# On triharmonic hypersurfaces in space forms

Yu Fu  School of Data Science and Artificial Intelligence, Dongbei University
of Finance and Economics, Dalian 116025, P. R. China<EMAIL_ADDRESS>and
Dan Yang  School of Mathematics and Statistics, Liaoning University, Shenyang,
110036, China<EMAIL_ADDRESS>
###### Abstract.

In this paper we study triharmonic hypersurfaces immersed in a space form
$N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the
sphere $\mathbb{S}^{n+1}$ has constant scalar curvature; any CMC triharmonic
hypersurface in the hyperbolic space $\mathbb{H}^{n+1}$ is minimal. Moreover,
we show that any CMC triharmonic hypersurface in the Euclidean space
$\mathbb{R}^{n+1}$ is minimal provided that the multiplicity of the principal
curvature zero is at most one. In particular, we are able to prove that every
CMC triharmonic hypersurface in the Euclidean space $\mathbb{R}^{6}$ is
minimal. These results extend some recent works due to Montaldo-Oniciuc-Ratto
and Chen-Guan, and give affirmative answer to the generalized Chen’s
conjecture.

###### Key words and phrases:

$k$-harmonic maps, Triharmonic hypersurfaces, constant mean curvature,
constant scalar curvature

###### 1991 Mathematics Subject Classification:

Primary 53C40, 58E20; Secondary 53C42

##  1\. Introduction

Let $\phi:(M,g)\rightarrow(N,{\bar{g}})$ be a smooth map between Riemannian
manifolds $M$ and $N$. A $k$-harmonic map $\phi$, proposed by Eells and
Lemaire [8], is a critical point of the $k$-energy functional

$\displaystyle
E_{k}(\phi)=\frac{1}{2}\int_{M}\big{|}(d+d^{*})^{k}\phi\big{|}^{2}v_{g}.$

The Euler-Lagrange equation is given by $\tau_{k}(\phi)\equiv 0$ , where
$\tau_{k}(\phi)$ is the $k$-tension field. The concept of $k$-harmonic map is
a natural generalization of harmonic map, and in particular for $k=2$ and
$k=3$, the critical points of $E_{2}$ or $E_{3}$ are biharmonic or triharmonic
maps, respectively (c.f.[24], [25],[11], [12]).

In recent years, biharmonic maps and biharmonic submanifolds have been widely
studied, see [10, 3, 19] and the references therein. There are also a lot of
results on triharmonic maps and triharmonic submanifolds (c.f.[12, 11, 13, 14,
17, 16, 1, 2]). Concerning triharmonic hypersurfaces in a space form
$N^{n+1}(c)$, Maeta [11] proved that any compact CMC triharmonic hypersurface
in $N^{n+1}(c)$ for $c\leq 0$ is minimal. Recently, Montaldo-Oniciuc-Ratto
[15] gave a systematic study of CMC triharmonic hypersurface in space forms.
The authors proved that

######  Theorem 1.1.

([15]) Let $M^{n}$ be a CMC triharmonic hypersurface in $N^{n+1}(c)$ with
$c\leq 0$ and assume that the squared norm of the second fundamental form $S$
is constant. Then $M^{n}$ is minimal.

In particular, the assumption $S$ being constant was removed for $n=2$ in
[15].

######  Theorem 1.2.

([15]) Let $M^{2}$ be a CMC triharmonic surface in $N^{3}(c)$. Then $M^{2}$ is
minimal if $c\leq 0$ and $M^{2}$ is an open part of the small hypersphere if
$c>0$.

Very recently, Chen-Guan investigated triharmonic CMC hypersurfaces in a space
form $N^{n+1}(c)$ under some assumptions on the number of distinct principal
curvatures in [6, 7].

######  Theorem 1.3.

([6]) Let $M^{n}$ ($n\geq 3$) be a CMC proper triharmonic hypersurface with at
most three distinct principal curvatures in $N^{n+1}(c)$. Then $M^{n}$ has
constant scalar curvature.

######  Theorem 1.4.

([7]) Let $M^{n}$ ($n\geq 4$) be a CMC proper triharmonic hypersurface with
four distinct principal curvatures in $N^{n+1}(c)$. If zero is a principal
curvature with multiplicity at most one, then $M^{n}$ has constant scalar
curvature.

We recall Chen’s conjecture in the literature of biharmonic submanifolds: any
biharmonic submanifold in the Euclidean space $\mathbb{R}^{n+1}$ is minimal.
There are some important progress in recent years to support the conjecture
under some geometric restrictions (see, for instances [9, 19]), however the
general case remains open. Taking into account Chen’s conjecture, Maeta [11]
further proposed the generalized Chen’s conjecture on $k$-harmonic
submanifolds:

Conjecture : Any $k$-harmonic submanifold in the Euclidean space
$\mathbb{R}^{n+1}$ is minimal.

In this paper, we are able to determine the geometry of CMC triharmonic
hypersurfaces in a space form $N^{n+1}(c)$ without the restrictions on the
number of principal curvatures. We will prove the following statements:

######  Theorem 1.5.

Let $M^{n}$ be a CMC triharmonic hypersurface in the hyperbolic space
$\mathbb{H}^{n+1}$. Then $M^{n}$ is minimal.

Moreover, restricting ourselves on the case of $c>0$, we get

######  Theorem 1.6.

Let $M^{n}$ be a CMC proper triharmonic hypersurface in the sphere
$\mathbb{S}^{n+1}$. Then $M^{n}$ has constant scalar curvature.

Combining our Theorem 1.6 with the results of Montaldo-Oniciuc-Ratto (Theorem
1.9, [15]) and Chen-Guan (Corollary 1.7, [7]), we provide a more general
result for CMC triharmonic hypersurfaces in $\mathbb{S}^{n+1}$.

######  Corollary 1.7.

Let $M^{n}$ be a CMC proper triharmonic hypersurface in the sphere
$\mathbb{S}^{n+1}$. Then either
(1) $H^{2}=2$ and $M^{n}$ is an open part of $S^{n}(1/\sqrt{3})$, or
(2) $H^{2}\in(0,t_{0}]$ and $H^{2}=t_{0}$ if and only $M^{n}$ is an open part
of $S^{n-1}(a)\times S^{1}(\sqrt{1-a^{2}})$, where $a$ is given by

$a^{2}=\frac{2(n-1)^{2}}{n^{2}H^{2}+2n(n-1)+nH\sqrt{n^{2}H^{2}+4(n-1)}}$

and $t_{0}$ is the unique real root belonging to $(0,2)$ of the polynomial

$f_{n}=n^{4}t^{3}-2n^{2}(n^{2}-5n+5)t^{2}-(n-1)(2n-5)(3n-5)t-(n-1)(n-2)^{2}.$

Let us recall the generalized Chern conjecture (c.f.[4, 5]), which says that:
any closed hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant
mean curvature and constant scalar curvature is isoparametric. Since the class
of CMC proper triharmonic hypersurfaces in a sphere have constant scalar
curvature, the next important problem is to study whether these hypersurfaces
are isoparametric. The problem remains open in its full generality. The
readers may refer to the recent important progress on the generalized Chern
conjecture due to Tang and Yan et al. [22, 23].

Considering the case $c=0$, we obtain a characterization under an assumption
on the multiplicity of zero principal curvature.

######  Theorem 1.8.

Let $M^{n}$ be a CMC triharmonic hypersurface in the Euclidean space
$\mathbb{R}^{n+1}$. If zero is a principal curvature with multiplicity at most
one, then $M^{n}$ is minimal.

In particular, we can prove

######  Theorem 1.9.

Any CMC triharmonic hypersurface in the Euclidean space $\mathbb{R}^{6}$ is
minimal.

######  Remark 1.10.

Note that Theorems 1.8 and 1.9 give partial affirmative answers to the
generalized Chen’s Conjecture.

######  Remark 1.11.

The assumption that the multiplicity of the principal curvature zero is at
most one was necessary in [7] for treating triharmonic hypersurfaces with four
distinct principal curvatures in space forms. In our results, we only need
this for $c=0$ and $n>5$.

At last, we point out that for a CMC proper triharmonic hypersurface in
$N^{n+1}(c)$, the two equations in (2.7) are quite similar to the equations of
a proper biharmonic hypersurface in a space form, see for instance [9]. This
is reasonable because the geometry property of triharmonicity is much weaker
than biharmonicity. Hence, it is expected that more geometric features of
triharmonic hypersurfaces could be found. Interestingly, we can achieve a
complete classification of CMC triharmonic hypersurfaces in $N^{n+1}(c)$ with
$c\neq 0$. This will benefit us in studying biharmonic hypersurfaces in
$N^{n+1}(c)$.

The paper is organized as follows. In Section 2, we recall some background on
the theory of triharmonic hypersurfaces in space forms and derive some useful
lemmas, which are very important for us to study the geometric properties of
triharmonic hypersurfaces. In Section 3, we give the proofs of Theorems 1.5
and 1.6. In Section 4, we finish the proofs of Theorems 1.8 and 1.9.

##  2\. Preliminaries

Let $N^{n+1}(c)$ be an $(n+1)$-dimensional Riemannian space form with constant
sectional curvature $c$. For an isometric immersion $\phi:M^{n}\rightarrow
N^{n+1}(c)$, we denote by $\nabla$ the Levi-Civita connection of $M^{n}$ and
$\widetilde{\nabla}$ the Levi-Civita connection of $N^{n+1}(c)$. The
Riemannian curvature tensors of $M^{n}$ are respectively given by

$\displaystyle
R(X,Y)Z=(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})Z,$
$\displaystyle R(X,Y,Z,W)=\langle R(X,Y)W,Z\rangle.$

The Gauss and Weingarten formulae are stated, respectively, as

$\displaystyle\widetilde{\nabla}_{X}Y$ $\displaystyle=\nabla_{X}Y+h(X,Y)\xi,$
$\displaystyle\widetilde{\nabla}_{X}\xi$ $\displaystyle=-AX.$

Here $X,Y,Z,W$ are tangent vector fields on $M$, $\xi$ is the unit normal
vector field on $M$, $h$ is the second fundamental form of $M$, and $A$ is the
shape operator.

Let us choose an orthonormal frame $\\{e_{i}\\}_{i=1}^{n}$ of $M$. With this
frame, define $\nabla_{e_{i}}e_{j}=\sum_{k}\Gamma_{ij}^{k}e_{k}$, where
$\Gamma_{ij}^{k}$ are the connection coefficients.

Denote by

$\displaystyle R_{ijkl}=$ $\displaystyle R(e_{i},e_{j},e_{k},e_{l}),\quad
h_{ij}=h(e_{i},e_{j}),$ $\displaystyle h_{ijk}=$ $\displaystyle
e_{k}(h_{ij})-h(\nabla_{e_{k}}e_{i},e_{j})-h(e_{i},\nabla_{e_{k}}e_{j}),$
$\displaystyle=$ $\displaystyle
e_{k}(h_{ij})-\sum_{l}(\Gamma_{ki}^{l}h_{lj}+\Gamma_{kj}^{l}h_{il}).$

From the definition of the Gauss curvature tensor we obtain

$\displaystyle R_{ijkl}$
$\displaystyle=e_{i}(\Gamma_{jl}^{k})-e_{j}(\Gamma_{il}^{k})+\sum_{m}\Big{(}\Gamma_{jl}^{m}\Gamma_{im}^{k}-\Gamma_{il}^{m}\Gamma_{jm}^{k}-(\Gamma_{ij}^{m}-\Gamma_{ji}^{m})\Gamma_{ml}^{k}\Big{)}.$
(2.1)

Moreover, the Gauss and Codazzi equations are given, respectively, by

$\displaystyle R_{ijkl}$
$\displaystyle=c(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+(h_{ik}h_{jl}-h_{il}h_{jk}),$
(2.2) $\displaystyle h_{ijk}$ $\displaystyle=h_{ikj}.$ (2.3)

The mean curvature function $H$ and the squared norm of the second fundamental
form $S$ are written respectively as

$\displaystyle H=\frac{1}{n}\sum_{i=1}^{n}h_{ii}\quad{\rm and}\quad
S=\sum_{i,j=1}^{n}h_{ij}^{2}.$ (2.4)

From the Gauss equation, the scalar curvature $R$ is given by

$\displaystyle R=n(n-1)c+n^{2}H^{2}-S.$ (2.5)

We recall a fundamental characterization result on CMC triharmonic
hypersurfaces in $N^{n+1}(c)$.

######  Proposition 2.1.

(c.f.[15]) A CMC hypersurface $\phi:M^{n}\rightarrow N^{n+1}(c)$ is
triharmonic if the mean curvature $H$ and the squared norm of the second
fundamental form $S$ on $M^{n}$ satisfy

$\left\\{\begin{split}&H({\rm\Delta}\,S+S^{2}-ncS-n^{2}cH^{2})=0,\\\ &HA\nabla
S=0.\end{split}\right.$ (2.6)

According to (2.6), it is clear that minimal hypersurfaces are automatically
triharmonic in $N^{n+1}(c)$. A triharmonic hypersurfaces in $N^{n+1}(c)$ is
called proper if it is not minimal.

In the following, we will consider a CMC proper hypersurface $M^{n}$ in a
space form $N^{n+1}(c)$. Then (2.6) becomes

$\left\\{\begin{split}&{\rm\Delta}\,S+S^{2}-ncS-n^{2}cH^{2}=0,\\\ &A\nabla
S=0.\end{split}\right.$ (2.7)

For a hypersurface $M^{n}$ in $N^{n+1}(c)$, we denote by $\lambda_{i}$ for
$1\leq i\leq n$ its principal curvatures. The number of distinct principal
curvatures is locally constant and the set of all points here is an open and
dense subset of $M^{n}$. Denote by $M_{A}$ this set. On a non-empty connected
component of $M_{A}$, which is open, the number of distinct principal
curvatures is constant. On that connected component, the multiplicities of the
distinct principal curvatures are constant and hence $\lambda_{i}$ are always
smooth and the shape operator $A$ is locally diagonalizable (see[18, 20, 21]).

Denote by $\mathcal{N}:=\\{p\in M:\nabla S(p)\neq 0\\}$ and
$\mathcal{N}\subset M_{A}$. If $S$ is constant, then $\mathcal{N}$ is an empty
set. From now on, we assume that $S$ is not constant, that is
$\mathcal{N}\neq\emptyset$. We will work in $\mathcal{N}$.

Observing from the second equation of (2.7), it is known that $\nabla S$ is a
principal direction with the corresponding principal curvature $0$. Hence, we
may choose an orthonormal frame $\\{e_{i}\\}_{i=1}^{n}$ such that $e_{1}$ is
parallel to $\nabla S$ and the shape operator $A$ is diagonalizable with
respect to $\\{e_{i}\\}$, i.e., $h_{ij}=\lambda_{i}\delta_{ij}$, where
$\lambda_{i}$ is the principal curvature and $\lambda_{1}=0$.

Suppose that $M^{n}$ has $d$ distinct principal curvatures
$\mu_{1}=0,\mu_{2},\cdots,\mu_{d}$ with $d\geq 4$, that is

$\lambda_{i}=\mu_{\alpha}\quad{\rm when}\quad i\in I_{\alpha},$

where

$I_{\alpha}=\Big{\\{}\sum_{0\leq\beta\leq\alpha-1}n_{\beta}+1,\cdots,\sum_{0\leq\beta\leq\alpha}n_{\beta}\Big{\\}}$

with $n_{0}=0$ and $n_{\alpha}\in\mathbb{Z}_{+}$ satisfying
$\sum_{1\leq\alpha\leq d}n_{\alpha}=n$, namely, $n_{\alpha}$ is the
multiplicity of $\mu_{\alpha}$. For convenience, we will use the range of
indices $1\leq\alpha,\beta,\gamma,\cdots\leq d$ except special declaration.

We collect a lemma for later use.

######  Lemma 2.2.

(c.f.[7]) The connection coefficients $\Gamma_{ij}^{k}$ satisfy:
(1) $\Gamma_{ij}^{k}=-\Gamma_{ik}^{j}$.
(2) $\Gamma_{ii}^{k}=\frac{e_{k}(\lambda_{i})}{\lambda_{i}-\lambda_{k}}$ for
$i\in I_{\alpha}$ and $k\notin I_{\alpha}$.
(3) $\Gamma_{ij}^{k}=\Gamma_{ji}^{k}$ if the indices satisfy one of the
following conditions:
(3a) $i,j\in I_{\alpha}$ but $k\notin I_{\alpha}$;
(3b) $i,j\geq 2$ and $k=1$.
(4) $\Gamma_{ij}^{k}=0$ if the indices satisfy one of the following
conditions:
(4a) $j=k$;
(4b) $i=j\in I_{1}$ and $k\notin I_{1}$;
(4c) $i,k\in I_{\alpha},i\neq k$ and $j\notin I_{\alpha}$;
(4d) $i,j\geq 2,i\in I_{\alpha},j\in I_{\beta}$ with $\alpha\neq\beta$ and
$k=1$.
(5)
$\Gamma_{ji}^{k}=\frac{\lambda_{j}-\lambda_{k}}{\lambda_{i}-\lambda_{k}}\Gamma_{ij}^{k},$
$\Gamma_{ki}^{j}=\frac{\lambda_{k}-\lambda_{j}}{\lambda_{i}-\lambda_{j}}\Gamma_{ik}^{j}$
for $\lambda_{i}$, $\lambda_{j}$ and $\lambda_{k}$ are mutually different.
(6)
$\Gamma_{ij}^{k}\Gamma_{ji}^{k}+\Gamma_{ik}^{j}\Gamma_{ki}^{j}+\Gamma_{jk}^{i}\Gamma_{kj}^{i}=0$
for $\lambda_{i}$, $\lambda_{j}$ and $\lambda_{k}$ are mutually different.

We first derive some crucial lemmas for studying CMC proper triharmonic
hypersurfaces in a space form.

######  Lemma 2.3.

Denoting by $P_{\alpha}=\frac{e_{1}(\mu_{\alpha})}{\mu_{\alpha}}$ for
$2\leq\alpha\leq d$, we have

$\displaystyle e_{1}(P_{\alpha})=P_{\alpha}^{2}+c+\sum_{m\in
I_{1}}\Gamma_{\alpha\alpha}^{m}\Gamma_{11}^{m},$ (2.8) $\displaystyle
e_{j}(P_{\alpha})=\Gamma_{\alpha\alpha}^{j}P_{\alpha}-\sum_{m\in
I_{1}}\Gamma_{\alpha\alpha}^{m}\Gamma_{jm}^{1}\,\,\,{\rm for}\,\,j\in
I_{1}\,\,{\rm and}\,\,j\neq 1.$ (2.9)

###### Proof.

For $i\in I_{\alpha}$ and $\alpha\neq 1$, it follows from (2.1) and the terms
(1), (2) and (4) in Lemma 2.2 that

$\displaystyle R_{1i1i}=$ $\displaystyle
e_{1}(\Gamma_{ii}^{1})-e_{i}(\Gamma_{1i}^{1})+\sum_{m}\Big{(}\Gamma_{ii}^{m}\Gamma_{1m}^{1}-\Gamma_{1i}^{m}\Gamma_{im}^{1}-(\Gamma_{1i}^{m}-\Gamma_{i1}^{m})\Gamma_{mi}^{1}\Big{)}$
$\displaystyle=$ $\displaystyle
e_{1}(\Gamma_{ii}^{1})+e_{i}(\Gamma_{11}^{i})-\sum_{m}\Big{(}\Gamma_{ii}^{m}\Gamma_{11}^{m}+\Gamma_{1i}^{m}\Gamma_{im}^{1}+(\Gamma_{1i}^{m}+\Gamma_{im}^{1})\Gamma_{mi}^{1}\Big{)}$
$\displaystyle=$ $\displaystyle e_{1}(\Gamma_{ii}^{1})-\sum_{m\in
I_{1}}\Big{(}\Gamma_{ii}^{m}\Gamma_{11}^{m}+\Gamma_{1i}^{m}\Gamma_{im}^{1}+(\Gamma_{1i}^{m}+\Gamma_{im}^{1})\Gamma_{mi}^{1}\Big{)}$
$\displaystyle=$ $\displaystyle
e_{1}(\Gamma_{ii}^{1})-(\Gamma_{ii}^{1})^{2}-\sum_{m\in
I_{1}}\Gamma_{ii}^{m}\Gamma_{11}^{m}.$

On the other hand, from the Gauss equation (2.2) we get $R_{1i1i}=c$.
Therefore, we obtain (2.8).

For $i\in I_{\alpha}$ and $\alpha\neq 1$, $j\in I_{1}$ and $j\neq 1$, it
follows from (2.1) and Lemma 2.2 that

$\displaystyle R_{iji1}$
$\displaystyle=e_{i}(\Gamma_{j1}^{i})-e_{j}(\Gamma_{i1}^{i})+\sum_{m}\Big{(}\Gamma_{j1}^{m}\Gamma_{im}^{i}-\Gamma_{i1}^{m}\Gamma_{jm}^{i}-(\Gamma_{ij}^{m}-\Gamma_{ji}^{m})\Gamma_{m1}^{i}\Big{)}$
$\displaystyle=-e_{i}(\Gamma_{ji}^{1})+e_{j}(\Gamma_{ii}^{1})+\sum_{m}\Big{(}\Gamma_{jm}^{1}\Gamma_{ii}^{m}+\Gamma_{im}^{1}\Gamma_{jm}^{i}+(\Gamma_{ij}^{m}-\Gamma_{ji}^{m})\Gamma_{mi}^{1}\Big{)}$
$\displaystyle=e_{j}(\Gamma_{ii}^{1})+\sum_{m\in
I_{1}}\Gamma_{jm}^{1}\Gamma_{ii}^{m}-\Gamma_{ii}^{j}\Gamma_{ii}^{1},$ (2.10)

which together with $R_{iji1}=0$ gives (2.9). Note that $R_{iji1}=0$ follows
from the Gauss equation (2.2) directly. We thus complete the proof. ∎

######  Lemma 2.4.

For any $q\in\mathbb{Z}_{+}$, we have

$\displaystyle\begin{split}f(q):=\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}P^{q}_{\alpha}=\left\\{\begin{array}[]{ll}\quad
0,&{\rm when}\,\,q\,\,{\rm is}\,\,{\rm odd};\\\
\frac{(q-1)!!}{q!!}(-c)^{\frac{q}{2}}nH,&{\rm when}\,\,q\,\,{\rm is}\,\,{\rm
even}.\end{array}\right.\end{split}$ (2.11)

###### Proof.

Since the case $n_{1}=1$ has been obtained in [7], we only need to prove it
for $n_{1}>1$.

Taking into account the definition of $H$ from the first expression of (2.4),
we have

$\displaystyle\sum_{2\leq\alpha\leq d}n_{\alpha}\mu_{\alpha}=nH.$ (2.12)

Since $H$ is constant, differentiating (2.12) with respect to $e_{1}$, we
obtain

$\displaystyle\sum_{2\leq\alpha\leq d}n_{\alpha}e_{1}(\mu_{\alpha})=0,$

which is equivalent to

$\displaystyle\sum_{2\leq\alpha\leq d}n_{\alpha}\mu_{\alpha}P_{\alpha}=0.$
(2.13)

Differentiating (2.12) with respect to $e_{m}$ for $m\in I_{1}$ and $m\neq 1$,
we obtain

$\displaystyle\sum_{2\leq\alpha\leq d}n_{\alpha}e_{m}(\mu_{\alpha})=0.$ (2.14)

Differentiating (2.13) with respect to $e_{1}$, from (2.8), (2.12),
$\mu_{\alpha}\Gamma_{\alpha\alpha}^{m}=e_{m}(\mu_{\alpha})$ and (2.14) we have

$\displaystyle 0$ $\displaystyle=\sum_{2\leq\alpha\leq
d}n_{\alpha}\Big{(}e_{1}(\mu_{\alpha})P_{\alpha}+\mu_{\alpha}e_{1}(P_{\alpha})\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}n_{\alpha}\Big{(}\mu_{\alpha}P_{\alpha}^{2}+\mu_{\alpha}(P_{\alpha}^{2}+c+\sum_{m\in
I_{1}}\Gamma_{\alpha\alpha}^{m}\Gamma_{11}^{m})\Big{)}$
$\displaystyle=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}P_{\alpha}^{2}+cnH+\sum_{2\leq\alpha\leq d}\sum_{m\in
I_{1}}n_{\alpha}\mu_{\alpha}\Gamma_{\alpha\alpha}^{m}\Gamma_{11}^{m}$
$\displaystyle=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}P_{\alpha}^{2}+cnH+\sum_{m\in
I_{1}}\Big{(}\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{m}(\mu_{\alpha})\Big{)}\Gamma_{11}^{m}$
$\displaystyle=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}P_{\alpha}^{2}+cnH,$

which is equivalent to

$\displaystyle\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}P_{\alpha}^{2}=-\frac{1}{2}cnH.$ (2.15)

Equations (2.13) and (2.15) imply that (2.11) holds for $q=1,2$. Next we will
prove that it holds for general $q$ by induction.

Differentiating (2.11) with respect to $e_{1}$ yields

$\displaystyle 0$ $\displaystyle=\sum_{2\leq\alpha\leq
d}n_{\alpha}\Big{(}e_{1}(\mu_{\alpha})P_{\alpha}^{q}+q\mu_{\alpha}P_{\alpha}^{q-1}e_{1}(P_{\alpha})\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}n_{\alpha}\Big{(}\mu_{\alpha}P_{\alpha}^{q+1}+q\mu_{\alpha}P_{\alpha}^{q-1}(P_{\alpha}^{2}+c+\sum_{m\in
I_{1}}\Gamma_{\alpha\alpha}^{m}\Gamma_{11}^{m})\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}\Big{(}(1+q)n_{\alpha}\mu_{\alpha}P_{\alpha}^{q+1}+cqn_{\alpha}\mu_{\alpha}P_{\alpha}^{q-1}\Big{)}+q\sum_{2\leq\alpha\leq
d}n_{\alpha}P_{\alpha}^{q-1}\sum_{m\in
I_{1}}e_{m}(\mu_{\alpha})\Gamma_{11}^{m}$
$\displaystyle=(1+q)f(q+1)+cqf(q-1)+q\sum_{m\in
I_{1}}\Big{\\{}\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{m}(\mu_{\alpha})P_{\alpha}^{q-1}\Big{\\}}\Gamma_{11}^{m}.$
(2.16)

On the other hand, since (2.11) holds for $f(q-1)$, we differentiate
$f(q-1)=\sum_{2\leq\alpha\leq d}n_{\alpha}\mu_{\alpha}P^{q-1}_{\alpha}$ with
respect to $e_{j}$ for $j\in I_{1}$ and $j\neq 1$. It follows from (2.9) that

$\displaystyle 0$ $\displaystyle=\sum_{2\leq\alpha\leq
d}\Big{(}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}+(q-1)n_{\alpha}\mu_{\alpha}P_{\alpha}^{q-2}e_{j}(P_{\alpha})\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}\Big{(}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}+(q-1)n_{\alpha}\mu_{\alpha}P_{\alpha}^{q-2}\big{(}\Gamma_{\alpha\alpha}^{j}P_{\alpha}-\sum_{m\in
I_{1}}\Gamma_{\alpha\alpha}^{m}\Gamma_{jm}^{1})\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}\Big{(}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}+(q-1)n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}-(q-1)\sum_{m\in
I_{1}}n_{\alpha}P_{\alpha}^{q-2}e_{m}(\mu_{\alpha})\Gamma_{jm}^{1}\Big{)}$
$\displaystyle=\sum_{2\leq\alpha\leq
d}\Big{(}qn_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}-(q-1)\sum_{m\in
I_{1}}n_{\alpha}P_{\alpha}^{q-2}e_{m}(\mu_{\alpha})\Gamma_{jm}^{1}\Big{)}.$

Hence the following relation holds for any $j\in I_{1}$ and $j\neq 1$

$\displaystyle q\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}=\sum_{m\in
I_{1}}\Big{\\{}(q-1)\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{m}(\mu_{\alpha})P_{\alpha}^{q-2}\Big{\\}}\Gamma_{jm}^{1}.$
(2.17)

Since (2.17) holds for any $q$, letting $q=2$, (2.17) reduces to

$\displaystyle 2\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}=\sum_{m\in
I_{1}}\Big{\\{}\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{m}(\mu_{\alpha})\Big{\\}}\Gamma_{jm}^{1},$

which together with (2.14) yields

$\displaystyle 2\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}=0.$ (2.18)

Letting $q=3$ and using (2.18), (2.17) reduces to

$\displaystyle 3\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{2}=\sum_{m\in
I_{1}}\Big{\\{}2\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{m}(\mu_{\alpha})P_{\alpha}\Big{\\}}\Gamma_{jm}^{1}=0.$

Similarly, we can gradually show

$\displaystyle q\sum_{2\leq\alpha\leq
d}n_{\alpha}e_{j}(\mu_{\alpha})P_{\alpha}^{q-1}=0.$ (2.19)

Hence, combing (2) with (2.19) gives

$\displaystyle(q+1)f(q+1)+cqf(q-1)=0.$ (2.20)

When $q$ is even, both of $q-1$ and $q+1$ are odd. From (2.20), $f(q-1)=0$ can
yield $f(q+1)=0$ as well.

When $q$ is odd, both of $q-1$ and $q+1$ are even. We conclude from (2.20)
that

$\displaystyle f(q+1)$ $\displaystyle=-\frac{cq}{q+1}f(q-1)$
$\displaystyle=-\frac{cq}{q+1}\times\frac{(q-2)!!}{(q-1)!!}(-c)^{(q-1)/2}nH$
$\displaystyle=\frac{q!!}{(q+1)!!}(-c)^{(q+1)/2}nH,$ (2.21)

which completes the proof of Lemma 2.4. ∎

######  Remark 2.5.

While assuming an additional condition that the multiplicity of zero principal
curvature is at most one, special cases of Lemmas 2.3 and 2.4 were derived in
[7].

######  Lemma 2.6.

The equation ${\rm\Delta}\,S+S^{2}-ncS-n^{2}cH^{2}=0$ is equivalent to

$\displaystyle-6\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}\Big{)}\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}P_{\alpha}+\sum_{m\in
I_{1}}\Gamma_{mm}^{1}\Big{)}+\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}^{2}$
$\displaystyle-(n+2)c\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}-c\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}\Big{)}^{2}=0.$

###### Proof.

Since $e_{i}(S)=0$ for $2\leq i\leq n$, it follows from Lemma 2.2 that

$\displaystyle\Delta S$
$\displaystyle=-\sum_{i=1}^{n}(\nabla_{e_{i}}\nabla_{e_{i}}S-\nabla_{\nabla_{e_{i}e_{i}}}S)$
$\displaystyle=-e_{1}e_{1}S+e_{1}(S)\sum_{i=2}^{n}\Gamma_{ii}^{1}$
$\displaystyle=-e_{1}e_{1}S+e_{1}(S)\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}P_{\alpha}+\sum_{m\in
I_{1}}\Gamma_{mm}^{1}\Big{)}.$ (2.22)

Noting $S=\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}$, it follows from
(2.8) that

$\displaystyle e_{1}(S)$
$\displaystyle=2\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}e_{1}(\mu_{\alpha})=2\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha},$
(2.23) $\displaystyle e_{1}e_{1}(S)$
$\displaystyle=4\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2c\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}+2\sum_{\alpha=2}^{d}\sum_{m\in
I_{1}}n_{\alpha}\mu_{\alpha}^{2}\Gamma_{\alpha\alpha}^{m}\Gamma_{11}^{m}$
$\displaystyle=6\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2c\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}+2\sum_{m\in
I_{1}}\Big{\\{}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Gamma_{\alpha\alpha}^{m}\Big{\\}}\Gamma_{11}^{m}.$
(2.24)

Differentiating $S=\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}$ with respect
to $e_{m}$ for $m\in I_{1}$ and $m\neq 1$, we get

$\displaystyle
0=e_{m}(S)=2\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}e_{m}(\mu_{\alpha})=2\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Gamma_{\alpha\alpha}^{m}.$
(2.25)

Combining (2.24) with (2.25) gives

$\displaystyle
e_{1}e_{1}(S)=6\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2c\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}.$
(2.26)

Substituting (2.23) and (2.26) into (2), we have

$\displaystyle\Delta S=$
$\displaystyle-6\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}\Big{)}\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}P_{\alpha}+\sum_{m\in
I_{1}}\Gamma_{mm}^{1}\Big{)}$
$\displaystyle-2c\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}.$

Hence, the proof has been completed. ∎

##  3\. Proof of Theorems 1.5 and 1.6

The proof of Theorems 1.5 and 1.6: We will prove Theorems 1.5 and 1.6 by
deriving a contradiction from the assumption that $\mathcal{N}=\\{p\in
M^{n}:\nabla S(p)\neq 0\\}\neq\emptyset$.

Taking $q=1,3,5,\cdots,2d-3$ in Lemma 2.4, we have

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}n_{2}\mu_{2}P_{2}+n_{3}\mu_{3}P_{3}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\
n_{2}\mu_{2}P_{2}^{3}+n_{3}\mu_{3}P_{3}^{3}+\cdots+n_{d}\mu_{d}P_{d}^{3}=0,\\\
\quad\,\vdots\\\
n_{2}\mu_{2}P_{2}^{2d-3}+n_{3}\mu_{3}P_{3}^{2d-3}+\cdots+n_{d}\mu_{d}P_{d}^{2d-3}=0,\end{array}\right.\end{split}$
(3.1)

which is a $(d-1)$-th order equation system with a non-zero solution. Hence on
$\mathcal{N}$ we have

$\displaystyle\begin{split}\left|\begin{array}[]{llll}P_{2}&P_{3}&\cdots&P_{d}\\\
P_{2}^{3}&P_{3}^{3}&\cdots&P_{d}^{3}\\\
\,\,\vdots&\,\,\vdots&\cdots&\,\,\vdots\\\
P_{2}^{2d-3}&P_{3}^{2d-3}&\cdots&P_{d}^{2d-3}\end{array}\right|=P_{2}\cdots
P_{d}\prod_{2\leq\alpha\leq\beta\leq
d}(P_{\alpha}^{2}-P_{\beta}^{2})=0.\end{split}$ (3.2)

Taking $q=2,4,6,\cdots,2d-2$ in Lemma 2.4, we obtain

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}n_{2}\mu_{2}P_{2}^{2}+n_{3}\mu_{3}P_{3}^{2}+\cdots+n_{d}\mu_{d}P_{d}^{2}=-\frac{1}{2}ncH,\\\
n_{2}\mu_{2}P_{2}^{4}+n_{3}\mu_{3}P_{3}^{4}+\cdots+n_{d}\mu_{d}P_{d}^{4}=\frac{3}{8}nc^{2}H,\\\
\quad\,\vdots\\\
n_{2}\mu_{2}P_{2}^{2d-2}+n_{3}\mu_{3}P_{3}^{2d-2}+\cdots+n_{d}\mu_{d}P_{d}^{2d-2}=\frac{(2d-3)!!}{(2d-2)!!}(-c)^{d-1}nH.\end{array}\right.\end{split}$
(3.3)

Next we consider all possible cases.

Case 1. $P_{2}P_{3}\cdots P_{d}\neq 0$ at some $p\in\mathcal{N}$. Then from
(3.2) we have that $\prod_{2\leq\alpha\leq\beta\leq
d}(P_{\alpha}^{2}-P_{\beta}^{2})=0$ at $p$. Without loss of generality, we
assume $P_{2}^{2}-P_{3}^{2}=0$, i.e., $P_{2}=\pm P_{3}$. Now the former $d-2$
equations in the system of equations (3.1) determine a new system of equations
as follows:

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}(n_{2}\mu_{2}\pm
n_{3}\mu_{3})P_{3}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\ (n_{2}\mu_{2}\pm
n_{3}\mu_{3})P_{3}^{3}+\cdots+n_{d}\mu_{d}P_{d}^{3}=0,\\\
\quad\quad\quad\vdots\\\ (n_{2}\mu_{2}\pm
n_{3}\mu_{3})P_{3}^{2d-5}+\cdots+n_{d}\mu_{d}P_{d}^{2d-5}=0,\end{array}\right.\end{split}$
(3.4)

which has a non-zero solution. Then we have

$\displaystyle\begin{split}\left|\begin{array}[]{llll}P_{3}&P_{4}&\cdots&P_{d}\\\
P_{3}^{3}&P_{4}^{3}&\cdots&P_{d}^{3}\\\
\,\,\vdots&\,\,\vdots&\cdots&\,\,\vdots\\\
P_{3}^{2d-5}&P_{4}^{2d-5}&\cdots&P_{d}^{2d-5}\end{array}\right|=P_{3}\cdots
P_{d}\prod_{3\leq\alpha\leq\beta\leq
d}(P_{\alpha}^{2}-P_{\beta}^{2})=0.\end{split}$

Since $P_{3}P_{4}\cdots P_{d}\neq 0$, we have that
$\prod_{3\leq\alpha\leq\beta\leq d}(P_{\alpha}^{2}-P_{\beta}^{2})=0$. Without
loss of generality, we assume that $P_{3}^{2}=P_{4}^{2}$. Proceeding in this
way, we obtain that $P_{2}^{2}=P_{3}^{2}=\cdots=P_{d}^{2}:=P^{2}$ at $p$. Now
(3.3) becomes

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}-\frac{1}{2}ncH=n_{2}\mu_{2}P_{2}^{2}+n_{3}\mu_{3}P_{3}^{2}+\cdots+n_{d}\mu_{d}P_{d}^{2}=nHP^{2},\\\
\frac{3}{8}nc^{2}H=n_{2}\mu_{2}P_{2}^{4}+n_{3}\mu_{3}P_{3}^{4}+\cdots+n_{d}\mu_{d}P_{d}^{4}=nHP^{4},\\\
\quad\,\,\vdots\\\
\frac{(2d-3)!!}{(2d-2)!!}(-c)^{d-1}nH=n_{2}\mu_{2}P_{2}^{2d-2}+n_{3}\mu_{3}P_{3}^{2d-2}+\cdots+n_{d}\mu_{d}P_{d}^{2d-2}=nHP^{2d-2}.\end{array}\right.\end{split}$

Since $nH\neq 0$, the first two equations of the above system imply that

$P^{2}=-\frac{1}{2}c\quad{\rm and}\quad P^{4}=\frac{3}{8}c^{2},$

and hence $c=P=0$. It is a contradiction, so this case is ruled out.

Case 2. $P_{\alpha}=0$ for all $\alpha=2,\cdots,d$ at some $p\in\mathcal{N}$.
In this case, we have

$e_{1}S=e_{1}\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}^{2}=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}e_{1}(\mu_{\alpha})=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}=0.$

This contradicts $\nabla S\neq 0$ at $p$ and this case is also ruled out.

Case 3. For any given point $p\in\mathcal{N}$, some terms of $P_{\alpha}$ are
zero and the others are not zero. In this case, without loss of generality,
assume $P_{\alpha}=0$ for $\alpha=2,\cdots,r$ and $P_{\alpha}\neq 0$ for
$\alpha=r+1,\cdots,d$. Then the first $d-r$ equations in (3.1) form a new
system of equations

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}n_{r+1}\mu_{r+1}P_{r+1}+n_{r+2}\mu_{r+2}P_{r+2}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\
n_{r+1}\mu_{r+1}P_{r+1}^{3}+n_{r+2}\mu_{r+2}P_{r+2}^{3}+\cdots+n_{d}\mu_{d}P_{d}^{3}=0,\\\
\quad\quad\quad\vdots\\\
n_{r+1}\mu_{r+1}P_{r+1}^{2(d-r)-1}+n_{r+2}\mu_{r+2}P_{r+2}^{2(d-r)-1}+\cdots+n_{d}\mu_{d}P_{d}^{2(d-r)-1}=0,\end{array}\right.\end{split}$
(3.5)

which is a $(d-r)$-th order equation system with non-zero solutions. So the
coefficient determinant is zero, that is $\prod_{r+1\leq\alpha\leq\beta\leq
d}(P_{\alpha}^{2}-P_{\beta}^{2})=0$. Without loss of generality, we assume
that $P_{r+1}^{2}=P_{r+2}^{2}$. Proceeding in this way, we can show that
$P_{r+1}^{2}=\cdots=P_{d}^{2}\neq 0$. Denote by
$P^{2}:=P_{r+1}^{2}=\cdots=P_{d}^{2}$. Then (3.3) becomes

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}(n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d})P^{2}=-\frac{1}{2}ncH,\\\
(n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d})P^{4}=\frac{3}{8}nc^{2}H,\\\
(n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d})P^{6}=-\frac{5}{16}nc^{3}H,\\\
\quad\quad\quad\vdots\\\
(n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d})P^{2d-2}=\frac{(2d-3)!!}{(2d-2)!!}(-c)^{d-1}nH.\end{array}\right.\end{split}$
(3.6)

The above system of equations means that
$n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d}\neq 0$ since $c\neq 0$ and $H\neq 0$.
The first two equations of (3.6) force that $P^{2}=-\frac{3}{4}c$, and the
second and the third equation of (3.6) force that $P^{2}=-\frac{5}{6}c$. Hence
we have $P=c=0$, a contradiction.

In conclusion, we have that $\mathcal{N}=\\{p\in M:\nabla S(p)\neq 0\\}$ is
empty and hence $S$ has to be a constant. From (2.5), we conclude that the
scalar curvature $R$ of $M^{n}$ is constant as well. But for $c<0$, the first
equation of (2.7) means that $S^{2}-ncS-n^{2}cH^{2}=0$, a contradiction. This
completes the proof of Theorems 1.5 and 1.6. $\hfill\square$

##  4\. Proofs of Theorems 1.8 and 1.9

In this section, we mainly concern CMC triharmonic hypersurfaces in the
Euclidean space $\mathbb{R}^{n+1}$. For any dimension $n$, we need another
assumption that the multiplicity of the zero principal curvature is at most
one as discussed in [7] for four distinct principal curvatures.

The proof of Theorem 1.8: Assume that $\mathcal{N}=\\{p\in M^{n}:\nabla
S(p)\neq 0\\}\neq\emptyset$. We will prove Theorems 1.8 by deriving a
contradiction.

Taking $q=1,2,3,\cdots,d-1$ in Lemma 2.4, we have

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}n_{2}\mu_{2}P_{2}+n_{3}\mu_{3}P_{3}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\
n_{2}\mu_{2}P_{2}^{2}+n_{3}\mu_{3}P_{3}^{2}+\cdots+n_{d}\mu_{d}P_{d}^{2}=0,\\\
\quad\,\vdots\\\
n_{2}\mu_{2}P_{2}^{d-1}+n_{3}\mu_{3}P_{3}^{d-1}+\cdots+n_{d}\mu_{d}P_{d}^{d-1}=0,\end{array}\right.\end{split}$
(4.1)

which is a $(d-1)$-th order equation system with a non-zero solution. Hence on
$\mathcal{N}$ we have

$\displaystyle\begin{split}\left|\begin{array}[]{llll}P_{2}&P_{3}&\cdots&P_{d}\\\
P_{2}^{2}&P_{3}^{2}&\cdots&P_{d}^{2}\\\
\,\,\vdots&\,\,\vdots&\cdots&\,\,\vdots\\\
P_{2}^{d-1}&P_{3}^{d-1}&\cdots&P_{d}^{d-1}\end{array}\right|=P_{2}\cdots
P_{d}\prod_{2\leq\alpha\leq\beta\leq d}(P_{\alpha}-P_{\beta})=0.\end{split}$
(4.2)

Next we consider three possible cases.

Case 1. $P_{2}P_{3}\cdots P_{d}\neq 0$ at some $p\in\mathcal{N}$. Then from
(4.2) we have that $\prod_{2\leq\alpha\leq\beta\leq
d}(P_{\alpha}-P_{\beta})=0$ at $p$. Without loss of generality, we assume
$P_{2}-P_{3}=0$. Now the former $d-2$ equations in the system of equations
(4.1) determine a new system of equations as follows:

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}(n_{2}\mu_{2}+n_{3}\mu_{3})P_{3}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\
(n_{2}\mu_{2}+n_{3}\mu_{3})P_{3}^{2}+\cdots+n_{d}\mu_{d}P_{d}^{2}=0,\\\
\quad\quad\quad\vdots\\\
(n_{2}\mu_{2}+n_{3}\mu_{3})P_{3}^{d-2}+\cdots+n_{d}\mu_{d}P_{d}^{d-2}=0,\end{array}\right.\end{split}$
(4.3)

which has a non-zero solution. Then we have

$\displaystyle\begin{split}\left|\begin{array}[]{llll}P_{3}&P_{4}&\cdots&P_{d}\\\
P_{3}^{2}&P_{4}^{2}&\cdots&P_{d}^{2}\\\
\,\,\vdots&\,\,\vdots&\cdots&\,\,\vdots\\\
P_{3}^{d-2}&P_{4}^{d-2}&\cdots&P_{d}^{d-2}\end{array}\right|=P_{3}\cdots
P_{d}\prod_{3\leq\alpha\leq\beta\leq d}(P_{\alpha}-P_{\beta})=0.\end{split}$

Since $P_{3}P_{4}\cdots P_{d}\neq 0$, we have that
$\prod_{3\leq\alpha\leq\beta\leq d}(P_{\alpha}-P_{\beta})=0$. Without loss of
generality, we assume that $P_{3}=P_{4}$. Similarly, we obtain that
$P_{3}=P_{4}=\cdots=P_{d}:=P$ at $p$. Since $nH\neq 0$, the first equation of
the above system (4.3) implies that

$\Big{(}\sum_{2\leq\alpha\leq d}{n_{\alpha}\mu_{\alpha}}\Big{)}P=nHP=0$

and hence $P=0$. It is a contradiction.

Case 2. $P_{\alpha}=0$ for all $\alpha=2,\cdots,d$ at some $p\in\mathcal{N}$.
Then

$e_{1}S=e_{1}\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}^{2}=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}e_{1}(\mu_{\alpha})=2\sum_{2\leq\alpha\leq
d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}=0,$

which contradicts $\nabla S\neq 0$ at $p$.

Case 3. For any given point $p\in\mathcal{N}$, some terms of $P_{\alpha}$ are
zero and the others are not zero. In this case, without loss of generality,
assume $P_{\alpha}=0$ for $\alpha=2,\cdots,r$ and $P_{\alpha}\neq 0$ for
$\alpha=r+1,\cdots,d$. Then the first $d-r$ equations in (4.1) form a new
system of equations

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}n_{r+1}\mu_{r+1}P_{r+1}+n_{r+2}\mu_{r+2}P_{r+2}+\cdots+n_{d}\mu_{d}P_{d}=0,\\\
n_{r+1}\mu_{r+1}P_{r+1}^{2}+n_{r+2}\mu_{r+2}P_{r+2}^{2}+\cdots+n_{d}\mu_{d}P_{d}^{2}=0,\\\
\quad\quad\quad\vdots\\\
n_{r+1}\mu_{r+1}P_{r+1}^{d-r}+n_{r+2}\mu_{r+2}P_{r+2}^{d-r}+\cdots+n_{d}\mu_{d}P_{d}^{d-r}=0,\end{array}\right.\end{split}$
(4.4)

which is a $(d-r)$-th order equation system with non-zero solutions. Thus the
coefficient determinant is zero, that is $\prod_{r+1\leq\alpha\leq\beta\leq
d}(P_{\alpha}-P_{\beta})=0$. Without loss of generality, we assume that
$P_{r+1}=P_{r+2}$. Similar discussion as the above yields
$P_{r+1}=\cdots=P_{d}\neq 0$. Denote by $P:=P_{r+1}=\cdots=P_{d}$.

It follows from (4.4) that $n_{r+1}\mu_{r+1}+\cdots+n_{d}\mu_{d}=0$, because
of $P\neq 0$. In this case, we have $n_{2}\mu_{2}+\cdots n_{r}\mu_{r}=nH$.
Therefore we deduce from Lemma 2.6 that

$\displaystyle-6\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}^{2}+2\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}P_{\alpha}\Big{)}\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}P_{\alpha}+\sum_{m\in
I_{1}}\Gamma_{mm}^{1}\Big{)}+\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}^{2}=0,$

that is

$\displaystyle
2\Big{(}\sum_{\alpha=r+1}^{d}n_{\alpha}-3\Big{)}P^{2}\sum_{\alpha=r+1}^{d}n_{\alpha}\mu_{\alpha}^{2}+2\Big{(}\sum_{\alpha=r+1}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}P\sum_{m\in
I_{1}}\Gamma_{mm}^{1}$
$\displaystyle+\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}^{2}=0,$
(4.5)

where we have used $P_{\alpha}=0$ for $\alpha=2,\ldots,r$.

Because we have assumed that the multiplicity of the zero principal curvature
is one, i.e. $I_{1}=\\{1\\}$, (4) becomes

$\displaystyle
2\Big{(}\sum_{\alpha=r+1}^{d}n_{\alpha}-3\Big{)}P^{2}\sum_{\alpha=r+1}^{d}n_{\alpha}\mu_{\alpha}^{2}+\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}^{2}=0.$
(4.6)

Differentiating with respect to $e_{1}$ on both sides of equation (4.6), it
follows from (2.8) that

$\displaystyle
8\Big{(}\sum_{\alpha=r+1}^{d}n_{\alpha}-3\Big{)}P^{3}\sum_{\alpha=r+1}^{d}n_{\alpha}\mu_{\alpha}^{2}+4\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}\Big{(}\sum_{\alpha=r+1}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}P=0.$
(4.7)

Dividing (4.7) by $4P$, and then subtracting (4.6), we have

$\displaystyle\Big{(}\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\Big{)}\Big{(}\sum_{\alpha=2}^{r}n_{\alpha}\mu_{\alpha}^{2}\Big{)}=0.$
(4.8)

Since $S=\sum_{\alpha=2}^{d}n_{\alpha}\mu_{\alpha}^{2}\neq 0$, it follows from
(4.8) that $\sum_{\alpha=2}^{r}n_{\alpha}\mu_{\alpha}^{2}=0$, and hence
$\mu_{\alpha}=0$ for $\alpha=2,\cdots,r$. This is a contradiction since
$\sum_{\alpha=2}^{r}n_{\alpha}\mu_{\alpha}=nH\neq 0$. $\hfill\square$

The proof of Theorem 1.9: Let us restrict the case of a CMC triharmonic
hypersurface $M^{5}$ in $\mathbb{R}^{6}$. We will still work on
$\mathcal{N}=\\{p\in M^{5}:\nabla S(p)\neq 0\\}\neq\emptyset$ and prove
Theorem 1.9 with a contradiction.

When the multiplicity of zero principal curvature is one, we can derive a
contradiction from Theorem 1.8.

By Theorem 1.3, we only need to deal with the case that the multiplicity of
zero principal curvature is two, i.e. $I_{1}=\\{1,2\\}$. In this case, the
principle curvatures on $M^{5}$ are respectively
$0,0,\lambda_{3},\lambda_{4},\lambda_{5}$ with
$\lambda_{3}+\lambda_{4}+\lambda_{5}=5H$.

According to the proof of Theorem 1.8, we can deduce a contradiction when
$P_{\alpha}\neq 0$ or $P_{\alpha}=0$ for all $P_{\alpha}$. Let us consider the
remaining case that $P_{3}=0$, $P_{4}=P_{5}\neq 0$. It follows from (4.4) that
$\lambda_{4}+\lambda_{5}=0$ and $\lambda_{3}=5H$. For simplicity, we denote
$\lambda_{4}=-\lambda_{5}=:\mu$ and $P_{4}=P_{5}=:P$. Then the squared norm of
the second fundamental form $S$ is given by $S=2\mu^{2}+25H^{2}$. Since
$e_{2}(S)=0$, we have $e_{2}(\mu)=0$ and hence
$\Gamma_{44}^{2}=\Gamma_{55}^{2}=0$. In this case, we deduce from (2.9) that
$e_{2}(P)=\Gamma_{44}^{2}P-\Gamma_{44}^{2}\Gamma_{22}^{1}=0$. Moreover,
$e_{1}(S)=4\mu e_{1}(\mu)=4\mu^{2}P$ and hence $e_{2}e_{1}(S)=0$. Since
$e_{1}(S)\neq 0$, $e_{i}(S)=0$ for $i\geq 2$ and $\Gamma_{21}^{1}=0$, we find

$\displaystyle
0=e_{1}e_{2}(S)-e_{2}e_{1}(S)=[e_{1},e_{2}](S)=(\nabla_{e_{1}}e_{2}-\nabla_{e_{2}}e_{1})S=\Gamma_{12}^{1}e_{1}(S),$

which means that $\Gamma_{12}^{1}=-\Gamma_{11}^{2}=0$. Then (2.8) turns into

$\displaystyle e_{1}(P)=P^{2}.$ (4.9)

One the other hand, taking into account the Gauss equation and (2.2), from
Lemma 2.2 we obtain

$\displaystyle 0=R_{1212}=$ $\displaystyle
e_{1}(\Gamma_{22}^{1})-e_{2}(\Gamma_{12}^{1})+\sum_{m}\Big{(}\Gamma_{22}^{m}\Gamma_{1m}^{1}-\Gamma_{12}^{m}\Gamma_{2m}^{1}-(\Gamma_{12}^{m}-\Gamma_{21}^{m})\Gamma_{m2}^{1}\Big{)}$
$\displaystyle=$ $\displaystyle e_{1}(\Gamma_{22}^{1})-(\Gamma_{22}^{1})^{2}.$
(4.10)

Based on the above discussion, (4) becomes

$\displaystyle-4\mu^{2}P^{2}+4\mu^{2}P\Gamma_{22}^{1}+(2\mu^{2}+25H^{2})^{2}=0,$
(4.11)

and hence

$\displaystyle\Gamma_{22}^{1}=\frac{4\mu^{2}P^{2}-(2\mu^{2}+25H^{2})^{2}}{4\mu^{2}P}.$
(4.12)

Noting $e_{1}(\mu)=\mu P$ and differentiating (4.11) with respect to $e_{1}$,
from (4.9) and (4.10) we have

$\displaystyle-4P^{2}+3P\Gamma_{22}^{1}+(\Gamma_{22}^{1})^{2}+2(2\mu^{2}+25H^{2})=0.$
(4.13)

Substituting (4.12) into (4.13) and eliminating the terms of $\Gamma_{22}^{1}$
one gets

$\displaystyle
16\mu^{4}P^{2}\Big{(}2(2\mu^{2}+25H^{2})-4P^{2}\Big{)}+12\mu^{2}P^{2}\Big{(}4\mu^{2}P^{2}-(2\mu^{2}+25H^{2})^{2}\Big{)}$
$\displaystyle+\Big{(}4\mu^{2}P^{2}-(2\mu^{2}+25H^{2})^{2}\Big{)}^{2}=0,$

that is

$\displaystyle
32\mu^{4}P^{2}(2\mu^{2}+25H^{2})-20\mu^{2}P^{2}(2\mu^{2}+25H^{2})^{2}+(2\mu^{2}+25H^{2})^{4}=0.$
(4.14)

Because $2\mu^{2}+25H^{2}\neq 0$, (4.14) becomes

$\displaystyle
32\mu^{4}P^{2}-20\mu^{2}P^{2}(2\mu^{2}+25H^{2})+(2\mu^{2}+25H^{2})^{3}=0.$
(4.15)

Differentiating with respect to $e_{1}$ on (4.15), from (4.9) and (4.10) we
have

$\displaystyle-12\mu^{2}P^{2}-500H^{2}P^{2}+3(2\mu^{2}+25H^{2})^{2}=0.$ (4.16)

Differentiating with respect to $e_{1}$ on (4.16) yields

$\displaystyle-12\mu^{2}P^{2}-250H^{2}P^{2}+6\mu^{2}(2\mu^{2}+25H^{2})=0.$
(4.17)

Eliminating the terms of $P^{2}$ between (4.16) and (4.17) gives

$\displaystyle
2\mu^{2}(12\mu^{2}+500H^{2})=(2\mu^{2}+25H^{2})(12\mu^{2}+250H^{2}),$

that is

$\displaystyle 4\mu^{2}H^{2}-125H^{4}=0,$ (4.18)

which implies that $\mu$ is constant and $S=2\mu^{2}+25H^{2}$ is constant as
well. This contradicts $\nabla S(p)\neq 0$.

In summary, we conclude that $\mathcal{N}=\\{p\in M:\nabla S(p)\neq
0\\}=\emptyset$ and according to (2.6) we know that $M^{n}$ is minimal. This
completes the proof of Theorem 1.9. $\hfill\square$

Acknowledgement: The authors are supported by the NSFC (No.11801246) and
Liaoning Provincial Education Department Project (No.LJKMZ20221561)

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