Let $ N^{n+1}_p(c) $ be the ($ n+1 $)-dimensional pseudo-Riemannian space form of constant sectional curvature $ c $ with index $ s\; (0\leq s\leq n+1) $, and $ \varphi $ : $ M^n_r\rightarrow N^{n+1}_p(c) $ be an isometric immersion from a pseudo-Riemannian mainfold $ M^n_r $ with index $ r $ into $ N^{n+1}_p(c) $. The hypersurface $ M^n_r $ is called $ \lambda $-biharmonic if the immersion $ \varphi $ is a critical point of the following functional (cf. [1-3]),

$ E_{2,\lambda}(\varphi)=E_2(\varphi)+\lambda E(\varphi),\; \; \; \; \; \lambda\in R, $

where $ E(\varphi) $ and $ E_2(\varphi) $ are the energy and bienergy functionals. The Euler-Lagrange equation of $ E_{2,\lambda}(\varphi) $ gives the $ \lambda $-biharmonic equation (cf. [1])

$ \begin{equation*} \tau_2(\varphi)=\lambda\tau(\varphi), \end{equation*} $

where $ \tau(\varphi):=\text{trace}(\nabla \text{d}\varphi) $ and $ \tau_2(\varphi):=-\Delta\tau(\varphi)-\text{trace}\tilde{R}(\text{d}\varphi,\tau(\varphi))\text{d}\varphi $ are the tension and bitension fields of $ \varphi $, and $ \tilde{R} $ is the curvature tensor of $ N^{n+1}_p(c) $. Specially, when $ \lambda=0 $, the hypersurface $ M^n_r $ is called biharmonic hypersurface (cf. [4, 5]).

In 1988, Chen Bang-yen [] initiated the study of $ \lambda $-biharmonic hypersurface $ M^n_r $ of $ \mathbb{E}^{n+1}_s $, and proved that the surface $ M^2 $ in $ \mathbb{E}^3 $ (i.e. $ \mathbb{E}^3_0 $) is minimal, or an open part of a circular cylinder. And then, Ferrández A and Lucus P [] classified such surfaces with $ s=1 $. For $ n=3 $, it has been proved that the hypersurface $ M^3_r $ has constant mean curvature (cf. [8] with $ s=0 $, [9] with $ s=1 $, and [] with $ s=2 $). Based on these results, Arvanitoyeorgos A and Kaimakamis G [10] conjectured that {any $ \lambda $-biharmonic hypersurface in $ \mathbb{E}^{n+1}_s $ has constant mean curvature.} For $ n=4 $, Fu Yu and Zhan Xin [11] gave an affirmative answer to this conjecture with $ s=0 $. Afterwards, Yang Chao, Liu Jiancheng and Du Li [12] showed this conjecture is also true for $ n=4 $ and $ s>0 $ and extended this result to hypersurfaces of non-flat pseudo-Riemannian space forms.

In this paper, we investigate the $ 5 $-dimensional $ \lambda $-biharmonic hypersurface $ M^5_r $ of $ N^6_p(c) $, and prove that the mean curvature is a constant under the assumption that $ M^5_r $ has diagonalizable shape operator. Applying this result, we show that some types of biharmonic hypersurfaces of $ N^6_p(c) $ are minimal.

Let $ M^5_r $ be a $ \lambda $-biharmonic hypersurface of $ N^6_p(c) $ with diagonalizable shape operator $ A $. In this section, we give some important equations and lemmas about the hypersurface $ M^5_r $ under the assumption that the mean curvature $ H $ is not a constant.

According to [13] and [14], the hypersurface $ M^5_r $ satisfies

$ \begin{equation} \begin{cases} \Delta H+\varepsilon H\text{tr}A^2-(5c-\lambda)H=0, \\ 2A\nabla H+5\varepsilon H\nabla H=0, \end{cases} \end{equation} $

(2.1)

where $ \Delta H=\text{div}(\nabla H) $ and $ \varepsilon=\langle \xi, \xi\rangle $, with $ \xi $ a unit normal vector field on $ M^5_r $. The assumption that $ H $ is not a constant tells us that $ \nabla H\neq0 $ on some open subset. Then, we learn from the second equation of (2.1) that $ \nabla H $ is an eigenvector of the shape operator $ A $, with corresponding eigenvalue $ -\frac{5}{2}\varepsilon H $. Considering that $ A $ is diagonalizable, we can choose a local orthonormal frame $ \{e_i\}_{i=1}^5 $ with $ \langle e_i, e_i\rangle=\varepsilon_i=\pm1 $, such that $ \nabla H $ is parallel to $ e_5 $ and $ A(e_i)=\mu_i e_i $ with $ i=1, 2, \cdots, 5 $. Here $ \mu_5=-\frac{5}{2}\varepsilon H $.

For simplicity, we write $ -\frac{5}{2}\varepsilon H $ as $ \mu $. And then, we have

$ \begin{equation} S:=\text{tr}A^2=\sum\limits_{i=1}^{4}\mu_i^2+\mu^2 \end{equation} $

(2.2)

and

$ \begin{equation} \sum\limits_{i=1}^{4}\mu_i=-3\mu \end{equation} $

(2.3)

by $ \text{tr}A=5\varepsilon H $. Since $ e_5 $ is parallel to $ \nabla H $, we get

$ \begin{equation} e_5(\mu)\neq0\ \text{and}\ e_i(\mu)=0,\ 1\leq i\leq 4. \end{equation} $

(2.4)

Let $ \nabla $ be the Levi-Civita connection of $ M^5_r $, and $ \nabla_{e_i}e_j=\sum\limits^5_{k=1}\Gamma^k_{ij}e_k $ with $ 1\leq i,j\leq 5 $. By compatibility and symmetry of the connection $ \nabla $, we obtain

$ \begin{equation} \Gamma^i_{ki}=0,\ \ \Gamma^j_{ki}=-\varepsilon_i\varepsilon_j\Gamma^i_{kj},\ 1\leq i, j, k\leq 5, \end{equation} $

(2.5)

and

$ \begin{equation} \Gamma^5_{ij}=\Gamma^5_{ji},\ 1\leq i,j\leq 4. \end{equation} $

(2.6)

Combining (2.4), (2.5) and (2.6), we deduce from the Codazzi equation $ \langle(\nabla_{e_i}A)e_j,e_k\rangle=\langle(\nabla_{e_j}A)e_i,e_k\rangle $ that

$ \begin{equation} \begin{cases} \Gamma_{5i}^{5}=\Gamma_{ij}^{5}=0,\\ e_i(\mu_j)=(\mu_i-\mu_j)\Gamma^j_{ji},\\ (\mu_i-\mu_j)\Gamma^j_{ki}=(\mu_k-\mu_j)\Gamma^j_{ik}, \end{cases} \end{equation} $

(2.7)

for distinct $ i,j,k $. From (2.4) and the second equation of (2.7), we find $ \mu_j\neq\mu,\ \text{for}\ 1\leq j\leq 4. $ By using (2.5) and the third equation of (2.7), we obtain that

$ \begin{equation} \begin{aligned} \varepsilon_k(\mu_j-\mu_k)\Gamma^k_{ij} =\varepsilon_k(\mu_i-\mu_k)\Gamma^k_{ji} =\varepsilon_i(\mu_j-\mu_i)\Gamma^i_{kj}, \end{aligned} \end{equation} $

(2.8)

for distinct $ i, j, k $ and $ 1\leq i, j, k\leq 4 $, which together with (2.5) implies that

$ (\mu_i-\mu_k)(\mu_j-\mu_k)(\mu_i-\mu_j) (\Gamma^k_{ij}\Gamma^k_{ji}+\Gamma^i_{jk}\Gamma^i_{kj}+\Gamma^j_{ik}\Gamma^j_{ki})=0, $

i.e.

$ \begin{equation} \begin{aligned} \Gamma^k_{ij}\Gamma^k_{ji}+\Gamma^i_{jk}\Gamma^i_{kj}+\Gamma^j_{ik}\Gamma^j_{ki}=0, \end{aligned} \end{equation} $

(2.9)

for distinct $ \mu_i, \mu_j, \mu_k $ and $ 1\leq i, j, k\leq 4 $. Applying Gauss equation for $ \langle R(e_i,e_j)e_k,e_5\rangle $ with distinct $ i, j, k $ and $ 1\leq i, j, k\leq 4 $, combining (2.5) and (2.7), we get

$ \begin{equation} \begin{aligned} &\varepsilon_k(\Gamma^j_{j5}-\Gamma^k_{k5})\Gamma^k_{ij} =\varepsilon_k(\Gamma^i_{i5}-\Gamma^k_{k5})\Gamma^k_{ji} =\varepsilon_i(\Gamma^j_{j5}-\Gamma^i_{i5})\Gamma^i_{kj}. \end{aligned} \end{equation} $

(2.10)

Let $ f $ be a smooth function on $ M^5_r $, and denote by $ f' $, $ f'' $ and $ f^{(k)} $ (the index $ k\geq3 $) the first, second and $ k $-th derivatives of $ f $ along $ e_5 $. It follows from the first equation of (2.1) and the second equation of (2.7) that

$ \begin{equation} \begin{cases} \mu''=-\mu'(\sum^{4}_{i=1}\Gamma^i_{i5})+\varepsilon_5\mu(\varepsilon S-5c+\lambda),\\ \mu_i'=(\mu-\mu_i)\Gamma^i_{i5},\; \; \; 1\leq i\leq4. \end{cases} \end{equation} $

(2.11)

By using the Gauss equation

$ R(e_5,e_i)e_5=c(\langle e_i,e_5\rangle e_5-\langle e_5,e_5\rangle e_i)+\varepsilon\langle A(e_i),e_5\rangle A(e_5)-\varepsilon\langle A(e_5),e_5\rangle A(e_i), $

combining (2.5), (2.6) and the first equation of (2.7), we derive that

$ \begin{equation} (\Gamma^i_{i5})'=-(\Gamma^i_{i5})^2-(\varepsilon\mu\mu_i+c)\varepsilon_5. \end{equation} $

(2.12)

Applying (2.11) and (2.12), $ \sum^{4}_{i=1}(\Gamma^i_{i5})^k $ with the index $ 1\leq k\leq 7 $ can be expressed by $ \mu $, $ \sum^{4}_{i=1}\Gamma^i_{i5} $ and $ \sum^{4}_{i=1}\mu^3_i $ and their derivatives (see Lemma 2.2). By substituting these expressions into the Murnaghan-Nakayama type formula (c.f. [15]) and employing a complex elimination process, we can demonstrate that $ e_j(\sum^{4}_{i=1}\Gamma^i_{i5})=0 $ for $ j = 1, 2, 3, 4 $ (see Lemma 2.3). Furthermore, it can be proved that $ e_j(\mu_i)=0 $ for $ 1 \leq i \leq 5 $ and $ 1 \leq j \leq 4 $ (see Lemma 2.4). This result will play a crucial role in the subsequent proof of our main theorems.

Let $ F_{r,0}=\sum^{4}_{i=1}\mu_i^r $, $ F_{0,s}=\sum^{4}_{i=1}(\Gamma^i_{i5})^s $ and $ F_{r,s}=\sum^{4}_{i=1}\mu_i^r(\Gamma^i_{i5})^s $ with $ r, s=1, 2, \cdots $. Differentiating $ F_{r,0} $, $ F_{0,s} $, and $ F_{r,s} $ with respect to $ e_5 $, combining the second equation of (2.11) and (2.12), we derive the recurrence formulas as Lemma 2.1.

Lemma 2.1  We have

$ \begin{equation} \begin{cases} rF_{r,1}=-F'_{r,0}+r\mu F_{r-1,1},\\ sF_{0,s+1}=-F'_{0,s}-s\varepsilon\varepsilon_5\mu F_{1,s-1}-s\varepsilon_5cF_{0,s-1},\\ (r+s)F_{r,s+1}=-F'_{r,s}+r\mu F_{r-1,s+1}-s\varepsilon\varepsilon_5\mu F_{r+1,s-1}-s\varepsilon_5cF_{r,s-1}, \end{cases} \end{equation} $

(2.13)

for positive integers $ r $ and $ s $, where $ F_{0, 0}=4 $.

Lemma 2.2  Denote $ T:=F_{0,1} $, then we have

$ \begin{equation} \begin{cases} F_{0,2}=-T'+3\varepsilon\varepsilon_5\mu^2-4\varepsilon_5c,\\ F_{0,3}=\frac{1}{2}T''-6\varepsilon\varepsilon_5\mu\mu'-\varepsilon_5(\varepsilon\mu^2+c)T,\\ F_{0,4}=-\frac{1}{6}T^{(3)}+\frac{4}{3}\varepsilon_5(\varepsilon\mu^2+c)T'+\frac{5}{3}\varepsilon\varepsilon_5\mu\mu'T+A_0,\\ F_{0,5}=\frac{1}{24}T^{(4)}-\frac{5}{6}\varepsilon_5(\varepsilon\mu^2+c)T''-\frac{25}{12}\varepsilon\varepsilon_5\mu\mu'T'+A_1T+A_2,\\ F_{0,6}=-\frac{1}{4}\varepsilon\varepsilon_5\mu^3 F_{3,0}-\frac{1}{120}T^{(5)}+\frac{1}{3}\varepsilon_5(\varepsilon\mu^2+c)T^{(3)} +\frac{5}{4}\varepsilon\varepsilon_5\mu\mu'T'' +A_3T'+A_4T+A_5,\\ F_{0,7}=\frac{7}{24}\varepsilon\varepsilon_5(\mu^2\mu'F_{3,0}+\mu^3F_{3,0}')+\frac{1}{720}T^{(6)} -\frac{7}{72}\varepsilon_5(\mu^2\varepsilon+c)T^{(4)}-\frac{35}{72}\varepsilon\varepsilon_5 \mu\mu'T^{(3)}\\ \; \; \; \; \; \; \; \; +A_6T''+A_7T'+A_8T+A_9, \end{cases} \end{equation} $

(2.14)

where the expressions for $ A_0, A_1, \cdots, A_9 $ can be found in (2.21), (2.24), (2.26) and (2.28).

Proof  Since $ F_{0,0}=4 $ and $ F_{1, 0}=-3\mu $, it follows from the second equation of (2.13) with $ s=1 $ that

$ \begin{equation} F_{0,2}=-T'+3\varepsilon\varepsilon_5\mu^2-4\varepsilon_5c. \end{equation} $

(2.15)

The first equation of (2.13) with $ r=1 $ tells us that

$ \begin{equation} F_{1,1}=\mu T+3\mu'. \end{equation} $

(2.16)

Substituting (2.15) and (2.16) into the second equation of (2.13) with $ s=2 $, we have

$ \begin{equation} F_{0,3}=\frac{1}{2}T''-6\varepsilon\varepsilon_5\mu\mu'-\varepsilon_5(\varepsilon\mu^2+c)T. \end{equation} $

(2.17)

We obtain from (2.2) and the first equation of (2.11) that

$ \begin{equation} F_{2,0}=S-\mu^2=\frac{\mu''+\mu'T}{\varepsilon_5\varepsilon\mu}-\mu^2+\frac{5c}{\varepsilon} -\frac{\lambda}{\varepsilon}. \end{equation} $

(2.18)

Putting (2.15), (2.16) and (2.18) into the third equation of (2.13) with $ r=s=1 $ gives that

$ \begin{equation} \begin{aligned} F_{1,2}=-\mu T'-\mu'T-2\mu''+2\varepsilon\varepsilon_5\mu^3-3\varepsilon_5\mu c+\frac{\varepsilon_5\lambda}{2}\mu. \end{aligned} \end{equation} $

(2.19)

Combining (2.15), (2.17) and (2.19), we get from the second equation of (2.13) with $ s=3 $ that

$ \begin{equation} \begin{aligned} F_{0,4}=-\frac{1}{6}T^{(3)}+\frac{4}{3}(\varepsilon\varepsilon_5\mu^2+\varepsilon_5c)T' +\frac{5}{3}\varepsilon\varepsilon_5\mu\mu'T+A_0, \end{aligned} \end{equation} $

(2.20)

where

$ \begin{equation} \begin{aligned} A_0=2\varepsilon\varepsilon_5\mu'^2+4\varepsilon\varepsilon_5\mu\mu''-2\mu^4+4c^2-\frac{\varepsilon\lambda}{2}\mu^2. \end{aligned} \end{equation} $

(2.21)

Since (2.16) and (2.18), it follows from the first equation of (2.13) with $ r=2 $ that

$ \begin{equation} \begin{aligned} F_{2,1}=-\frac{\mu'}{2\varepsilon\varepsilon_5\mu}T'+(\mu^2-\frac{\mu''\mu-\mu'^2} {2\varepsilon\varepsilon_5\mu^2})T+4\mu\mu'+\frac{\mu'\mu''-\mu\mu^{(3)}}{2\varepsilon\varepsilon_5\mu^2}. \end{aligned} \end{equation} $

(2.22)

Combining (2.16), (2.17), (2.19) and (2.22), we derive from the third equation of (2.13) with $ r=1,s=2 $ that

$ \begin{equation} \begin{aligned} F_{1,3}=&\frac{1}{2}\mu T''+\mu'T'+\frac{T}{3}(2\mu''-3\varepsilon\varepsilon_5\mu^3 -\frac{\mu'^2}{\mu}-3\varepsilon_5c\mu)\\ &+\mu^{(3)}-\frac{20}{3}\varepsilon\varepsilon_5\mu^2\mu' -\frac{\mu'\mu''}{3\mu}-\varepsilon_5c\mu' -\frac{\varepsilon_5}{6}\mu'\lambda. \end{aligned} \end{equation} $

(2.23)

Substitute (2.17), (2.20) and (2.23) into the second equation of (2.13) with $ s=4 $, we have

$ \begin{equation*} \begin{aligned} F_{0,5}=\frac{1}{24}T^{(4)}-\frac{5}{6}\varepsilon_5(\varepsilon\mu^2+c)T''-\frac{25}{12}\varepsilon\varepsilon_5\mu\mu'T' +A_1T+A_2, \end{aligned} \end{equation*} $

where

$ \begin{equation} \begin{cases} A_1=\mu^4-\frac{13}{12}\varepsilon\varepsilon_5\mu\mu''-\frac{1}{12}\varepsilon\varepsilon_5\mu'^2+2\varepsilon c\mu^2+c^2,\\ A_2=-2\varepsilon\varepsilon_5\mu\mu^{(3)}-\frac{5}{3}\varepsilon\varepsilon_5\mu'\mu''+\frac{26}{3}\mu^3\mu'+7\varepsilon c\mu\mu'+\frac{5}{12}\varepsilon\mu\mu'\lambda. \end{cases} \end{equation} $

(2.24)

Putting (2.18), (2.19) and (2.22) into the third equation of (2.13) with $ r=2,s=1 $ gives that

$ \begin{equation} \begin{aligned} F_{2,2}=&-\frac{1}{3}\varepsilon\varepsilon_5\mu F_{3,0} +\frac{1}{6\varepsilon\varepsilon_5\mu}\mu'T''-(\mu^2+\frac{\mu'^2-\mu\mu''}{3\varepsilon\varepsilon_5\mu^2})T'\\& -\frac{T}{6\varepsilon\varepsilon_5\mu^4} (8\mu^5\mu'\varepsilon\varepsilon_5+2\varepsilon_5c\mu^3\mu'-\mu^3\mu^{(3)}+3\mu^2\mu'\mu'' -2\mu\mu'^3)\\&+\frac{1}{6\varepsilon\varepsilon_5\mu^3} (-16\varepsilon\varepsilon_5\mu^4\mu''-8\varepsilon\varepsilon_5\mu^3\mu'^2+8\mu^7 -10\varepsilon c\mu^5-10c^2\mu^3 \\ &-2\varepsilon_5c\mu^2\mu''+\mu^2\mu^{(4)}-\mu\mu''^2-2\mu\mu'\mu^{(3)}+2\mu'^2\mu'') +\frac{\varepsilon_5}{3\varepsilon}(\varepsilon\mu^2\lambda+c\lambda). \end{aligned} \end{equation} $

(2.25)

By use of (2.19), (2.20), (2.23), (2.25) and the third equation of (2.13) with $ r=1,s=3 $, we can express $ F_{1,4} $ in terms of $ F_{3,0} $, $ T $ and $ \mu $. And then, applying the second equation of (2.13) with $ s=5 $, we can write $ F_{0,6} $ as following:

$ \begin{equation*} \begin{aligned} F_{0,6}=-\frac{\mu^3}{4}\varepsilon\varepsilon_5F_{3,0}-\frac{1}{120}T^{(5)}+\frac{1}{3}\varepsilon_5(\varepsilon\mu^2+c)T^{(3)} +\frac{5\varepsilon\varepsilon_5}{4}\mu\mu'T''+A_3T'+A_4T+A_5, \end{aligned} \end{equation*} $

where

$ \begin{equation} \begin{cases} A_3=-\frac{1}{30}(46\mu^4+92\varepsilon c\mu^2 -39\varepsilon\varepsilon_5\mu\mu'' -3\varepsilon\varepsilon_5\mu'^2+46c^2),\\ A_4=\frac{1}{120\mu}(61\varepsilon\varepsilon_5\mu\mu^{(3)}-35\varepsilon\varepsilon_5\mu\mu'\mu''-356\mu^4\mu'-446\varepsilon c\mu^2\mu'+40\varepsilon\varepsilon_5\mu'^3),\\ A_5=\frac{1}{120\mu}(93\varepsilon\varepsilon_5\mu^2\mu^{(4)}+48\varepsilon\varepsilon_5\mu\mu'\mu^{(3)} +15\varepsilon\varepsilon_5\mu\mu''^2+40\varepsilon\varepsilon_5\mu'^2\mu''-1204\mu^3\mu'^2\\ \; \; \; \; \; \; \; -888\varepsilon c\mu^2\mu''-768\mu^4\mu''-408\varepsilon c\mu\mu'^2+180\varepsilon\varepsilon_5\mu^7 +270\varepsilon_5c\mu^5-540\varepsilon\varepsilon_5c^2\mu^3\\ \; \; \; \; \; \; \; -480\varepsilon_5c^3\mu) +\frac{\lambda}{24}(9\varepsilon_5\mu^4+27\varepsilon\varepsilon_5\mu^2 -3\varepsilon\mu\mu''-2\varepsilon\mu'^2). \end{cases} \end{equation} $

(2.26)

Substituting (2.22) into the first equation (2.13) with $ r=3 $ yields that

$ \begin{equation} \begin{aligned} F_{3,1}=-\frac{1}{3}F_{3,0}'-\frac{\mu'}{2\varepsilon\varepsilon_5}T'+(\mu^3+\frac{\mu'^2-\mu\mu''} {2\varepsilon\varepsilon_5\mu})T+4\mu^2\mu' +(\frac{\mu'\mu''-\mu\mu^{(3)}}{2\varepsilon\varepsilon_5\mu}). \end{aligned} \end{equation} $

(2.27)

Combining (2.22), (2.23), (2.25) and (2.27), we can obtain the expression of $ F_{2,3} $ from the third equation of (2.13) with $ r=s=2 $. Then, the third equation of (2.13) with $ r=1,s=4 $ gives the expression of $ F_{1,5} $. It follows from the second equation of (2.13) with $ s=6 $ that

$ \begin{equation*} \begin{aligned} F_{0,7}=&\frac{7}{24}\varepsilon\varepsilon_5\mu^2\mu'F_{3,0}+\frac{7}{24}\varepsilon\varepsilon_5\mu^3F_{3,0}'+\frac{1}{720}T^{(6)} -\frac{7\varepsilon_5(\mu^2\varepsilon+c)}{72}T^{(4)}\\&-\frac{35\varepsilon\varepsilon_5\mu\mu'}{72}T^{(3)} +A_6T''+A_7T'+A_8T+A_9, \end{aligned} \end{equation*} $

where

$ \begin{equation} \begin{cases} A_6=\frac{1}{360}(-273\varepsilon\varepsilon_5\mu\mu''-21\varepsilon\varepsilon_5\mu'^2+392\mu^4+784\varepsilon c\mu^2+392c^2),\\ A_7=\frac{1}{720\mu}(-427\varepsilon\varepsilon_5\mu^2\mu^{(3)}+245\varepsilon\varepsilon_5\mu\mu'\mu'' -280\varepsilon\varepsilon_5\mu'^3+3192\mu^4\mu' +3822\varepsilon c\mu^2\mu'),\\ A_8=\frac{1}{720\mu^2}(-127\varepsilon\varepsilon_5\mu^3\mu^{(4)}+148\varepsilon\varepsilon_5\mu^2\mu'\mu^{(3)} +185\varepsilon\varepsilon_5\mu^2\mu''^2-630\varepsilon\varepsilon_5\mu\mu'^2\mu''\\ \; \; \; \; \; \; \; \; \; +1352\mu^5\mu''+1982\varepsilon c\mu^3\mu''+122\varepsilon c\mu^2\mu'^2+280\varepsilon\varepsilon_5\mu'^4 +1656\mu^4\mu'^2\\ \; \; \; \; \; \; \; \; \; -720\varepsilon\varepsilon_5\mu^8-2160\varepsilon_5c\mu^6-2160\varepsilon\varepsilon_5c^2\mu^4 -720\varepsilon_5c^3\mu^2),\\ A_9=\frac{1}{720\mu^2}(-171\varepsilon\varepsilon_5\mu^3\mu^{(5)} -21\varepsilon\varepsilon_5\mu^2\mu'\mu^{(4)}+126\varepsilon\varepsilon_5\mu^2\mu''\mu^{(3)} -280\varepsilon\varepsilon_5\mu\mu'^2\mu^{(3)}\\ \; \; \; \; \; \; \; \; \; +2544\mu^5\mu^{(3)}+3384\varepsilon c\mu^3\mu^{(3)}+2520\varepsilon c\mu^2\mu'\mu'' +8216\mu^4\mu'\mu''-350\varepsilon\varepsilon_5\mu\mu'\mu''^2\\ \; \; \; \; \; \; \; \; \; +280\varepsilon\varepsilon_5\mu'^3\mu''+3104\mu^3\mu'^3-3888\varepsilon\varepsilon_5c^2\mu^3\mu'-13236\varepsilon_5 c\mu^5\mu'-7248\varepsilon\varepsilon_5\mu^7\mu')\\ \; \; \; \; \; \; \; \; \; +\frac{\lambda}{720}(21\varepsilon\mu\mu^{(3)} +35\varepsilon\mu'\mu''-546\varepsilon_5\mu^3\mu'-756\varepsilon\varepsilon_5c\mu\mu'). \end{cases} \end{equation} $

(2.28)

Lemma 2.3  For $ i=1,2,3,4 $, the function $ T $ satisfies $ e_i(T)=0 $.

Proof  Denote $ F_k:=F_{0,k} $, then Murnaghan-Nakayama type formula (c.f. [15]) yields

$ \begin{equation} \begin{cases} 0=F^5_1-10F^3_1F_2+20F^2_1F_3+15F_1F^2_2-30F_1F_4-20F_2F_3+24F_5,\\ 0=F^6_1-9F^4_1F_2+16F^3_1F_3+9F_1^2F^2_2-18F^2_1F_4+3F^3_2-18F_2F_4-8F^2_3+24F_6,\\ 0=F^7_1-7F^5_1F_2+14F^4_1F_3-7F_1^3F^2_2-14F^3_1F_4+28F^2_1F_2F_3+21F_1F^3_2\\ \; \; \; \; \; -42F_1F_2F_4-14F^2_2F_3-28F_3F_4+48F_7. \end{cases} \end{equation} $

(2.29)

It follows from Lemma 2.2 and (2.29) that

$ \begin{equation} \begin{aligned} &T^{(4)}+5TT^{(3)}+10T'T''+10T^2T''-50\varepsilon\varepsilon_5\mu^2T''+20\varepsilon_5cT'' +15TT'^2+10T^3T'\\&-150\varepsilon\varepsilon_5\mu^2TT'-170\varepsilon\varepsilon_5\mu\mu'T' +60\varepsilon_5cTT'+T^5+20\varepsilon_5cT^3-170\varepsilon\varepsilon_5\mu\mu'T^2\\& -50\varepsilon\varepsilon_5\mu^2T^3-62\varepsilon\varepsilon_5\mu'^2T -146\varepsilon\varepsilon_5\mu\mu''T-332\varepsilon c\mu^2T+279\mu^4T +15\lambda\varepsilon\mu^2T\\&+64c^2T-48\varepsilon\varepsilon_5\mu\mu^{(3)} -40\varepsilon\varepsilon_5\mu'\mu''+568\mu^3\mu'-312\varepsilon c\mu\mu' +10\varepsilon\lambda\mu\mu'=0, \end{aligned} \end{equation} $

(2.30)

$ \begin{equation} \begin{aligned} &30\varepsilon\varepsilon_5\mu^4F_{3,0}+\mu T^{(5)}+5(3T'-3T^2 -17\varepsilon\varepsilon_5\mu^2+4\varepsilon_5c)\mu T^{(3)}+10\mu T''^2 -10(4\varepsilon_5 T^3T''\\&+4\varepsilon\mu^2T+4c T +39\varepsilon\mu\mu')\varepsilon_5\mu T''+15\mu T'^3+15(4\varepsilon_5c-3 T^2 -17\varepsilon\varepsilon_5\mu^2) \mu T'^2-45\mu T^4T'\\& +30(13\varepsilon\mu^2-8c)\varepsilon_5\mu T^2T'-150\varepsilon\varepsilon_5\mu^2\mu'TT'+(1129\mu^4 -516\varepsilon\varepsilon_5\mu\mu'' -192\varepsilon\varepsilon_5\mu'^2+64c^2\\ &-832\varepsilon c\mu^2 +45\varepsilon\lambda\mu^2)\mu T'-5\mu T^6 +5(43\varepsilon\mu^2-20c)\varepsilon_5\mu T^4+630\varepsilon\varepsilon_5\mu^2\mu'T^3 +1286\mu^4\mu'T\\&+5(72\varepsilon\varepsilon_5\mu\mu''-109\mu^4-9\varepsilon\lambda\mu^2 -64c^2+36\varepsilon\varepsilon_5\mu'^2+232\varepsilon c\mu^2)\mu T^2 -61\varepsilon\varepsilon_5\mu^2\mu^{(3)}T\\& -40\varepsilon\varepsilon_5\mu'^3T+326\varepsilon c\mu^2\mu'T +35\varepsilon\varepsilon_5\mu\mu'\mu''T-93\varepsilon\varepsilon_5\mu^2\mu^{(4)} -48\varepsilon\varepsilon_5\mu\mu'\mu^{(3)}-15\varepsilon\varepsilon_5\mu\mu''^2\\ & -40\varepsilon\varepsilon_5\mu'^2\mu'' +3(5\lambda-184 c)\varepsilon\mu^2\mu''+1848\mu^4\mu''+3184\mu^3\mu'^2 +2(5\lambda-156 c)\varepsilon\mu\mu'^2\\& -1125\varepsilon\varepsilon_5\mu^7+90(23c-2\lambda)\varepsilon_5\mu^5 +45(\lambda-12c)\varepsilon\varepsilon_5c\mu^3=0, \end{aligned} \end{equation} $

(2.31)

and

$ \begin{equation} \begin{aligned} &70\varepsilon_5(\varepsilon\mu^2 +c)\mu^2T^{(4)}-210\varepsilon\varepsilon_5\mu^5F_{3,0}'-210\varepsilon\varepsilon_5\mu^4\mu'F_{3,0}-\mu^2T^{(6)} -35(T''-3TT')\mu^2T^{(3)}\\& -35\varepsilon_5\mu^2(\varepsilon_5T^3+7\varepsilon\mu^2T-13c T -22\varepsilon\mu\mu')T^{(3)} +7\mu^2(120\varepsilon_5cT^2-15T^4-90\varepsilon\varepsilon_5\mu^2T^2\\& -37\mu^4+50\varepsilon\varepsilon_5\mu\mu'T-15\varepsilon\lambda\mu^2 -584\varepsilon c\mu^2+248c^2+66\varepsilon\varepsilon_5\mu'^2 +198\varepsilon\varepsilon_5\mu\mu'')T''+35(6T^2\\&+3T' -10\varepsilon\varepsilon_5\mu^2+32\varepsilon_5c)\mu^2T'T''+(315T-8144\mu)\mu^2T'^3 +105(\varepsilon_5 T^3+26cT-37\varepsilon\mu^2T\\& -12\varepsilon\mu\mu')\varepsilon_5\mu^2T'^2 +7(100\varepsilon_5c\mu T^3-15\mu T^5 -110\varepsilon\varepsilon_5\mu^3T^3-510\varepsilon\varepsilon_5\mu^2\mu'T^2 +1855\mu^5T\\&+1000c^2\mu T-3580\varepsilon c\mu^3T -360\varepsilon\varepsilon_5\mu^2\mu''T-180\varepsilon\varepsilon_5\mu\mu'^2T -2466\varepsilon c\mu^2\mu'-35\varepsilon\varepsilon_5\mu\mu'\mu''\\& +144\mu^4\mu' +40\varepsilon\varepsilon_5\mu'^3+61\varepsilon\varepsilon_5\mu^2\mu^{(3)} +45\varepsilon\lambda\mu^3T)\mu T' -15\mu^2T^7+105\varepsilon_5 (5\varepsilon\mu^2 -2c)\mu^2T^5\\&+1610\varepsilon\varepsilon_5\mu^3\mu'T^4 +105(17\mu^4-\varepsilon\lambda\mu^2-28\varepsilon c\mu^2+8c^2 +4\varepsilon\varepsilon_5\mu'^2+8\varepsilon\varepsilon_5\mu\mu'')\mu^2T^3\\& +70(143\mu^2-214\varepsilon c)\mu^3\mu'T^2 +127\varepsilon\varepsilon_5\mu^3\mu^{(4)}T-148\varepsilon\varepsilon_5\mu^2\mu'\mu^{(3)}T +630\varepsilon\varepsilon_5\mu\mu'^2\mu''T\\& +2(2264\mu^2 -6871\varepsilon c)\mu^3\mu''T -185\varepsilon\varepsilon_5\mu^2\mu''^2T-(280\varepsilon\varepsilon_5\mu'^2 +6002\varepsilon c\mu^2+2916\mu^4)\mu'^2T\\&-15(841\varepsilon\mu^6 -3014 c\mu^4+49\lambda\mu^4-98\varepsilon c\lambda\mu^2 +2376\varepsilon c^2\mu^2-960c^3)\varepsilon_5\mu^2T +171\varepsilon\varepsilon_5\mu^3\mu^{(5)}\\& +21\varepsilon\varepsilon_5\mu^2\mu'\mu^{(4)}-126\varepsilon\varepsilon_5\mu^2\mu''\mu^{(3)} -3384\varepsilon c\mu^3\mu^{(3)}+280\varepsilon\varepsilon_5\mu\mu'^2\mu^{(3)} -2520\varepsilon c\mu^2\mu'\mu''\\&-2544\mu^5\mu^{(3)}-21\varepsilon\lambda\mu^3\mu^{(3)}+350\varepsilon\varepsilon_5\mu\mu'\mu''^2 -280\varepsilon\varepsilon_5\mu'^3\mu''-(18296\mu^2+35\varepsilon\lambda)\mu^2\mu'\mu'' \\&+948\varepsilon\varepsilon_5\mu^7\mu' +(43476c+1806\lambda)\varepsilon_5\mu^5\mu'-(26352c -756\lambda)\varepsilon\varepsilon_5c\mu^3\mu'=0. \end{aligned} \end{equation} $

(2.32)

Differentiating (2.31) along $ e_5 $, and combining (2.31) and (2.32), we can eliminate $ F_{3,0} $ and $ F_{3,0}' $, and get

$ \begin{equation} \begin{aligned} &6\mu^2T^{(6)}-14\mu\mu'T^{(5)}+5(21T'-21T^2 -105\varepsilon\varepsilon_5\mu^2+42\varepsilon_5c)\mu^2T^{(4)} +210\mu^2T''T^{(3)}\\&-105(2\mu'+\mu T)\mu T'T^{(3)} +5\mathcal{A}_1 T^{(3)} -140\mu\mu'T''^2+420\mu^2T'^2T''+14\mathcal{A}_2\mu T''+7\mathcal{A}_3T'\\&-420( 3T^2-4\varepsilon_5c +10\varepsilon\varepsilon_5\mu^2)\mu^2T'T'' -105(3\mu T+2\mu')\mu T'^3+105\mathcal{A}_4T'^2 +\mathcal{A}_5=0, \end{aligned} \end{equation} $

(2.33)

where

$ \begin{equation*} \begin{aligned} &\mathcal{A}_1=-63\mu^2 T^3 +42 \mu\mu'T^2-105\varepsilon\varepsilon_5\mu^4T +42\varepsilon_5c\mu^2 T-392\varepsilon\varepsilon_5\mu^3\mu' -56\varepsilon_5c\mu\mu',\\& \mathcal{A}_2=40\mu'T^3-30\mu T^4-60\varepsilon_5c\mu T^2+150\varepsilon\varepsilon_5\mu^3T^2 -50\varepsilon\varepsilon_5\mu^2\mu'T+40\varepsilon_5c \mu'T-64c^2 \mu'\\& \; \; \; \; \; \; \; +156c^2\mu -354\varepsilon\varepsilon_5\mu^2\mu''-708\varepsilon c\mu^3 +132\varepsilon\varepsilon_5\mu \mu'^2+546\mu^5 +15\varepsilon\lambda\mu^3,\\& \mathcal{A}_3=90\mu\mu'T^4-45\mu^2T^5+750\varepsilon\varepsilon_5\mu^4T^3 -300\varepsilon_5c\mu^2T^3+(480\varepsilon_5c\mu+1380\varepsilon\varepsilon_5\mu^3)\mu'T^2\\& \; \; \; \; \; \; \; +(210\varepsilon\varepsilon_5\mu^3\mu''+765\mu^6 +330\varepsilon\varepsilon_5\mu^2\mu'^2-1260\varepsilon c\mu^4+360c^2\mu^2-45\varepsilon\lambda\mu^4)T\\& \; \; \; \; \; \; \; +3688\mu^5\mu'-516\varepsilon\varepsilon_5\mu^3\mu^{(3)}+384\varepsilon\varepsilon_5\mu\mu'^3 +132\varepsilon\varepsilon_5\mu^2\mu'\mu''-2140\varepsilon c\mu^3\mu',\\& \mathcal{A}_4=-11\mu^2 T^3 +15\varepsilon_5\varepsilon\mu^4T-6\varepsilon_5c\mu^2 T -8\varepsilon_5c \mu\mu'T+6 \mu\mu'T^2-22\varepsilon_5\varepsilon\mu^3\mu',\\& \mathcal{A}_5=-15\mu^2T^7+70\mu\mu'T^6+(525\varepsilon\mu^2 -210c)\varepsilon_5\mu^2 T^5+1400\varepsilon_5c\mu\mu'T^4-2940\varepsilon c\mu^4T^3\\&\; \; \; \; \; \; \; +\cdots-30552\varepsilon\varepsilon_5\mu^7\mu' -26352\varepsilon\varepsilon_5c^2\mu^3\mu'+756\varepsilon\varepsilon_5\lambda c\mu^3\mu' -714\varepsilon_5\lambda\mu^5\mu'. \end{aligned} \end{equation*} $

By applying (2.30), we eliminate $ T^{(6)}, T^{(5)} $ gradually from (2.33) and derive

$ \begin{equation} \begin{aligned} 5\mu\mathcal{B}_1T^{(4)}+5\mu\mathcal{B}_2T^{(3)} +5\mu\mathcal{B}_3T''+225\mu^2TT'^3+5\mu\mathcal{B}_4T'^2+5\mu\mathcal{B}_5T'+\mathcal{B}_6=0, \end{aligned} \end{equation} $

(2.34)

where

$ \begin{equation*} \begin{aligned} &\mathcal{B}_1=3\mu T'+3\mu T^2-14\mu'T+45\varepsilon\varepsilon_5\mu^3 -18\varepsilon_5c\mu,\\& \mathcal{B}_2=15\mu TT'+15\mu T^3-70\mu'T^2-90T\varepsilon_5c\mu +225\varepsilon\varepsilon_5\mu^3T+88\varepsilon\varepsilon_5\mu^2\mu',\\& \mathcal{B}_3=30\mu T'^2-140\mu'T'T-120\mu\varepsilon_5cT'+60\mu T'T^2 +300\varepsilon\varepsilon_5\mu^3T'+30\mu T^4-140\mu'T^3\\&\; \; \; \; \; \; +300\varepsilon\varepsilon_5\mu^3T^2-120\varepsilon_5c\mu T^2 +1052\varepsilon\varepsilon_5T\mu^2\mu'-280\varepsilon_5c\mu'\mu T -216\varepsilon\varepsilon_5\mu\mu'^2\\&\; \; \; \; \; \; +1584\varepsilon c\mu^4 -24\varepsilon\lambda\mu^3+720\varepsilon\varepsilon_5\mu^2\mu''-1194\mu^5-360c^2\mu,\\& \mathcal{B}_4=75\mu T^3-210\mu'T^2+225\varepsilon\varepsilon_5\mu^3T-90\varepsilon_5c\mu T -246\varepsilon\varepsilon_5\mu^2\mu',\\& \mathcal{B}_5=33\mu T^5-140\mu'T^4-150\varepsilon\varepsilon_5\mu^3T^3+60\varepsilon_5c\mu T^3 +1608\varepsilon\varepsilon_5\mu^2\mu'T^2-840\varepsilon_5c\mu'T^2\\&\; \; \; \; \; \; +\cdots -560\varepsilon\varepsilon_5\mu\mu'\mu''+112\varepsilon\varepsilon_5\mu'^3 +2332\varepsilon c\mu^2\mu'+30\varepsilon\lambda\mu^2\mu',\\& \mathcal{B}_6=15\mu^2T^7-70\mu\mu'T^6-525\varepsilon\varepsilon_5\mu^4T^5 +210\varepsilon\varepsilon_5c\mu^2T^5-1400\varepsilon_5c\mu'\mu T^4\\& \; \; \; \; \; \; +\cdots-336\varepsilon\varepsilon_5\mu^2\mu''\mu^{(3)}+560\varepsilon\varepsilon_5\mu\mu'\mu''^2 +5600\varepsilon\varepsilon_5\mu\mu'^2\mu^{(3)}-560\varepsilon\varepsilon_5\mu'^3\mu''. \end{aligned} \end{equation*} $

From (2.30) and (2.34), we may eliminate $ T^{(4)} $ and obtain

$ \begin{equation} \begin{aligned} &-55\varepsilon\varepsilon_5\mu^3\mu'T^{(3)}+5\mathcal{C}_1\mu^2 T''+5\mathcal{C}_2\mu T'+\mathcal{C}_3=0, \end{aligned} \end{equation} $

(2.35)

where

$ \begin{equation*} \begin{aligned} &\mathcal{C}_1=-44\varepsilon\varepsilon_5\mu\mu'T-132\mu^4+27\varepsilon c\mu^2+ 3\lambda\varepsilon\mu^2-36\varepsilon\varepsilon_5\mu\mu'' +27\varepsilon\varepsilon_5\mu'^2,\\& \mathcal{C}_2=-66\varepsilon\varepsilon_5\mu^2\mu'T^2+(81\varepsilon c\mu^2-376\mu^4 +9\lambda\varepsilon\mu^2-108\varepsilon\varepsilon_5\mu\mu'' +81\varepsilon\varepsilon_5\mu'^2)\mu T\\& \; \; \; \; \; \; \; -335\mu^4\mu'-39\varepsilon\varepsilon_5\mu^2\mu^{(3)}-26\varepsilon c\mu^2\mu' +55\varepsilon\varepsilon_5\mu\mu'\mu''-14\varepsilon\varepsilon_5\mu'^3,\\& \mathcal{C}_3=-660\mu^6T^3+15\varepsilon\lambda\mu^4T^3+135\varepsilon c\mu^4T^3 -180\varepsilon\varepsilon_5\mu^3\mu''T^3+135\varepsilon\varepsilon_5\mu^2\mu'^2T^3\\& \; \; \; \; \; \; \; +\cdots+228\varepsilon c\mu^2\mu'\mu'' -5\varepsilon\lambda\mu^2\mu'\mu''-477\mu^5\mu^{(3)}-24\varepsilon\varepsilon_5\mu^3\mu^{(5)}. \end{aligned} \end{equation*} $

Differentiating (2.35) and combining (2.30), we arrive at

$ \begin{equation} \begin{aligned} &5\mu^2\mathcal{D}_1T^{(3)}+5\mu^2\mathcal{D}_2T''+5\mu^2\mathcal{D}_3T'^2+\mathcal{D}_4T'+\mathcal{D}_5=0, \end{aligned} \end{equation} $

(2.36)

where

$ \begin{equation*} \begin{aligned} &\mathcal{D}_1=3\varepsilon\lambda\mu^4-132\mu^4+27c\varepsilon\mu^2+11\varepsilon\varepsilon_5\mu\mu'T -47\varepsilon\varepsilon_5\mu\mu''-6\varepsilon\varepsilon_5\mu'^2,\\& \mathcal{D}_2=44\varepsilon\varepsilon_5\mu\mu'T^2-396\mu^4T+81\varepsilon c\mu^2T +9\varepsilon\lambda\mu^2T-152\varepsilon\varepsilon_5\mu\mu''T -1677\mu^3\mu'\\& \; \; \; \; \; \; \; +81\varepsilon c\mu^2+302\varepsilon c\mu\mu' +12\varepsilon\lambda\mu\mu'-75\varepsilon\varepsilon_5\mu\mu^{(3)} +\varepsilon\varepsilon_5\mu'\mu'',\\ & \mathcal{D}_3=33\varepsilon\varepsilon_5\mu\mu'T-396\mu^4+81\varepsilon c\mu^2 +9\varepsilon\lambda\mu^2-141\varepsilon\varepsilon_5\mu\mu''-18\varepsilon\varepsilon_5\mu'^2, \end{aligned} \end{equation*} $

$ \begin{equation*} \begin{aligned} & \mathcal{D}_4=330\varepsilon\varepsilon_5\mu^3\mu'T^3-1980\mu^6T^2+45\varepsilon\lambda\mu^4T^2 +405\varepsilon c\mu^4T^2-870\varepsilon\varepsilon_5\mu^3\mu''T^2 \\&\; \; \; \; \; \; \; +\cdots-20\varepsilon\lambda\mu^2\mu'^2-98\varepsilon\varepsilon_5\mu^2\mu'\mu^{(3)} +130\varepsilon\varepsilon_5\mu\mu'^2\mu''+670\varepsilon\varepsilon_5\mu\mu'^3,\\& \mathcal{D}_5=-55\varepsilon\varepsilon_5\mu^3\mu''T^4-165\varepsilon\varepsilon_5\mu^2\mu'^2T^4 -6710\mu^5\mu'T^3+60\varepsilon\lambda\mu^3\mu'T^3 \\&\; \; \; \; \; \; \; +\cdots+102\varepsilon\varepsilon_5\mu^2\mu''\mu^{(4)} -70\varepsilon\varepsilon_5\mu\mu''^3+140\varepsilon\varepsilon_5\mu'^2\mu''^2. \end{aligned} \end{equation*} $

Combining (2.35) and (2.36), we can eliminate $ T^{(3)} $. Then, using the similar methods as the above, we can eliminate $ T^{(2)} $ and derive

$ \begin{equation} \begin{cases} K_1T'+K_2T^2+K_3T+K_4=0,\\ (P_1T+P_2)T'+P_3T^3+P_4T^2+P_5T+P_{6}=0, \end{cases} \end{equation} $

(2.37)

where

$ \begin{equation*} \begin{aligned} &K_1=1102743180\varepsilon\varepsilon_5\mu^{17}\mu'-528660000\varepsilon_5c\mu^{15}\mu' +199809720\mu^{15}\mu^{(3)}\\&\; \; \; \; \; \; \; +\cdots+7406700\varepsilon\varepsilon_5\mu^3\mu'^5\mu''^2 -2617300\varepsilon\varepsilon_5\mu^2\mu'^7\mu''+476000\varepsilon\varepsilon_5\mu\mu'^9,\\& K_2=1102743180\varepsilon\varepsilon_5\mu^{17}\mu'-528660000\varepsilon_5c\mu^{15}\mu' +199809720\mu^{15}\mu^{(3)}\\&\; \; \; \; \; \; \; +\cdots+7406700\varepsilon\varepsilon_5\mu^2\mu'^5\mu''^2 -2617300\varepsilon\varepsilon_5\mu^2\mu'^7\mu''+476000\varepsilon\varepsilon_5\mu\mu'^9,\\& K_3=6080540400\mu^{20}-10535094900\varepsilon c\mu^{18} +7668498420\varepsilon\varepsilon_5\mu^{17}\mu''\\&\; \; \; \; \; \; \; +\cdots -11295900\varepsilon\varepsilon_5\mu'^6\mu''^2+3414600\varepsilon\varepsilon_5\mu\mu'^8\mu'' -476000\varepsilon\varepsilon_5\mu'^{10},\\& K_4=10231198560\mu^{19}\mu'-13403764440\varepsilon c\mu^{17}\mu' +17903160\varepsilon\varepsilon_5\mu^{17}\mu^{(3)}\\&\; \; \; \; \; \; \; +\ldots +476000\varepsilon\varepsilon_5\mu\mu'^8\mu^{(3)}+2462600\varepsilon\varepsilon_5\mu\mu'^7\mu''^2 -476000\varepsilon\varepsilon_5\mu'^9\mu''. \end{aligned} \end{equation*} $

and

$ \begin{equation*} \begin{aligned} &P_1=-96070985841600\varepsilon\varepsilon_5\mu^{26}\mu'+85358626135200\varepsilon_5c\mu^{24}\mu'\\& \; \; \; \; \; \; \; +\cdots+6118007000\varepsilon\varepsilon_5\mu^3\mu'^{11}\mu'' -661640000\varepsilon\varepsilon_5\mu^2\mu'^{13},\\& P_2=529736679648000\mu^{29}-1134527927544000\varepsilon c\mu^{27}\\&\; \; \; \; \; \; +\cdots +3062906000\varepsilon\varepsilon_5\mu^2\mu'^{12}\mu''-199920000\varepsilon\varepsilon_5\mu\mu'^{14},\\& P_3=-96070985841600\varepsilon\varepsilon_5\mu^{26}\mu'+85358626135200\varepsilon_5c\mu^{24}\mu'\\& \; \; \; \; \; \; \; +\cdots+6118007000\varepsilon\varepsilon_5\mu^3\mu'^{11}\mu'' -661640000\varepsilon\varepsilon_5\mu^2\mu'^{13},\\& P_4=96070985841600\varepsilon\varepsilon_5\mu^{26}\mu'' +1313334045084600\varepsilon\varepsilon_5\mu^{25}\mu'^2\\& \; \; \; \; \; \; \; +\cdots-4163348000\varepsilon\varepsilon_5\mu^2\mu'^{12}\mu'' +461720000\varepsilon\varepsilon_5\mu\mu'^{14},\\& P_5=11507407958455200\mu^{28}\mu'-486534246046200\varepsilon\lambda\mu^{26}\mu'\\&\; \; \; \; \; \; \; +\cdots -2736132000\varepsilon\varepsilon_5\mu\mu'^{13}\mu''+199920000\varepsilon\varepsilon_5\mu'^{15},\\& P_{6}=891342018547200\mu^{28}\mu''+18704951237079360\mu^{27}\mu'^2\\&\; \; \; \; \; \; \; +\cdots -2997932000\varepsilon\varepsilon_5\mu\mu'^{12}\mu''^2+199920000\varepsilon\varepsilon_5\mu'^{14}\mu''. \end{aligned} \end{equation*} $

From (2.37), we obtain the following algebraic polynomial equation

$ \begin{equation} \begin{aligned} G_1(\mu,\mu',\mu'',\cdots,\mu^{(7)})T+G_2(\mu,\mu',\mu'',\cdots,\mu^{(8)})=0, \end{aligned} \end{equation} $

(2.38)

where $ G_1(\mu,\mu',\mu'',\cdots,\mu^{(7)}) $ and $ G_2(\mu,\mu',\mu'',\cdots,\mu^{(8)}) $ are polynomials of $ \mu $ and its derivatives. By (2.4), the first equation of (2.7) and the symmetry of connection $ \nabla $, we conclude that

$ \begin{equation} e_i(\mu)=e_i(\mu')=e_i(\mu'')=e_i(\mu^{(k)})=0,\ k\geq 3. \end{equation} $

(2.39)

Acting on (2.38) by $ e_i $, with $ 1\leq i\leq 4 $, combining (2.39), we know

$ \begin{equation} G_1(\mu,\mu',\mu'',\cdots,\mu^{(7)})e_i(T)=0. \end{equation} $

(2.40)

Assume that $ e_j(T)\neq0 $ for some $ 1\leq j\leq 4 $ on some open subset, then (2.40) implies $ G_1=0 $. It follows from (2.38) that $ G_2=0 $. We can eliminate $ \mu', \mu'', \cdots, \mu^{(8)} $ from $ G_1=0 $ and $ G_2=0 $ step by step, and get a non-trival polynomial equation of $ \mu $. So, $ \mu $ is a constant, a contradiction. Therefore, $ e_j(T)=0 $ for any $ 1\leq j\leq 4 $.

Lemma 2.4  For $ 1\leq i\leq5 $ and $ 1\leq j\leq4 $, we have

$ e_j(\mu_i)=0 $

on some open subset.

Proof  For the case that $ M^5_r $ has at most three distinct principal curvatures, the conclusion has been obtained in [13]. We suppose that $ M^5_r $ has five or four distinct principal curvatures. According to Lemma 2.3, we find that $ e_i(T)=e_i(T')=e_i(T'')=e_i(T^{(k)})=0 $ for $ k\geq 3 $ and $ 1\leq i\leq 4 $. It follows from (2.14) that $ e_i(F_k)=0 $ for $ 1\leq i, k\leq4 $, that is

$ \begin{equation} \begin{aligned} \sum^4_{l=1}(\Gamma^l_{l5})^{k-1}e_i(\Gamma^l_{l5})=0. \end{aligned} \end{equation} $

(2.41)

When $ M^5_r $ has five distinct principal curvatures, i.e. $ \mu_1, \mu_2, \mu_3 $ and $ \mu_4 $ are distinct, we know from (2.12) that $ \Gamma^1_{15}, \Gamma^2_{25}, \Gamma^3_{35} $ and $ \Gamma^4_{45} $ are distinct on some open subset. Then, the coefficient determinant of the system (2.41)

$ \begin{equation*} \begin{aligned} \left| \begin{array}{cccc} 1 & 1 & 1 & 1 \\ \Gamma^1_{15} & \Gamma^2_{25} & \Gamma^3_{35} & \Gamma^4_{45} \\ (\Gamma^1_{15})^2& (\Gamma^2_{25})^2 & (\Gamma^3_{35})^2 & (\Gamma^4_{45})^2 \\ (\Gamma^1_{15})^3 & (\Gamma^2_{25})^3 & (\Gamma^3_{35})^3 & (\Gamma^4_{45})^3 \\ \end{array} \right| =\prod\limits_{1\leq i<j\leq4}(\Gamma^j_{j5}-\Gamma^i_{i5})\neq0. \end{aligned} \end{equation*} $

Therefore, (2.41) admits only zero solutions, i.e.,

$ \begin{equation} e_i(\Gamma^1_{15})=e_i(\Gamma^2_{25})=e_i(\Gamma^3_{35})=e_i(\Gamma^4_{45})=0,\; \; \; \; 1\leq i\leq4. \end{equation} $

(2.42)

Furthermore, we have

$ \begin{equation} \begin{aligned} e_je_5(\Gamma^i_{i5})=0. \end{aligned} \end{equation} $

(2.43)

Differentiating (2.12) along $ e_j $, $ 1\leq j\leq4 $, combining (2.42) and (2.43), we obtain $ e_j(\mu_i)=0 $ for $ 1\leq i, j\leq4 $, which together with (2.4) leads to the result.

For the case that $ M^5_r $ has four distinct principal curvatures, without loss of generality, we suppose that $ \mu_1, \mu_2 $ and $ \mu_3 $ are distinct and $ \mu_4=\mu_3 $. It follows from the second equation of (2.11) and (2.12) that $ \Gamma_{45}^4=\Gamma_{35}^3 $ and $ \Gamma^1_{15}, \Gamma^2_{25}, \Gamma^3_{35} $ are distinct on some open subset. The system (2.41) gives that

$ \begin{align*} \left\{ \begin{array}{ll} e_i(\Gamma^1_{15})+e_i(\Gamma^2_{25})+2e_i(\Gamma^3_{35})=0,\\ \Gamma^1_{15}e_i(\Gamma^1_{15})+\Gamma^2_{25}e_i(\Gamma^2_{25})+2\Gamma^3_{35}e_i(\Gamma^3_{35})=0,\\ (\Gamma^1_{15})^2e_i(\Gamma^1_{15})+(\Gamma^2_{25})^2e_i(\Gamma^2_{25})+2(\Gamma^3_{35})^2e_i(\Gamma^3_{35})=0, \end{array} \right. \end{align*} $

which have nonzero coefficient determinant. So, $ e_i(\Gamma^j_{j5})=0 $, $ 1\leq i, j\leq4 $. Furthermore, $ e_j(\mu_i)=0 $ for $ 1\leq i\leq5 $ and $ 1\leq j\leq4 $.

Theorem 3.1   {Let $ M^5_r $ be a $ \lambda $-biharmonic hypersurface of $ N^6_p(c) $ with diagonalizable shape operator, then it has constant mean curvature.

Proof  We employ the method of contradiction to prove this Theorem. Assume that $ H $ is not a constant, now we use the equations and lemmas in Section 2 to derive contradictions. For the case that the number of distinct principal curvatures is not more than three, the contradiction has been derived in [13]. We only need to consider the case that $ M^5_r $ has four or five distinct principal curvatures.

Applying Lemma 2.4, we obtain from the second equation of (2.7) that

$ \begin{equation} \begin{aligned} \Gamma^i_{ij}=0 \; \; \; \text{for}\; \; \; \; 1\leq i,j\leq4. \end{aligned} \end{equation} $

(3.1)

By using Gauss equation for $ \langle R(e_i, e_j)e_i, e_j \rangle $, and combining (2.5), (2.9) and (3.1), we derive that

$ \begin{equation} \begin{aligned} \varepsilon_5\varepsilon_i\varepsilon_j\Gamma^i_{i5}\Gamma^j_{j5} -2\sum\limits_{k\neq i,j}\varepsilon_k\Gamma^k_{ij}\Gamma^k_{ji} =-\varepsilon\varepsilon_i\varepsilon_j\mu_i\mu_j-\varepsilon_i\varepsilon_jc, \end{aligned} \end{equation} $

(3.2)

for distinct $ i, j $, and $ 1\leq i, j\leq 4 $.

Case 1: The terms of $ \{\Gamma^1_{23},\ \Gamma^1_{24},\ \Gamma^1_{34},\ \Gamma^2_{34}\} $ are all zero.

In this case, (3.2) is reduced to

$ \begin{equation} \begin{aligned} \varepsilon_5\Gamma^i_{i5}\Gamma^j_{j5}=-\varepsilon\mu_i\mu_j-c, \end{aligned} \end{equation} $

(3.3)

for distinct $ i,j $ and $ 1\leq i,j\leq4 $, which implies that

$ \begin{equation} \varepsilon_5\Gamma^k_{k5}(\Gamma^i_{i5}-\Gamma^j_{j5})=-\varepsilon\mu_k(\mu_i-\mu_j), \end{equation} $

(3.4)

i.e.

$ \begin{equation} \Gamma^k_{k5}=\varphi\mu_k,\ \ 1\leq k\leq4, \end{equation} $

(3.5)

where $ \varphi=-\varepsilon\varepsilon_5\frac{\mu_i-\mu_j}{\Gamma^i_{i5}-\Gamma^j_{j5}} $ for $ 1\leq i,j\leq4 $, $ i, j\neq k $ and $ \mu_i\neq \mu_j $. Notice that $ \varphi $ does not depend on the indices $ i $, $ j $, or $ k $, and satisfies that

$ \varepsilon_5\varphi^2+\varepsilon=0. $

When $ c\neq0 $, it follows from (3.3) and (3.5) that

$ 0=(\varepsilon_5\varphi^2+\varepsilon)\mu_i\mu_j=-c\neq 0,\ \ \text{for}\ i\neq j\ \text{and}\ i, j=1, 2, 3, 4, $

a contradiction.

When $ c=0 $, we obtain $ \varepsilon\varepsilon_5=-1 $ and $ \varphi^2=1 $. Differentiating both sides of (2.3) with respect to $ e_5 $, combining the second equation of (2.11) and (3.5), we have

$ \begin{equation} \begin{aligned} 3\mu'=12\varphi\mu^2+6\mu\varphi\sum^{3}_{i=1}\mu_i+2\varphi\sum^{3}_{i=1}\mu_i^2 +2\varphi\sum\limits_{1\leq i<j\leq3}\mu_i\mu_j. \end{aligned} \end{equation} $

(3.6)

By differentiating (3.6) along $ e_5 $ and applying the second equation of (2.11), we obtain

$ \begin{equation} \begin{aligned} 3\mu''&=96\mu^3+12\mu_1\mu_2\mu_3+78\mu^2\sum^{3}_{i=1}\mu_i+26\mu\sum^{3}_{i=1}\mu_i^2 +6\sum\limits_{c_1}\mu_i^2\mu_j+44\mu\sum\limits_{1\leq i<j\leq3}\mu_i\mu_j, \end{aligned} \end{equation} $

(3.7)

where $ c_1 $ means $ i,j $ are distinct and $ 1\leq i,j\leq3 $. Using (2.3) and (3.6), the first equation of (2.11) turns into

$ \begin{equation} \begin{aligned} \mu''&=2\mu^3+\varepsilon_5\lambda\mu. \end{aligned} \end{equation} $

(3.8)

Combining (3.7) and (3.8) gives

$ \begin{equation} \begin{aligned} 90\mu^3+12\mu_1\mu_2\mu_3+78\mu^2\sum^{3}_{i=1}\mu_i+26\mu\sum^{3}_{i=1}\mu_i^2 +6\sum\limits_{c_1}\mu_i^2\mu_j +22\mu\sum\limits_{c_1}\mu_i\mu_j-3\varepsilon_5\lambda\mu=0. \end{aligned} \end{equation} $

(3.9)

Since $ \varphi^2=1 $, we know $ \Gamma^i_{i5}=\pm\mu_i $. If $ \Gamma^i_{i5}=\mu_i $, then differentiating (3.9) two times along $ e_5 $, applying (3.6), the second equation of (2.11) and (3.5), we have

$ \begin{equation} \begin{aligned} &3240\mu^4+3726\mu^3\sum\limits_{i=1}^3\mu_i+1710\mu^2\sum\limits_{i=1}^3\mu_i^2 +1602\mu^2\sum\limits_{c_1}\mu_i\mu_j +312\mu\sum\limits_{i=1}^3\mu_i^3+52\sum\limits_{i=1}^3\mu_i^4\\& +104\sum\limits_{c_1}\mu_i^3\mu_j +966\mu\sum\limits_{c_1}\mu_i^2\mu_j -36\varepsilon_5\lambda\mu^2-18\varepsilon_5\lambda\mu\sum\limits_{i=1}^3\mu_i-6\varepsilon_5\lambda\sum\limits_{i=1}^3\mu_i^2 \\&+78\sum\limits_{c_1}\mu_i^2\mu_j^2+140\sum\limits_{c_2}\mu_i^2\mu_j\mu_k -3\varepsilon_5\lambda\sum\limits_{c_1}\mu_i\mu_j+1836\mu\mu_1\mu_2\mu_3=0, \end{aligned} \end{equation} $

(3.10)

and

$ \begin{equation} \begin{aligned} &51840\mu^5+74358\mu^4\sum\limits_{i=1}^3\mu_i+44370\mu^3\sum\limits_{i=1}^3\mu_i^2 +42696\mu^3\sum\limits_{c_1}\mu_i\mu_j+13056\mu^2\sum\limits_{i=1}^3\mu_i^3\\& +38118\mu^2\sum\limits_{c_1}\mu_i\mu_j^2+73656\mu^2\mu_1\mu_2\mu_3 +2176\mu\sum\limits_{i=1}^3\mu_i^4+7592\mu\sum\limits_{c_1}\mu_i\mu_j^3\\& +5694\mu\sum\limits_{c_1}\mu_i^2\mu_j^2+10958\mu\sum\limits_{c_2}\mu_i^2\mu_j\mu_k +540\sum\limits_{c_1}\mu_i\mu_j^4+1080\sum\limits_{c_2}\mu_i\mu_j\mu_k^3\\& +1080\sum\limits_{c_1}\mu_i^2\mu_j^3 +1620\sum\limits_{c_2}\mu_i\mu_j^2\mu_k^2 -234\varepsilon_5\lambda\mu^2\sum\limits_{i=1}^3\mu_i-78\varepsilon_5\lambda\mu\sum\limits_{i=1}^3\mu_i^2\\& -288\varepsilon_5\lambda\mu^3-132\varepsilon_5\lambda\mu\sum\limits_{c_1}\mu_i\mu_j -18\varepsilon_5\lambda\sum\limits_{c_1}\mu_i^2\mu_j-36\varepsilon_5\lambda\mu_1\mu_2\mu_3=0, \end{aligned} \end{equation} $

(3.11)

where $ c_2 $ means $ i, j, k $ are distinct and $ 1\leq i,j,k\leq3 $. When $ M^5_r $ has four distinct principal curvatures, we suppose $ \mu_1=\mu_2 $. By using (3.9)–(3.11), we may eliminate $ \mu_1, \mu_2,\mu_3 $ and get a 165th-degree polynomial equation of $ \mu $ with constant coefficients. Thus $ \mu $ is a constant, a contradiction.

When $ M^5_r $ has five distinct principal curvatures, i.e. $ \mu_1, \mu_2, \mu_3, \mu_4 $ are distinct, we differentiate (3.11) along $ e_5 $ and obtain that

$ \begin{equation} \begin{aligned} &3110400\mu^6+347458\mu^5\sum\limits_{i=1}^3\mu_i+3943458\mu^4\sum\limits_{i=1}^3\mu_i^2+3837024\mu^4\sum\limits_{c_1}\mu_i\mu_j -10368\varepsilon_5\lambda\mu^4\\&+1558152\mu^3\sum\limits_{i=1}^3\mu_i^3 +4527162\mu^3\sum\limits_{c_1}\mu_i\mu_j^2+8826408\mu^3\mu_1\mu_2\mu_3 -11502\varepsilon_5\lambda\mu^3\sum\limits_{i=1}^3\mu_i\\&+\cdots +4352\sum\limits_{i=1}^3\mu_i^6+13056\sum\limits_{c_1}\mu_i\mu_j^5+30972\sum\limits_{c_1}\mu_i^2\mu_j^4+20092\sum\limits_{c_1}(\mu_i\mu_j)^3\\& -156\varepsilon_5\lambda\sum\limits_{i=1}^3\mu_i^4-132\varepsilon_5\lambda\sum\limits_{c_1}\mu_i\mu_j^3 -234\varepsilon_5\lambda\sum\limits_{c_1}(\mu_i\mu_j)^2=0. \end{aligned} \end{equation} $

(3.12)

By using (3.9)–(3.12), we may eliminate $ \mu_1, \mu_2 $ and $ \mu_3 $, and finally derive a 96th-degree polynomial equation of $ \mu $ with constant coefficients, which yields that $ \mu $ is a constant, a contradiction. If $ \Gamma^i_{i5}=-\mu_i $, we can similarly deduce a contradiction.

Case 2: At least two terms of $ \{\Gamma^1_{23},\ \Gamma^1_{24},\ \Gamma^1_{34},\ \Gamma^2_{34}\} $ are nonzero.

Suppose $ \Gamma^1_{23} $ and $ \Gamma^1_{24} $ are nonzero, then $ \mu_1, \mu_2, \mu_3 $ are distinct and $ \mu_1, \mu_2, \mu_4 $ are also distinct by (2.8). It follows from (2.8) and (2.10) that

$ \begin{equation} \begin{aligned} \frac{\Gamma^2_{25}-\Gamma^1_{15}}{\mu_2-\mu_1} =\frac{\Gamma^3_{35}-\Gamma^1_{15}}{\mu_3-\mu_1} =\frac{\Gamma^3_{35}-\Gamma^2_{25}}{\mu_3-\mu_2}, \end{aligned} \end{equation} $

(3.13)

and

$ \begin{equation*} \begin{aligned} \frac{\Gamma^2_{25}-\Gamma^1_{15}}{\mu_2-\mu_1} =\frac{\Gamma^4_{45}-\Gamma^1_{15}}{\mu_4-\mu_1} =\frac{\Gamma^4_{45}-\Gamma^2_{25}}{\mu_4-\mu_2}. \end{aligned} \end{equation*} $

Since $ e_i(\Gamma_{j5}^{j})=e_i(\mu_j)=0 $, $ 1\leq i, j\leq 4 $, we conclude from the above two equations that there exists two smooth functions $ \xi $ and $ \eta $, with $ e_i(\xi)=e_i(\eta)=0,\ 1\leq i\leq 4 $, such that

$ \begin{equation} \begin{aligned} \Gamma^i_{i5}=\xi\mu_i+\eta. \end{aligned} \end{equation} $

(3.14)

Since $ \mu_1\neq\mu_2 $ and $ \Gamma_{15}^1\neq \Gamma_{25}^2 $, we know $ \xi\neq0 $. Differentiating both sides of (3.14) with respect to $ e_5 $, and combining the second equation of (2.11) and (2.12), we obtain that

$ (\xi'+\xi^2\mu+\varepsilon\varepsilon_5\mu+\xi\eta)\mu_i+\eta'+\xi\eta\mu+\eta^2+\varepsilon_5c=0,\ i=1, 2, 3, 4, $

which gives that

$ \begin{equation} \begin{cases} \xi'=-\xi^2\mu-\varepsilon\varepsilon_5\mu-\xi\eta,\\ \eta'=-\xi\eta\mu-\eta^2-\varepsilon_5c. \end{cases} \end{equation} $

(3.15)

Applying (2.2), (2.3), the second equation of (2.11) and (3.14), we deduce that

$ \begin{equation} \begin{aligned} \sum^4_{i=1}\Gamma^i_{i5}=-3\mu\xi+4\eta, \end{aligned} \end{equation} $

(3.16)

and

$ \begin{equation} \begin{aligned} -3\mu'=&-\xi(S+2\mu^2)+7\mu\eta. \end{aligned} \end{equation} $

(3.17)

Using (3.16) and (3.17), the first equation of (2.11) can be written as

$ \begin{equation} \begin{aligned} \mu''=(\mu\xi-\frac{4}{3}\eta)\{\xi(2\mu^2+S)-7\mu\eta\} +\varepsilon_5\mu(\varepsilon S-5c+\lambda). \end{aligned} \end{equation} $

(3.18)

Acting on (3.17) by $ e_5 $, and using (3.15) and (3.17), we derive that

$ \begin{align*} -3\mu''=-\xi S'+\xi\eta(9\mu^2+\frac{10}{3}S)-\frac{1}{3}\xi^2\mu(S+2\mu^2) +\mu(\varepsilon\varepsilon_5S+2\varepsilon\varepsilon_5\mu^2-7\varepsilon_5c-\frac{70}{3}\eta^2), \end{align*} $

which together with (3.18) yields

$ \begin{equation} 3\xi S'-8\xi^2\mu(2\mu^2+S)+2\xi\eta(30\mu^2+S)-\varepsilon_5\mu(14\varepsilon_5\eta^2 +12\varepsilon S-66c+6\varepsilon\mu^2+9\lambda)=0. \end{equation} $

(3.19)

Taking the sum over the index $ i,j $ for $ 1\leq i<j\leq4 $ in (3.2), and combining (3.14) and (2.9), we have

$ \begin{equation} \begin{aligned} (\varepsilon_5\xi^2+\varepsilon)\sum\limits_{1\leq i<j\leq4}\mu_i\mu_j +3\xi\eta\varepsilon_5\sum^4_{i=1}\mu_i+6\varepsilon_5\eta^2+6c=0. \end{aligned} \end{equation} $

(3.20)

Since

$ \sum\limits_{1\leq i<j\leq4}\mu_i\mu_j =5\mu^2-\frac{S}{2}, $

(3.20) turns into

$ \begin{equation} \begin{aligned} (\varepsilon_5\xi^2+\varepsilon)(5\mu^2-\frac{1}{2}S)-9\varepsilon_5\xi\eta\mu+6\varepsilon_5\eta^2 +6c=0. \end{aligned} \end{equation} $

(3.21)

By differentiating (3.21) along $ e_5 $, combining (3.15) and (3.17), we have

$ \begin{equation} \begin{aligned} &(\varepsilon_5\xi^2+\varepsilon)(20\xi\mu^3-26\xi\mu S+3S'+86\eta\mu^2)+6\varepsilon_5\xi^2\eta(2S+7\mu^2)\\& +18\varepsilon_5\eta^2(4\eta-9\varepsilon_5\xi\mu)+18c(4\eta-3\xi\mu)=0. \end{aligned} \end{equation} $

(3.22)

If $ \varepsilon_5\xi^2+\varepsilon=0 $, then it follows from (3.15) that $ \eta=0 $. And then, (3.21) gives $ c=0 $. Follow the process of Case 1, we can deduce a contradiction. If $ \varepsilon_5\xi^2+\varepsilon\neq0 $ on some open subset, we can eliminate $ S' $ from (3.19) and (3.22), and obtain that

$ \begin{equation} \begin{aligned} &(\varepsilon_5\xi^2+\varepsilon)[68\mu^2\xi\eta +10\xi\eta S+6\mu^3(6\xi^2+\varepsilon\varepsilon_5) +6\mu S(2\varepsilon_5\varepsilon-3\xi^2) -120\varepsilon_5c\mu+9\varepsilon_5\lambda\mu\\&-148\eta^2\mu ]-6\xi\eta(7\varepsilon\mu^2+2\varepsilon S -12c)+18\eta^2(9\varepsilon\mu+4\xi\eta\varepsilon_5) +54c\mu\varepsilon\varepsilon_5=0. \end{aligned} \end{equation} $

(3.23)

By (3.21), (3.17) and (3.23) become

$ \begin{equation} \begin{aligned} (\varepsilon_5\xi^2+\varepsilon)\mu'=4\varepsilon_5\xi^3\mu^2 -\frac{25}{3}\varepsilon_5\xi^2\eta\mu+4\varepsilon\xi\mu^2 +4\varepsilon_5\xi\eta^2+4\xi c-\frac{7}{3}\varepsilon\eta\mu, \end{aligned} \end{equation} $

(3.24)

and

$ \begin{equation} \begin{aligned} &-48\varepsilon\varepsilon_5\xi\eta^3-192\varepsilon_5\xi^3\eta^3+544\xi^4\eta^2\mu +170\varepsilon\varepsilon_5\xi^2\eta^2\mu-158\eta^2\mu-492\xi^5\eta\mu^2 +210\xi\eta\mu^2\\&-24(3+5\varepsilon_5)c\xi^3\eta-282\varepsilon\varepsilon_5\xi^3\eta\mu^2 -48\varepsilon \xi\eta+144\xi^6\mu^3+162\varepsilon\varepsilon_5\xi^4\mu^3 +3\varepsilon_5(112c-3\lambda)\xi^4\mu\\&+6\varepsilon(43c-3\lambda)\xi^2\mu-108\xi^2\mu^3 -126\varepsilon\varepsilon_5\mu^3-3\varepsilon_5(26c+3\lambda)\mu=0. \end{aligned} \end{equation} $

(3.25)

Differentiating (3.25) along $ e_5 $ two times, combining (3.15) and (3.24), we deduce that

$ \begin{equation} \begin{aligned} &6072\varepsilon\xi^3\eta^4+9984\varepsilon_5\xi^5\eta^4-1302\varepsilon_5\xi\eta^4 +2198\varepsilon\eta^3\mu-3888\varepsilon_5\xi\mu^4-31744\varepsilon_5\xi^6\eta^3\mu+6660c\xi^7\mu^2\\& +9156\varepsilon_5\xi^2\eta^3\mu-20898\varepsilon\xi^4\eta^3\mu-7650\varepsilon\varepsilon_5c\xi\mu^2 +35376\varepsilon_5\xi^7\eta^2\mu^2+29268\varepsilon\xi^5\eta^2\mu^2+4608c^2\xi^5\\&-12(176c+9\lambda)\varepsilon_5\xi\eta^2 +12(1216c-9\lambda)\varepsilon_5\xi^5\eta^2-16812\varepsilon_5\xi^3\eta^2\mu^2 +24(412c-9\lambda)\varepsilon_5\varepsilon\xi^3\eta^2\\&-10704\varepsilon\xi\eta^2\mu^2 -16344\varepsilon_5\xi^8\eta\mu^3+9(37\lambda-2800c)\xi^6\eta\mu-18666\varepsilon\xi^6\eta\mu^3 +13716\varepsilon_5\xi^4\eta\mu^3\\&+27\varepsilon\varepsilon_5\xi^4\eta\mu(9\lambda-782c) +27(17\lambda+212c)\xi^2\eta\mu+2016\varepsilon_5\eta\mu^3 +63(26c+\lambda+120\varepsilon_5c)\varepsilon\varepsilon_5\eta\mu\\& +2592\varepsilon_5\xi^9\mu^4+3888\varepsilon\xi^7\mu^4 -108\varepsilon_5c\lambda\xi^5-3888\varepsilon_5\xi^5\mu^4 +5670\varepsilon\varepsilon_5c\xi^5\mu^2-8640c\xi^3\mu^2\\&+18054\varepsilon\xi^2\eta\mu^3 -9072\varepsilon\xi^3\mu^4+72\varepsilon\xi^3c(53c-3\lambda)-36\varepsilon_5c\xi(22c+3\lambda)=0, \end{aligned} \end{equation} $

(3.26)

and

$ \begin{equation} \begin{aligned} &54432\xi^{12}\mu^5-476280\xi^{11}\eta\mu^4+1685376\xi^{10}\eta^2\mu^3 +89424\varepsilon\varepsilon_5\xi^{10}\mu^5 +193428\varepsilon_5c\xi^{10}\mu^3\\&-2836176\xi^9\eta^3\mu^2-2997\varepsilon_5\lambda\xi^9\eta\mu^2 -1046700c\xi^9\eta\mu^2-1020780\varepsilon_5c\xi^9\eta\mu^2 -647838\varepsilon\varepsilon_5\xi^9\eta\mu^4\\& +2001852\varepsilon\varepsilon_5\xi^8\eta^2\mu^3+2147328c\varepsilon_5\xi^8\eta^2\mu +\cdots+270426\xi\eta^3\mu^2 +78732\varepsilon\varepsilon_5\xi\eta\mu^4\\&-2187\varepsilon_5\lambda\xi\eta\mu^2 +34704\varepsilon\varepsilon_5c^2\xi\eta+1728\varepsilon\xi\eta^3\lambda +1080\varepsilon\varepsilon_5c\lambda\xi\eta+324\lambda\eta^2\mu +46176\varepsilon\varepsilon_5\xi\eta^5\\& +11664\mu^5+16902c\varepsilon\mu^3+135c\lambda\mu-306\varepsilon_5\lambda\eta^2\mu=0. \end{aligned} \end{equation} $

(3.27)

We can eliminate $ \xi $ and $ \eta $ from (3.25), (3.26) and (3.27), and obtain an algebraic polynomial equation of $ \mu $. Thus, $ \mu $ is a constant, which leads to a contradiction.

Case 3: Only one term of $ \{\Gamma^1_{23},\ \Gamma^1_{24},\ \Gamma^1_{34},\ \Gamma^2_{34}\} $ is nonzero.

Suppose $ \Gamma^1_{23}\neq 0 $ and $ \Gamma^1_{24}=\Gamma^1_{34}=\Gamma^2_{34}=0 $, then (3.13) holds. And then, we have

$ \begin{equation} \begin{aligned} \Gamma^i_{i5}=\xi\mu_i+\eta,\; \; \; i=1,2,3. \end{aligned} \end{equation} $

(3.28)

Here the smooth functions $ \xi $ and $ \eta $ satisfy (3.15) and $ \xi\neq 0 $ on some open subset. Letting $ i=4, j=2,3 $ respectively in (3.2), we obtain

$ \begin{equation} \begin{cases} \varepsilon_5\Gamma^4_{45}\Gamma^2_{25}=-\varepsilon\mu_4\mu_2-c,\\ \varepsilon_5\Gamma^4_{45}\Gamma^3_{35}=-\varepsilon\mu_4\mu_3-c. \end{cases} \end{equation} $

(3.29)

It follows that

$ \varepsilon_5\Gamma^4_{45}(\Gamma^2_{25}-\Gamma^3_{35})=-\varepsilon\mu_4(\mu_2-\mu_3), $

which together with (3.28) shows that

$ \begin{equation} \begin{aligned} \varepsilon_5\xi\Gamma^4_{45}=-\varepsilon\mu_4. \end{aligned} \end{equation} $

(3.30)

Substituting (3.28) into (3.29), and combining (3.30) gives

$ \begin{equation} \begin{aligned} \varepsilon\eta\mu_4=c\xi. \end{aligned} \end{equation} $

(3.31)

Taking the sum over the index $ i,j $ for $ 1\leq i<j\leq3 $ in (3.2), and combining (2.9) and (3.28), we have

$ \begin{equation} \begin{aligned} (\varepsilon_5\xi^2+\varepsilon)(\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3)+2\varepsilon_5\xi\eta (\mu_1+\mu_2+\mu_3)+3\varepsilon_5\eta^2+3c=0. \end{aligned} \end{equation} $

(3.32)

By using (2.2) and (2.3), (3.32) turns into

$ \begin{equation} \begin{aligned} (\varepsilon_5\xi^2+\varepsilon)(10\mu^2+6\mu\mu_4+2\mu_4^2-S)+2\varepsilon_5\eta(3\eta-2\xi\mu_4-6\xi\mu) +6c=0. \end{aligned} \end{equation} $

(3.33)

If $ \varepsilon_5\xi^2+\varepsilon=0 $, then $ \eta=0 $ and $ c=0 $ by (3.15) and (3.31). We can derive a contradiction as Case 1. In the following, we treat the case that $ \varepsilon_5\xi^2+\varepsilon\neq0 $ on some open subset. From (2.3), (2.11), (3.28) and (3.30), we deduce that

$ \begin{equation} \begin{aligned} 3\xi\mu'=&2\xi^2\mu^2+\xi^2\mu\mu_4-\xi^2\mu_4^2+\xi^2 S-6\xi\eta\mu-\xi\eta\mu_4+\varepsilon\varepsilon_5\mu\mu_4 -\varepsilon\varepsilon_5\mu_4^2, \end{aligned} \end{equation} $

(3.34)

and

$ \begin{equation} \begin{aligned} 3\xi^2\mu''=&6\xi^4\mu^3+5\xi^4\mu^2\mu_4-2\xi^4\mu\mu_4^2-\xi^4\mu_4^3 +3\xi^4\mu S+\xi^4\mu_4S-24\xi^3\eta\mu^2 -12\xi^3\eta\mu\mu_4\\&+2\xi^3\eta\mu_4^2-3\xi^3\eta S+3\varepsilon\varepsilon_5\xi^2\mu S+5\varepsilon\varepsilon_5\xi^2\mu^2\mu_4+18\xi^2\eta^2\mu -\varepsilon\varepsilon_5\xi^2\mu\mu_4^2+3\xi^2\eta^2\mu_4\\& +3(\lambda-5c)\varepsilon_5\xi^2\mu -2\varepsilon\varepsilon_5\xi^2\mu_4^3+\varepsilon\varepsilon_5\xi^2\mu_4S -9\varepsilon\varepsilon_5\xi\eta\mu\mu_4 +2\varepsilon\varepsilon_5\xi\eta\mu_4^2+\mu\mu_4^2-\mu_4^3. \end{aligned} \end{equation} $

(3.35)

Differentiating (3.34) along $ e_5 $, combining (3.15), (3.31) and (3.35), we get

$ \begin{equation} \begin{aligned} &16\xi^4\mu^3-2\xi^4\mu_4^3+12\xi^4\mu^2\mu_4 +2(4\mu +\mu_4)\xi^4 S-6\xi^4\mu\mu_4^2-20\xi^3\eta\mu\mu_4 -48\xi^3\eta\mu^2\\&-2\xi^3\eta\mu_4^2+(3+9\varepsilon\varepsilon_5)\xi^2\mu^2\mu_4 -(9+\varepsilon\varepsilon_5)\xi^2\mu\mu_4^2 +(9\lambda-63c)\varepsilon_5\xi^2\mu-3\varepsilon_5c\xi^2\mu_4 \\&-3\xi^3S'+(6-4\varepsilon\varepsilon_5)\xi^2\mu_4^3 +12\varepsilon\varepsilon_5\xi^2\mu S+2\varepsilon\varepsilon_5\xi^2\mu_4S +6\varepsilon\varepsilon_5\xi^2\mu^3 +(3+4\varepsilon\varepsilon_5)\xi\eta\mu_4^2 \\&-3(1+6\varepsilon\varepsilon_5)\xi\eta\mu\mu_4 +3(\varepsilon\varepsilon_5-1)\mu^2\mu_4 +(5-9\varepsilon\varepsilon_5)\mu\mu_4^2 +2(3\varepsilon\varepsilon_5-1)\mu_4^3=0. \end{aligned} \end{equation} $

(3.36)

Acting on (3.33) by $ e_5 $ and combining (3.36), we derive that

$ \begin{equation} \begin{aligned} &36\xi^6\mu^3+4\xi^6\mu_4^3+20\xi^6\mu\mu_4^2+16\xi^6\mu^2\mu_4-4\xi^6\mu_4S -18\xi^6\mu S+4\xi^5\eta\mu_4^2+6\xi^5\eta S+84\xi^5\eta\mu^2\\& +60\xi^5\eta\mu\mu_4+6(\varepsilon\varepsilon_5-1)\xi^4\mu_4^3 -36(\mu_4+3\mu)\xi^4\eta^2 +(21+5\varepsilon\varepsilon_5)\xi^4\mu^2\mu_4-15\varepsilon_5c\xi^4\mu_4\\& +9(\lambda-11c)\varepsilon_5\xi^4\mu-(6\mu +2\mu_4)\varepsilon\varepsilon_5\xi^4S +(33\varepsilon\varepsilon_5-15 )\xi^4\mu\mu_4^2 +42\varepsilon\varepsilon_5\xi^4\mu^3 +36\varepsilon\varepsilon_5\xi^3\eta\mu^2\\&+3(6\varepsilon\varepsilon_5 -5)\xi^3\eta\mu\mu_4 +(15-4\varepsilon\varepsilon_5)\xi^3\eta\mu_4^2 +36\varepsilon_5c\xi^3\eta +36\xi^3\eta^3+(24\varepsilon\varepsilon_5-14)\xi^2\mu^2\mu_4\\& +6(3-4\varepsilon\varepsilon_5)\xi^2\mu\mu_4^2 +2(6\mu+\mu_4)\xi^2 S +9(\lambda-7c)\varepsilon\xi^2\mu-3\varepsilon c\xi^2\mu_4 +(4+3\varepsilon\varepsilon_5)\xi\eta\mu_4^2\\& +6\xi^2\mu^3+(6-2\varepsilon\varepsilon_5)\mu_4^3 +3(1-\varepsilon\varepsilon_5)\mu^2\mu_4+(5\varepsilon\varepsilon_5-9)\mu\mu_4^2 -3(6+\varepsilon\varepsilon_5)\xi\eta\mu\mu_4 =0. \end{aligned} \end{equation} $

(3.37)

By using (3.31) and (3.33), (3.34) and (3.37) reduce to

$ \begin{equation} \begin{aligned} 3\eta^2(\xi^2+\varepsilon\varepsilon_5)\mu'=& \xi^5c^2+7\varepsilon c\xi^4\eta\mu+12\xi^3\eta^2\mu^2-5\varepsilon c\xi^3\eta^2 -18\xi^2\eta^3\mu+8\varepsilon_5c\xi^2\eta\mu\\& +12\varepsilon\varepsilon_5\xi\eta^2\mu^2+6\xi\eta^4 +5\varepsilon_5c\xi\eta^2-\xi c^2-6\varepsilon\varepsilon_5\eta^3\mu+\varepsilon c\eta\mu, \end{aligned} \end{equation} $

(3.38)

and

$ \begin{equation} \begin{aligned} &4\varepsilon_5\xi^{10}c^3+40\varepsilon\varepsilon_5c^2\xi^{9}\eta\mu+2(\varepsilon +3\varepsilon_5)c^3\xi^{8}-32\varepsilon\varepsilon_5c^2\xi^{8}\eta^2 -216\varepsilon_5c\xi^7\eta^3\mu+144\varepsilon\varepsilon_5\xi^7\eta^3\mu^3\\& +(19-15\varepsilon\varepsilon_5)c^2\xi^6\eta^2 +132\varepsilon_5c\xi^{8}\eta^2\mu^2 +(15\varepsilon\varepsilon_5+31)c^2\xi^7\eta\mu +3(61\varepsilon -7\varepsilon_5)c\xi^6\eta^2\mu^2\\&+6(\varepsilon -\varepsilon_5)c^3\xi^6-360\varepsilon\varepsilon_5\xi^6\eta^4\mu^2 +84\varepsilon_5c\xi^6\eta^4+(39-63\varepsilon\varepsilon_5)c^2\xi^5\eta\mu +288\varepsilon\varepsilon_5\xi^5\eta^5\mu\\& +15\varepsilon_5c\xi^5\eta^3+9\varepsilon(5c-\lambda)\xi^5\eta^3\mu +162\xi^5\eta^3\mu^3-2(3\varepsilon_5+\varepsilon)c^3\xi^4 +2(19\varepsilon\varepsilon_5-9)c^2\xi^4\eta^2\\&-9(5\varepsilon +3\varepsilon_5)c\xi^4\eta^2\mu^2 -252\xi^4\eta^4\mu^2-72\varepsilon\varepsilon_5\xi^4\eta^6 -24\varepsilon c\xi^4\eta^4+144\xi^3\eta^5\mu -108\varepsilon\varepsilon_5\xi^3\eta^3\mu^3\\&+(33\varepsilon\varepsilon_5-59)c^2\xi^3\eta\mu +18(\varepsilon c+15\varepsilon_5c-\lambda)\xi^3\eta^3\mu -36\xi^2\eta^6-48\varepsilon_5c\xi^2\eta^4+108\varepsilon\varepsilon_5\xi^2\eta^4\mu^2\\& -3c(25\varepsilon+9\varepsilon_5)\xi^2\eta^2\mu^2 +2(\varepsilon_5-3\varepsilon)c^3\xi^2 -72\varepsilon\varepsilon_5\xi\eta^5\mu +9\varepsilon(c-\lambda)\xi\eta^3\mu+3\varepsilon_5c\xi\eta^3\\& +(9-5\varepsilon\varepsilon_5)c^2\xi\eta\mu-126\xi\eta^3\mu^3 -(13+3\varepsilon\varepsilon_5)c^2\xi^2\eta^2+3c(\varepsilon_5-\varepsilon)\eta^2\mu^2=0. \end{aligned} \end{equation} $

(3.39)

Differentiating (3.39) two times along $ e_5 $ and using (3.15) and (3.38), we have

$ \begin{equation} \begin{aligned} &40\varepsilon\varepsilon_5c^4\xi^{14}+424\varepsilon_5c^3\xi^{13}\eta\mu+31c^4\xi^{12} +15\varepsilon\varepsilon_5c^4\xi^{12}-536\varepsilon_5c^3\xi^{12}\eta^2\\& +1560\varepsilon\varepsilon_5c^2\xi^{12}\eta^2\mu^2 +495\varepsilon c^3\xi^{11}\eta\mu -81\varepsilon_5c^3\xi^{11}\eta\mu+2232\varepsilon_5c\xi^{11}\eta^3\mu^3\\& +\cdots+702\varepsilon\lambda\xi^3\eta^5\mu+18\varepsilon_5c\xi^2\eta^6 +45\varepsilon_5c\eta^4\mu^2+96c^2\xi\eta^3\mu+1944\xi\eta^7\mu\\& -45\varepsilon\varepsilon_5c\lambda\xi^2\eta^4-15c^2\eta^2\mu^2 +9\varepsilon c^2\lambda\xi^2\eta^2+72c\lambda\xi\eta^3\mu -9c^4\xi^2+27\varepsilon\lambda\eta^4\mu^2\\&+1224c^2\xi^2\eta^2\mu^2 -6156\xi^2\eta^6\mu^2+108\varepsilon_5c\xi\eta^3\mu^3-81\varepsilon c\eta^4\mu^2=0, \end{aligned} \end{equation} $

(3.40)

and

$ \begin{equation} \begin{aligned} &424c^5\xi^{18}+4408\varepsilon c^4\xi^{17}\eta\mu +495\varepsilon\varepsilon_5c^5\xi^{16}-8312\varepsilon c^4\xi^{16}\eta^2 -81c^5\xi^{16}\\&+15816c^3\xi^{16}\eta^2\mu^2 +5501\varepsilon_5c^4\xi^{15}\eta\mu-2055\varepsilon c^4\xi^{15}\eta\mu -74160c^3\xi^{15}\eta^3\mu\\&-217944\varepsilon c^2\xi^{14}\eta^4\mu^2 -1216c^5\xi^{14}-4692c^3\xi^{14}\eta^2\mu^2 +10014\varepsilon\varepsilon_5c^3\xi^{14}\eta^2\mu^2\\& +\cdots+630c^2\lambda\xi^2\eta^4 +3024\varepsilon_5c\lambda\xi^2\eta^6-14742\varepsilon\varepsilon_5\lambda\xi^2\eta^6\mu^2 -1227c^3\xi^2\eta^2\mu^2\\&-2898\varepsilon\varepsilon_5c^3\xi^2\eta^4 -684\varepsilon\varepsilon_5c^2\lambda\xi\eta^4\mu+48c^3\xi\eta^3\mu -126\varepsilon_5c^3\lambda\xi^2\eta^2+36\varepsilon_5c^4\xi\eta\mu\\& -16362\varepsilon_5c^2\xi\eta^3\mu^3-1134\lambda\eta^6\mu^2 -486\varepsilon_5c\lambda\eta^4\mu^2+8c^5\xi^2+468\varepsilon_5c^2\eta^4\mu^2=0. \end{aligned} \end{equation} $

(3.41)

Eliminating $ \xi $ and $ \eta $ from (3.39)–(3.41), we obtain a polynomial equation of $ \mu $. Therefore $ \mu $ is a constant, which is a contradiction.

Applying Theorem 3.1, we have the following corollary for biharmonic hypersurfaces of $ N^6_p(c) $.

Corollary 3.2  Let $ M^5_r $ be a biharmonic hypersurface of $ N^6_p(c) $ with diagonalizable shape operator and $ c\varepsilon\leq 0 $, where $ \varepsilon=\langle \xi, \xi\rangle $ with $ \xi $ an unit normal vector field, then it must be minimal.

Proof   According to Theorem 3.1, the mean curvature $ H $ is a constant. It follows from the first equation of (2.1) that

$ \begin{equation} H(\text{tr}A^2-5c\varepsilon)=0. \end{equation} $

(3.42)

Let $ \mu_1, \mu_2, \cdots, \mu_5 $ are principal curvatures of $ M^5_r $. Assume that $ H\neq 0 $, then $ \mu_k\neq 0 $ for some $ 1\leq k\leq 5 $. Considering that the shape operator $ A $ is diagonalizable, we know $ \text{tr}A^2=\sum_{i=1}^5\mu_i^2> 0 $, which together with $ c\varepsilon\leq 0 $ tells us that $ \text{tr}A^2-5c\varepsilon>0 $. However, (3.42) implies that $ \text{tr}A^2-5c\varepsilon=0 $, a contradiction.

Remark   From Corollary 3.1, the hypersurface $ M^5_r $ of $ N^6_p(c) $ is minimal when $ c=0 $, or $ c>0 $ and the normal vector field is time-like, or $ c<0 $ and the normal vector field is space-like. Specially, when $ r=p=0 $, the above results degenerate into the results in [15].