In the late 1960s, Chern proposed the following notable conjecture[1, 2]:

Conjecture 1.1   Let $ {M^n} $ be a closed immersed minimal hypersurface of the unit sphere $ {\mathbb{S}^{n + 1}} $ with constant scalar curvature $ {R_M} $. Then for any dimension $ n $, the set of all possible values of $ {R_M} $ is discrete.

With further study of the above conjecture, mathematicians not only realized the importance of Conjecture 1.1, but also proposed a strengthened version:

Conjecture 1.2 (Chern's Conjecture)   Let $ {M^n} $ be a closed immersed minimal hypersurface of the unit sphere $ {\mathbb{S}^{n + 1}} $ with constant scalar curvature, then $ {M^n} $ is isoparametric.

Up to now, the conjecture remains far from being completely resolved. S. T. Yau has reintroduced it as the 105th problem in reference[3]. For the latest research on this conjecture, see the works of Scherfner-Weiss[4], Scherfner-Weiss-Yau[5], Ge-Tang[6], and others. In 1968, J. Simons proved the following theorem in reference[7]:

Theorem 1.3 (Simon's Inequality)   Let $ {M^n} $ be a closed immersed minimal hypersurface in the unit sphere $ {\mathbb{S}^{n + 1}} $ with squared norm $ S $ of the second fundamental form. Then the inequality

$ \begin{equation} \int_{{M^n}} {S(S - n)dV_M \ge 0} \end{equation} $

(1.1)

holds eternally on $ {M^n} $. In particular, if $ 0 \le S \le n $, then $ S \equiv 0 $ or $ S \equiv n $ on $ {M^n} $.

In Theorem 1.3, it is evident that the minimal hypersurface satisfying $ S \equiv 0 $ is the equatorial sphere. The classification of $ S \equiv n $ is proven by Chern-do Carmo-Kobayshi[8, 9] and Lawson[10]: the Clifford torus $ {S^k}(\sqrt {{k \over n}} ) \times {S^{n - k}}(\sqrt {{{n - k} \over n}} )(1 \le k \le n - 1) $ is the only closed minimal hypersurface in $ {S^{n + 1}} $ with $ S \equiv n $. By classifying isoparametric hypersurfaces with constant principal curvatures, it can be seen that the above two hypersurfaces are isoparametric hypersurfaces with multiplicity of principal curvatures being $ 1 $ and $ 2 $, respectively. For closed immersed minimal hypersurfaces in $ {\mathbb{S}^{n + 1}} $, it is easy to observe the following relationship from the Gauss equation:

$ \begin{equation} {R_M} = n(n - 1) - S. \end{equation} $

(1.2)

Therefore, Simons inequality is regarded as the first breakthrough for Conjecture 1.1.

In 1983, Peng and Terng made significant progress on Chern's Conjecture 1.1, proving that if \(S > n \), then \(S > n + \frac{1}{12n} \). In 1993, Chang[11] completed the proof of Chern's Conjecture 1.2 for the case \(n = 3 \): if \(S > 3 \), then \(S \ge 6 \). In recent years, Yang-Cheng[12] and Suh-Yang[13] have further improved the constant term from \(\frac{1}{12n} \) to \(\frac{3n}{7} \), and Cheng-Wei[14] proved that if \(S > n \), then \(S > 1.8252n - 0.712898 \) under the additional condition that $ trh^3 $ is constant. However, the question for higher dimension, "If \(S \equiv \) constant \(> n \), then \(S \ge 2n \)?" remains open.

In Section 3, this article systematically studies the height functions \({\varphi ^A} \) and \({\psi ^A} \) (defined below) and their important properties.

Definition 1.4 (Height Function)   Let $ {M^n} $ be a hypersurface isometrically immersed in $ {\mathbb{S}^{n + 1}}( \subset {\mathbb{R}^{n + 2}}) $, $ a $ be a fixed unit vector in $ {\mathbb{R}^{n + 2}} $, and $ \left\{ {\left. x \right|{e_1}, \cdots , {e_n}, {e_{n + 1}}} \right\} $ be the Darboux frame of the hypersurface $ {M^n} $. Define smooth functions $ \varphi $ and $ \psi $: $ {M^n} \to \mathbb{R} $, satisfying

$ \varphi \left( p \right) = \langle {x\left( p \right), a} \rangle , \psi \left( p \right) = \langle {{e_{n + 1}}(p), a} \rangle , \; \forall p \in M. $

Then $ \varphi $ and $ \psi $ are regarded as the height functions of the position vector $ x $ and the normal vector $ e_{n+1} $ in the direction of the vector $ a $. In particular, when $ a $ is the natural basis coordinate vector $ {E_A} = ( {0, \cdots , 0, \mathop 1\limits_A , 0, \cdots 0} ) $, $ {\varphi ^A} = \langle {x, {E_A}} \rangle $ and $ {\psi ^A} = \langle {{e_{n + 1}}, {E_A}} \rangle $ are the coordinate functions of $ x $ and $ e_{n+1} $ in $ {\mathbb{R}^{n + 2}} $ at the $ A $-th position ($ A=1, 2, \cdots, n+2 $).

In Section 4, utilizing the important properties of the height functions $ {\varphi ^A} $ and $ {\psi ^A} $, the motion equations of hypersurfaces in $ {\mathbb{S}^{n + 1}} $, the integrability conditions (Gauss-Codazzi equations), and the implications of Green's theorem, we calculate the Laplacians of the second and third covariant differential forms of $ {\varphi ^A} $, and obtain two integral equations for the first, second, and third gap point terms that simultaneously include the second fundamental form squared norm $ S $.

Theorem 1.5   Let $ {M^n} $ be a closed immersed minimal hypersurface in the unit sphere $ {\mathbb{S}^{n + 1}} $ with constant scalar curvature. Then the integrals

$ \begin{equation} \int_{{M^n}} {{{S(n - S) + \left\| {\nabla h} \right\|}^2}} dV_M= 0, \end{equation} $

(1.3)

$ \begin{equation} \int_{{M^n}} {S(S - n)(2n - S) + {{\left\| {{\nabla ^2}h} \right\|}^2} - 3\left[ {Str{h^4} - {{\left( {tr{h^3}} \right)}^2} - S(S - n) - {S^2}} \right]dV_M} = 0, \end{equation} $

(1.4)

holds true on $ M^n $, where $ S $ is the squared norm of the second fundamental form $ h $ of the hypersurface, and $ tr{h^3} $ and $ tr{h^4} $ are represented in a local orthogonal coordinate system respectively as $ \sum\nolimits_{i, j, k} {{h_{ij}}{h_{jk}}{h_{ki}}} $ and $ \sum\nolimits_{i, j, k, l} {{h_{ij}}{h_{jk}}{h_{kl}}{h_{li}}} $.

Through the non-negativity of the square of the gradient norm of the second fundamental form, it is easy to see that formula (1.3) can be derived similarly to the Simons inequality: If $ 0 \le S \le n $, then $ S \equiv 0 $ or $ S \equiv n $; for formula (1.4), if the non-negativity of integral

$ \int_{{M^n}} {{{\left\| {{\nabla ^2}h} \right\|}^2} - 3\left[ {Str{h^4} - {{\left( {tr{h^3}} \right)}^2} - S(S - n) - {S^2}} \right]dV_M} $

can be proven, it can be concluded that: If $ S > n $, then $ S \geq 2n $. Therefore, formulas (1.3) and (1.4) are collectively referred to as Simons-type integrals.

Let $ {M^n} $ be a closed immersed minimal hypersurface with constant scalar curvature that is isometrically immersed in the unit sphere $ {\mathbb{S}^{n + 1}}\; ( \subset {\mathbb{R}^{n + 2}}) $, $ h $ denotes the second fundamental form of the hypersurface with respect to the unit normal vector field $ e_{n+1} $. In $ {\mathbb{S}^{n + 1}} $, choose a set of orthogonal coordinate systems $ {e_1}, \cdots , {e_n}, {e_{n + 1}} $, such that their restriction on $ {M^n} $, $ {e_1}, \cdots , {e_n} $ are tangent to $ M^n $ and $ {e_{n + 1}} $ is orthogonal to $ M^n $, let $ {\omega _1}, \cdots , {\omega _n}, {\omega _{n + 1}} $ be the dual frame fields. In this article, we use the following range of indices:

$ 1 \le i, j, k, \cdots \le n + 1, \; 1 \le A, B, C, \cdots \le n + 2. $

Thus, the motion equations of the hypersurface $ M^n $ concerning the Darboux frame $ \left\{ {\left. x \right|{e_1}, \cdots , }\right. $ $ \left.{{e_n}, {e_{n + 1}}} \right\} $ can be written as

$ \begin{equation} \left\{ \begin{aligned} dx &= {\omega _i}{e_i}, \hfill \\ d{e_i} &= {\omega _{ij}}{e_j} + {\omega _{in + 1}}{e_{n + 1}} - {\omega_i}x, \hfill \\ \; d{e_{n + 1}} &= {\omega _{n + 1i}}{e_i}, \hfill \end{aligned} \right. \end{equation} $

(2.1)

The second fundamental form of $ M^n $ is

$ \begin{equation} h = - \langle {dx, d{e_{n + 1}}} \rangle = {\omega _i}{\omega _{in + 1}} = {h_{ij}}{\omega _i}{\omega _j}, \end{equation} $

(2.2)

where $ \langle { \cdot , \cdot } \rangle $ is the inner product of $ {M^n} $, $ x $ is the position vector of point $ p $ on $ M^n $ in $ \mathbb{R}^{n+2} $ satisfying $ {\left\| x \right\|^2} = 1 $, $ \left\{ {{\omega _{ij}}} \right\} $ is the connection form of $ {M^n} $ with respect to $ \left\{ {{\omega _i}} \right\} $, satisfying the following structural equation:

$ \begin{equation} d{\omega _i} = {\omega _j} \wedge {\omega _{ji}}, \; {\omega _{ij}} + {\omega _{ji}} = 0, \end{equation} $

(2.3)

$ \begin{equation} {\omega _{ij}} = {\omega _{ik}} \wedge {\omega _{kj}} - {1 \over 2}{R_{ijkl}}{\omega _k} \wedge {\omega _l}, \end{equation} $

(2.4)

where $ {R_{ijkl}} $ is the component of the Riemann curvature tensor on $ {M^n} $. The covariant derivative $ \nabla h $ with components $ {h_{ij, k}} $ is defined as

$ \begin{equation} {h_{ij, k}}{\omega _k} = d{h_{ij}} + {h_{kj}}{\omega _{ki}} + {h_{ik}}{\omega _{kj}}. \end{equation} $

(2.5)

Furthermore, the Gauss-Codazzi equations and Ricci formulas are given by

$ \begin{equation} {R_{ijkl}} = {h_{ik}}{h_{jl}} - {h_{il}}{h_{jk}} + {\delta _{ik}}{\delta _{jl}} - {\delta _{il}}{\delta _{jk}}, \; {h_{ij, k}} = {h_{ik, j}}, \end{equation} $

(2.6)

$ \begin{equation} h_{ij, kl}-h_{ij, lk}=\sum\limits_{m}h_{im}R_{mjkl}+\sum\limits_{m}h_{mj}R_{mikl}. \end{equation} $

(2.7)

Finally, let's review the divergence theorem in Riemann geometry and its important applications:

Lemma 2.1 (Divergence Theorem)   Let $ \left( {M, g} \right) $ be a compact oriented $ n $ dimensional Riemannian manifold with boundary, and $ \overrightarrow{n} $ be the unit normal vector pointing inward to $ \partial M $ in $ {M^n} $, then for any $ X \in \mathfrak{X}(M) $, the following integral formula holds:

$ \int_M {div X} dV_{M} = - \int_{\partial M} {g(\overrightarrow{n}, X)} dV_{\partial M}, $

where $ \partial M $ has a direction induced by $ M $, and $ dV_{M} $ is a volume element of $ \partial M $.

Corollary 2.2 (Green's Formula)   Assume the same as Lemma 2.1, then for any $ f \in {C^\infty }\left( M \right) $, the following integral formula holds:

$ \int_M {\Delta f} dV_{M} = - \int_{\partial M} {\overrightarrow{n}\left( f \right)} dV_{\partial M}. $

In particular,

$ \int_M {\Delta f} dV_{M} = 0 $

holds true when $ \partial M = \emptyset $.

In this section, we introduce three important properties of height functions on hypersurfaces $ M^n $ in spherical space. First, define the smooth function $ {\varphi ^A} $ to have first and second order covariant derivatives $ \nabla {\varphi ^A} $, $ {\nabla ^2}{\varphi ^A} $ of the components $ \varphi _i^A $ and $ \varphi _{i, j}^A $, respectively, as

$ \begin{equation} \varphi _i^A{\omega _i} = d{\varphi ^A}, \; \varphi _{i, j}^A{\omega _j} = d\varphi _i^A + \varphi _j^A{\omega _{ji}}. \end{equation} $

(3.1)

Thus, there exists the following property:

Property 3.1   Let $ {M^n} $ be a minimal hypersurface in $ {\mathbb{S}^{n + 1}}\left( { \subset {\mathbb{R}^{n + 2}}} \right) $, $ {\varphi ^A} $ and $ {\psi ^A} $ be the height functions of the position vector $ x $ and the normal vector $ {e_{n + 1}} $ defined on $ {M^n} $ in the direction of the natural basis coordinate vectors $ {E_A} $. Then $ {\varphi ^A} $ satisfies

$ \begin{equation} \varphi _i^A = \langle {{e_i}, {E_A}} \rangle , \; \varphi _{i, j}^A = {\psi ^A}{h_{ij}} - {\varphi ^A}{\delta _{ij}}, \; \Delta {\varphi ^A} = - n{\varphi ^A}. \end{equation} $

(3.2)

Proof   According to Definition 1.4, formula (3.1), (2.1), (2.2), and the compatibility of the connection $ \nabla $, it is calculated that

$ \begin{equation} \varphi _i^A{\omega _i} = d\langle {x, {E_A}} \rangle = \langle {dx, {E_A}} \rangle = \langle {{e_i}, {E_A}} \rangle {\omega _i}, \end{equation} $

(3.3)

and

$ \begin{equation} \begin{aligned} \varphi _{i, j}^A{\omega _j} & = d\langle {{e_i}, {E_A}} \rangle + \langle {{e_j}, {E_A}} \rangle {\omega _{ji}} \\ & = \langle {{\omega _{ij}}{e_j} + {h_{ij}}{\omega _j}{e_{n + 1}} - {\omega _i}x + {e_j}{\omega _{ji}}, {E_A}} \rangle \\ & = \left( {{\psi ^A}{h_{ij}} - {\varphi ^A}{\delta _{ij}}} \right){\omega _j}. \end{aligned} \end{equation} $

(3.4)

Furthermore, since $ {M^n} $ is minimal hypersurface ($ H = {1 \over n}{\sum_{i}h_{ii}} \equiv 0 $), we have $ \Delta {\varphi ^A} = - n{\varphi ^A} $, which completes the proof of Property 3.1.

Define the smooth function $ {\psi ^A} $ with components $ \psi _i^A $ and first and second order covariant derivatives $ \nabla {\psi ^A} $ and $ {\nabla ^2}{\psi ^A} $ as

$ \begin{equation} \psi _i^A{\omega _i} = d{\psi ^A}, \; \psi _{i, j}^A{\omega _j} = d\psi _i^A + \psi _j^A{\omega _{ji}}. \end{equation} $

(3.5)

Thus, there exists the following property:

Property 3.2   Let $ {M^n} $ be a minimal hypersurface in $ {\mathbb{S}^{n + 1}}\left( { \subset {\mathbb{R}^{n + 2}}} \right) $, $ {\varphi ^A} $ and $ {\psi ^A} $ be the height functions of the position vector $ x $ and the normal vector $ e_{n+1} $ in the direction of the natural basis coordinate vectors $ E_A $ defined on $ {M^n} $, and $ \varphi _i^A $ be the component of the covariant derivative $ \nabla {\varphi ^A} $. Then $ {\psi ^A} $ satisfies

$ \begin{equation} \psi _i^A = - \varphi _j^A{h_{ij}}, \; \psi _{i, j}^A = - \varphi _k^A{h_{ij, k}} + {\varphi ^A}{h_{ij}} - {\psi ^A}{h_{ik}}{h_{kj}}, \; \Delta {\psi ^A} = - S{\psi ^A}. \end{equation} $

(3.6)

Proof   According to Definition 1.4, formulas (2.1), (2.2), (2.5), (2.6), (3.2), (3.5) and the compatibility of the connection $ \nabla $, it is calculated that

$ \begin{equation} \psi _i^A{\omega _i} = d\langle {{e_{n + 1}}, {E_A}} \rangle = \langle {d{e_{n + 1}}, {E_A}} \rangle = \langle { - {h_{ij}}{\omega _i}{e_j}, {E_A}} \rangle = - \varphi _j^A{h_{ij}}{\omega _i}, \end{equation} $

(3.7)

and

$ \begin{equation} \begin{aligned} \psi _{i, j}^A{\omega _j} & = - d\left( {\varphi _j^A} \right){h_{ij}} - \varphi _j^Ad\left( {{h_{ij}}} \right) - \varphi _k^A{h_{jk}}{\omega _{ji}} \\ & = \varphi _k^A{h_{ij}}{\omega _{kj}} - \varphi _{j, k}^A{h_{ij}}{\omega _k} + \varphi _j^A{h_{kj}}{\omega _{ki}} + \varphi _j^A{h_{ik}}{\omega _{kj}} - \varphi _j^A{h_{ij, k}}{\omega _k} - \varphi _k^A{h_{jk}}{\omega _{ji}} \\ & = \left( { - \varphi _k^A{h_{ij, k}} + {\varphi ^A}{h_{ij}} - {\psi ^A}{h_{ik}}{h_{kj}}} \right){\omega _j}. \end{aligned} \end{equation} $

(3.8)

Furthermore, since $ {M^n} $ is a minimal hypersurface, we have $ \Delta {\psi ^A} = - S{\psi ^A} $, which completes the proof of Property 3.2.

Through Property 3.1 and Property 3.2, it is not difficult to obtain the following property:

Property 3.3   Let $ {M^n} $ be a minimal hypersurface in $ {\mathbb{S}^{n + 1}}\left( { \subset {\mathbb{R}^{n + 2}}} \right) $, and $ {\varphi ^A} $, $ {\psi ^A} $ are the height functions of the position vector $ x $ and the normal vector $ e_{n+1} $ in the direction of the natural basis coordinate vectors $ {E_A} $, while $ \varphi _i^A $, $ \psi _i^A $ are the components of the covariant derivatives $ \nabla {\varphi ^A} $ and $ \nabla {\psi ^A} $, respectively, then $ \varphi _i^A $, $ \psi _i^A $ satisfy

$ \begin{equation} \begin{aligned} {\varphi ^A}{\varphi ^A} = {\psi ^A}{\psi ^A} & = 1, \; \varphi _i^A\varphi _j^A = {\delta _{ij}}, \; \varphi _i^A\psi _j^A = - {h_{ij}}, \; \psi _i^A\psi _j^A = {h_{ik}}{h_{kj}}, \\ {\varphi ^A}{\psi ^A} & = {\varphi ^A}\varphi _i^A = {\varphi ^A}\psi _i^A = {\psi ^A}\varphi _i^A = {\psi ^A}\psi _i^A = 0. \end{aligned} \end{equation} $

(3.9)

Proof   Using Definition 1.4, Property 3.1, and Property 3.2, as well as the fact that $ x $, $ {e_i}\left( {1 \le i \le n} \right) $, and $ {e_{n + 1}} $ are mutually orthogonal unit vectors, we can complete the proof of Property 3.3.

Before addressing the two Simons type integrals, it is necessary to use the divergence theorem to establish the following lemma:

Lemma 4.1   Let $ {M^n} $ be a closed immersed minimal hypersurface in the unit sphere with constant scalar curvature $ {R_M} $, $ {h_{ij}} $ be the components of the second fundamental form of $ {M^n} $. Then the following integral formulas

$ \begin{equation} \int_M {\left( {2{h_{ij}}{h_{kl}}{h_{ik, m}}{h_{jl, m}} + {h_{ij}}{h_{jk}}{h_{lm}}{h_{ik, lm}}} \right)} dV_M = 0, \end{equation} $

(4.1)

$ \begin{equation} \int_M {\left[ {2{h_{ij}}{h_{kl}}{h_{lm, i}}{h_{lm, k}} + {h_{ij}}{h_{kl}}{h_{il, m}}{h_{jk, m}}} + (n-S)tr{h^4}\right]} dV_M = 0, \end{equation} $

(4.2)

$ \begin{equation} \int_M {\left[ {{h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, k}} + {h_{ij}}{h_{jk}}{h_{lm}}{h_{ik, lm}} + {{\left( {tr{h^3}} \right)}^2} - Str{h^4} + {S^2}} \right]} dV_M = 0, \end{equation} $

(4.3)

$ \begin{equation} \int_M {\left[ {2{h_{ij}}{h_{kl, i}}{h_{kl, j}}} + (n-S)tr{h^3}\right]} dV_M = 0, \end{equation} $

(4.4)

$ \begin{equation} \int_M {\left( {{h_{ij}}{h_{kl, i}}{h_{kl, j}}} + {{h_{ij}}{h_{kl}}{h_{ij, kl}}}\right)} dV_M = 0, \end{equation} $

(4.5)

$ \begin{equation} \int_M {\left[ {{\|\nabla h \|}^2 + (n-S)S}\right]} dV_M = 0 \end{equation} $

(4.6)

hold true on $ {M^n} $.

Proof   Define six 1-forms on $ M^n $: $ {\Phi ^{\lambda}}= \Phi _m^{\lambda}{\omega _m}\; (\; {\lambda}=1, 2, \cdots, 6\; ) $, where the components $ \Phi _m^{\lambda} $ satisfy:

$ \begin{equation} \begin{aligned} \Phi _m^1= {h_{ij}}{h_{jk}}{h_{lm}}{h_{ik, l}}, \; \Phi _m^2 & = {h_{ij}}{h_{jk}}{h_{kl}}{h_{il, m}}, \; \Phi _m^3 = {h_{ij}}{h_{mk}}{h_{kl}}{h_{li, j}}, \\ \Phi _m^4= {h_{ij}}{h_{jk}}{h_{ik, m}}, \; \Phi _m^5 & = {h_{ij}}{h_{km}}{h_{ij, k}}, \; \Phi _m^6 = {h_{ij}}{h_{ij, m}}. \end{aligned} \end{equation} $

(4.7)

Using formulas (2.6), (2.7), $ {R_M} \equiv $ constant, and the minimal condition, we obtain

$ \begin{equation} div \left( {{\Phi ^1}} \right) = \Phi _{m, m}^1 = 2{h_{ij}}{h_{kl}}{h_{ik, m}}{h_{jl, m}} + {h_{ij}}{h_{jk}}{h_{lm}}{h_{ik, lm}}, \end{equation} $

(4.8)

$ \begin{equation} div \left( {{\Phi ^2}} \right) = \Phi _{m, m}^2 = {2{h_{ij}}{h_{kl}}{h_{lm, i}}{h_{lm, k}} + {h_{ij}}{h_{kl}}{h_{il, m}}{h_{jk, m}}} + (n-S)tr{h^4}, \end{equation} $

(4.9)

$ \begin{equation} div \left( {{\Phi ^3}} \right) = \Phi _{m, m}^3 = {h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, k}} + {h_{ij}}{h_{jk}}{h_{lm}}{h_{ik, lm}} + {\left( {tr{h^3}} \right)^2} - Str{h^4} + {S^2}, \end{equation} $

(4.10)

$ \begin{equation} div \left( {{\Phi ^4}} \right) = \Phi _{m, m}^4 = {2{h_{ij}}{h_{kl, i}}{h_{kl, j}}} + (n-S)tr{h^3}, \end{equation} $

(4.11)

$ \begin{equation} div \left( {{\Phi ^5}} \right) = \Phi _{m, m}^5 = {{h_{ij}}{h_{kl, i}}{h_{kl, j}}} + {{h_{ij}}{h_{kl}}{h_{ij, kl}}}, \end{equation} $

(4.12)

$ \begin{equation} div \left( {{\Phi ^6}} \right) = \Phi _{m, m}^6 = {{\|\nabla h \|}^2 + (n-S)S}. \end{equation} $

(4.13)

Since $ {M^n} $ is closed, it follows that $ {M^n} $ is compact without boundary. By further utilizing the divergence theorem, the proof of Lemma 4.1 is completed.

The third, fourth, and fifth order covariant derivatives $ {\nabla ^3}{\varphi ^A} $, $ {\nabla ^4}{\varphi ^A} $, and $ {\nabla ^5}{\varphi ^A} $, and the second order covariant derivatives $ {\nabla ^2}h $, which have components $ \varphi _{i, jk}^A $, $ \varphi _{i, jkl}^A $, $ \varphi _{i, jklm}^A $, and $ {h_{ij, kl}} $, are defined as follows:

$ \begin{equation} \varphi _{i, jk}^A{\omega _k} = d\varphi _{i, j}^A + \varphi _{k, j}^A{\omega _{ki}} + \varphi _{i, k}^A{\omega _{kj}}, \end{equation} $

(4.14)

$ \begin{equation} \varphi _{i, jkl}^A{\omega _l} = d\varphi _{i, jk}^A + \varphi _{l, jk}^A{\omega _{ki}} + \varphi _{i, lk}^A{\omega _{kj}} + \varphi _{i, jl}^A{\omega _{lk}}, \end{equation} $

(4.15)

$ \begin{equation} \varphi _{i, jklm}^A{\omega _m} = d\varphi _{i, jkl}^A + \varphi _{m, jkl}^A{\omega _{mi}} + \varphi _{i, mkl}^A{\omega _{mj}} + \varphi _{i, jml}^A{\omega _{mk}} + \varphi _{i, jkm}^A{\omega _{ml}}, \end{equation} $

(4.16)

$ \begin{equation} {h_{ij, kl}}{\omega _l} = d{h_{ij, k}} + {h_{lj, k}}{\omega _{li}} + {h_{il, k}}{\omega _{lj}} + {h_{ij, l}}{\omega _{lk}}. \end{equation} $

(4.17)

Finally, we present the proof of Theorem 1.5:

Proof   First, for the proof of formula (1.3): Based on formulas (4.14) and (4.15), combined with formula (3.2), it can be calculated that

$ \begin{equation} \varphi _{i, jk}^A = \psi _k^A{h_{ij}} + {\psi ^A}{h_{ij, k}} - \varphi _k^A{\delta _{ij}}, \end{equation} $

(4.18)

$ \begin{equation} \varphi _{i, jkl}^A = \psi _{k, l}^A{h_{ij}} + \psi _k^A{h_{ij, l}} + \psi _l^A{h_{ij, k}} + {\psi ^A}{h_{ij, kl}} - \varphi _{k, l}^A{\delta _{ij}}, \end{equation} $

(4.19)

Using formulas (3.2) and (3.6), we have

$ \begin{equation} \varphi _{i, jkk}^A = - {\psi ^A} \cdot S{h_{ij}} + \psi _k^A \cdot 2{h_{ij, k}} + {\varphi ^A} \cdot n{\delta _{ij}} + {\psi ^A}{h_{ij, kk}}. \end{equation} $

(4.20)

From $ {R_M} \equiv $ constant, it can be inferred that $ S \equiv $ constant. Combining this with formulas (2.6), (2.7), and the minimal condition, we calculate to obtain

$ \begin{equation} {h_{ij}}{h_{ij, kk}} = {h_{ij}}\left[ {\left( {{h_{ki, jk}} - {h_{ki, kj}}} \right) + {h_{kk, ij}}} \right] = {h_{ij}}\left( {{h_{pi}}{R_{pj}} + {h_{kp}}{R_{pijk}}} \right) = S\left( {n - S} \right). \end{equation} $

(4.21)

Therefore, from formulas (2.6), (3.9), (4.18), (4.20), (4.21) and the minimal condition, we have

$ \begin{equation} \begin{aligned} {1 \over 2}\Delta \sum\limits_A {{{\left\| {{\nabla ^2}{\varphi ^A}} \right\|}^2}} & = {\left( {\varphi _{i, jk}^A} \right)^2} + \varphi _{i, j}^A\varphi _{i, jkk}^A \\ & = \left( {{S^2} + {{\left\| {\nabla h} \right\|}^2} + {n^2}} \right) + \left( { - {S^2} - {n^2} + {h_{ij}}{h_{ij, kk}}} \right) \\ & = {\left\| {\nabla h} \right\|^2} + S(n - S). \end{aligned} \end{equation} $

(4.22)

Since $ {M^n} $ is closed, the formula $ (1.3) $ holds using a corollary of the Green formula.

Next, we prove formula (1.4): substituting formula (3.2) into formula (4.19), we can see

$ \begin{equation} \begin{aligned} \varphi _{i, jkl}^A = & - \varphi _m^A{h_{kl, m}}{h_{ij}} + {\varphi ^A}{h_{ij}}{h_{kl}} - {\psi ^A}{h_{km}}{h_{ml}}{h_{ij}} + \psi _k^A{h_{ij, l}} \\ & + \psi _l^A{h_{ij, k}} + {\psi ^A}{h_{ij, kl}} - {\psi ^A}{h_{kl}}{\delta _{ij}} + {\varphi ^A}{\delta _{ij}}{\delta _{kl}}. \end{aligned} \end{equation} $

(4.23)

According to formulas (1.4), (3.6), (4.16), $ S \equiv $ constant and the minimal condition, it is noted that

$ \begin{equation} \begin{aligned} {h_{ij, k}}{h_{ij, kll}} & = {h_{ij, k}}\left[ {{{\left( {{h_{ij, kl}} - {h_{ij, lk}}} \right)}_l} + \left( {{h_{li, jkl}} - {h_{li, jlk}}} \right) + {{\left( {{h_{li, jl}} - {h_{li, lj}}} \right)}_k} + {h_{ll, ijk}}} \right] \\ & = {h_{ij, k}}\left[ {{{\left( {{h_{pj}}{R_{pikl}} + {h_{ip}}{R_{pjkl}}} \right)}_l} + \left( {{h_{pi, j}}{R_{pk}} + {h_{lp, j}}{R_{pikl}} + {h_{li, p}}{R_{pjkl}}} \right)} \right.\\ & \; \; \; \; \; \; \; \; \; \; \; \; \; \; \left. { + {{({h_{pi}}{R_{pj}} + {h_{lp}}{R_{pijl}})}_k}} \right]\\ & = 6{h_{ij}}{h_{kl}}{h_{ik, m}}{h_{jl, m}} - 3{h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, k}} + \left( {2n - S + 3} \right){\left\| {\nabla h} \right\|^2}. \end{aligned} \end{equation} $

(4.24)

Furthermore, we calculated that

$ \begin{equation} \begin{aligned} & {1 \over 2}\Delta \sum\limits_A {{{\left\| {{\nabla ^3}{\varphi ^A}} \right\|}^2}} = {\left( {\varphi _{i, jkl}^A} \right)^2} + \varphi _{i, jk}^A\varphi _{i, jkll}^A \\= & ( {{{\left\| {{\nabla ^2}h} \right\|}^2} + 3S{{\left\| {\nabla h} \right\|}^2} + Str{h^4} + {S^2} + nS + {n^2} - 2{h_{ij}}{h_{km}}{h_{ml}}{h_{ij, kl}} + 2{h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, j}}} )\\ & + [ {2{S^2}(n - S) + 2{h_{ij}}{h_{km}}{h_{ml}}{h_{ij, kl}} - Str{h^4} - {S^2} - 2{h_{ij}}{h_{jk}}{h_{ij, k}}{h_{ij, l}} - S{{\left\| {\nabla h} \right\|}^2}} \\ & { + {h_{ij, k}}{h_{ij, kll}} - nS - {n^2}}] \\ = & {\left\| {{\nabla ^2}h} \right\|^2} + \left( {2n + S + 3} \right){\left\| {\nabla h} \right\|^2} + 2{S^2}(n - S) + 6{h_{ij}}{h_{kl}}{h_{ik, m}}{h_{jl, m}} - 3{h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, k}}. \end{aligned} \end{equation} $

(4.25)

Finally, since $ {M^n} $ is closed, it can be inferred from the Green's formula that there holds an integral equation

$ \begin{equation} \begin{aligned} \int_M & {{{\left\| {{\nabla ^2}h} \right\|}^2} + \left( {2n + S + 3} \right){{\left\| {\nabla h} \right\|}^2} + 2{S^2}(n - S)} \\ & {+ 6{h_{ij}}{h_{kl}}{h_{ik, m}}{h_{jl, m}} - 3{h_{ij}}{h_{jk}}{h_{lm, i}}{h_{lm, k}}} dV_M = 0 \end{aligned} \end{equation} $

(4.26)

on $ {M^n} $, Using the formulas (4.1), (4.3), and (4.6) to simplify the above expression, we obtain the formula (1.4), thus completing the proof of Theorem 1.5.