat pseudo-Riemannian space forms, arxiv: 2403. 08205v1, 2024, 1–16.
  * [21] S.-T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.