Gap Phenomenon of an Abstract Willmore Type Functional of Hypersurface in Unit Sphere

Abstract

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called 𝑊_(𝑛,𝐹)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the 𝑊_(𝑛,𝐹)-Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples of 𝑊_(𝑛,𝐹)-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

1. Introduction

Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be an 𝑛-dimensional compact without boundary hypersurface in unit sphere. Choose an orthonormal frames field {𝑒₁,…,𝑒_𝑛,𝑒_𝑛+1} along 𝑀 such that {𝑒₁,…,𝑒_𝑛} are tangent to 𝑀 and {𝑒_𝑛+1} is normal to 𝑀. Their dual frames are {𝜃¹,…,𝜃^𝑛} and {𝜃^𝑛+1}, respectively; obviously, 𝜃^𝑛+1=0 when it is restricted over 𝑀. Let ℎ denote the second fundamental form of the immersion 𝜑∶𝑀→𝑆^𝑛+1(1). Write ℎ=ℎ_𝑖𝑗𝜃^𝑖⊗𝜃^𝑗 (1) and define 1𝐻=𝑛^𝑛_𝑖=1ℎ_𝑖𝑖,𝑆=_𝑖𝑗ℎ_𝑖𝑗²,𝜌=𝑆−𝑛𝐻²,𝑃_𝑘=𝑡𝑟ℎ^𝑘. (2) One calls 𝐻, 𝑆, 𝑃_𝑘 to be mean curvature, square of the length, and 𝑘th power polynomial of second fundamental form, respectively. It is well known that hypersurface 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) is minimal when 𝐻≡0 and is totally geodesic when 𝑆≡0. For invariant 𝜌, it is obvious that 𝜌(𝑄)≥0 for all 𝑄∈𝑀, 𝜌(𝑄)=0 if and only if 𝑄 is an umbilical point of 𝑀, and 0≤𝜌(𝑄)≤𝐶 for all 𝑄∈𝑀 and some positive constant 𝐶 since 𝑀 is compact.

In differential geometry, there is a famous classical Willmore functional of hypersurface which is defined as 𝑊_{(𝑛,𝑛/2)}𝜑=_𝑀𝜌^𝑛/2𝑑𝑣. (3) See [1–4] for the details. It is well known that this functional is invariant under conformal transformation. Its critical point is called classical Willmore or 𝑊_{(𝑛,𝑛/2)}-Willmore hypersurface. The famous Willmore conjecture says that 𝑊_(2,1)(𝜑)≥4𝜋² holds for all immersed tori 𝜑∶𝑀→𝑆³; recently, it has been completely solved by Marques and Neves in article [5] using Min-Max theory for two dimensional case, but higher dimensional case is still an open problem. When 𝑛≥3, the functional 𝑊_{(𝑛,𝑛/2)}(𝜑) was studied extensively in [6–11].

Due to the importance of Willmore conjecture, many geometric experts generalized the classic Willmore functional to a wide range and some interesting results have been obtained.

In [12], Cai studied the 𝑊_(2,𝑝)-Willmore functional for surfaces in 3-dimensional unit sphere 𝑆³(1): 𝑊_(2,𝑝)𝜑=_𝑀𝑘₁−𝑘₂^𝑝𝑑𝑣, (4) where 𝑘₁ and 𝑘₂ denote the principal curvatures of 𝑀 and 𝑘₁≥𝑘₂. Under some proper conditions, Cai obtained some interesting inequalities.

In [13], Guo and Li considered the 𝑊_(𝑛,1)-Willmore functional for submanifolds in unit sphere 𝑆^𝑛+1(1): 𝑊_(𝑛,1)𝜑=_𝑀𝜌𝑑𝑣. (5) They calculated the first variation and obtained Simons’ type inequalities and finally classified the point-wise gap phenomenon. In [14], Xu and Yang derived the global but not point-wise gap phenomenon for the same functional.

In [15], Wu researched the 𝑊_(𝑛,𝑟)-Willmore functional for submanifolds in unit sphere 𝑆^𝑛+1(1): 𝑊_(𝑛,𝑟)𝜑=_𝑀𝜌^𝑟𝑑𝑣. (6) The first variation formula, Simons’ type inequalities, and point-wise gap phenomenon are obtained.

Motivated by the work mentioned above, in this paper, we study an abstract Willmore type called 𝑊_(𝑛,𝐹)-Willmore functional: 𝑊_(𝑛,𝐹)𝜑=_𝑀𝐹𝜌𝑑𝑣, (7) where we always assume that the abstract function satisfies 𝐹∶𝑢⟶𝐹(𝑢),𝐹∈𝐶³([0,+∞)⟶𝑅). (8) Obviously, functionals 𝑊_{(𝑛,𝑛/2)},𝑊_(2,𝑝),𝑊_(𝑛,1), and 𝑊_(𝑛,𝑟) are special cases of 𝑊_(𝑛,𝐹), so the construction and investigation of 𝑊_(𝑛,𝐹) are meaningful. As usual, the critical point of functional 𝑊_(𝑛,𝐹)(𝜑) is called 𝑊_(𝑛,𝐹)-Willmore hypersurface. In Section 3, we calculate the 1st variation formula of functional 𝑊_(𝑛,𝐹).

Theorem 1. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be an 𝑛-dimensional hypersurface in an (𝑛+1)-dimensional unit sphere 𝑆^𝑛+1(1); then 𝑀 is a 𝑊_(𝑛,𝐹)-Willmore hypersurface if and only if _𝑖𝑗𝐹^′𝜌ℎ_{𝑖𝑗,𝑗𝑖}−Δ𝐹^′𝜌𝐻+𝑃₃−𝑆𝐻𝐹^′𝜌−𝑛2𝐹𝜌𝐻=0. (9) In particular, if 𝑀 is an isoparametric (all principal curvatures are constant) hypersurface, then 𝑀 is a 𝑊_(𝑛,𝐹)-Willmore hypersurface if and only if 𝑃₃−𝑆𝐻𝐹^′𝜌−𝑛2𝐹𝜌𝐻=0. (10)

Remark 2. When 𝐹(𝜌)=𝜌^𝑛/2,𝜌,𝜌^𝑟, Theorem 1 coincides with corresponding results in [6, 13, 15], respectively.
It is well known that Simons’ integral inequality plays an important role in the study of minimal hypersurface. It says that if 𝑀 is an 𝑛-dimensional compact minimal hypersurface in (𝑛+1)-dimensional unit sphere 𝑆^𝑛+1(1), then _𝑀𝑆(𝑛−𝑆)𝑑𝑣≤0. (11)
For classic Willmore hypersurface, Li [6] proved a Simons’ type integral result which says if 𝑀 is an 𝑛-dimensional compact classical Willmore hypersurface in unit sphere 𝑆^𝑛+1(1), then _𝑀𝜌^𝑛/2𝑛−𝜌𝑑𝑣≤0. (12) Similarly, for functionals 𝑊_(𝑛,1) and 𝑊_(𝑛,𝑟), [13, 15] obtained integral inequalities, respectively. A natural question is that whether a Simons’ type inequality or equality can be established for 𝑊_(𝑛,𝐹)-Willmore functional. We give a confirm answer in this study.

Theorem 3. Let 𝑀 be an 𝑛-dimensional compact 𝑊_(𝑛,𝐹)-Willmore hypersurface in sphere 𝑆^𝑛+1(1); then we have Simons’ type equality and can give a discussion according to the sign of 𝐹, 𝐹^′, and 𝐹^′′: 0=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²−2𝜌𝜌−𝑛𝐹^′𝜌+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌𝑑𝑣. (13)(1) When 𝜌≡𝑛, there holds 0=_𝑀2𝐹^′(𝑛)|∇ℎ|²−𝑛|∇𝐻|²+𝑛𝐻²𝑛𝐹(𝑛)−2𝑛𝐹^′(𝑛)𝑑𝑣. (14)(2) When 𝜌≡0, there holds 0=_𝑀2𝐹^′(0)|∇ℎ|²−𝑛|∇𝐻|²+𝑛²𝐻²𝐹(0)𝑑𝑣. (15)(3) When 𝐹^′≡0, 𝐹≡𝑐≠0, there holds 0=_𝑀𝑐𝑛²𝐻²𝑑𝑣. (16)(4) When 𝑛𝐹−2𝑢𝐹^′≥0, 𝐹^′≥0, 𝐹^′′≥0, there holds 2𝜌𝜌−𝑛𝐹_𝑀^′𝜌=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌𝑑𝑣. (17)(5) When 𝑛𝐹−2𝑢𝐹^′≤0, 𝐹^′≥0, 𝐹^′′≥0, there holds 2𝜌𝜌−𝑛𝐹_𝑀^′𝜌−𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²𝑑𝑣. (18)

Remark 4. When 𝐹(𝜌)=𝜌^𝑛/2,𝜌,𝜌^𝑟, Theorem 3 coincides with corresponding results in [6, 13, 15], respectively. Similar equalities also hold for the other sign cases of 𝐹, 𝐹^′, and 𝐹^′′.
Using the integral equalities in Theorem 3, together with two famous Chern et al. results in [16], we can obtain some conclusions of gap phenomenon.

Theorem 5. Let 𝑀 be an 𝑛-dimensional closed 𝑊_(𝑛,𝐹)-Willmore hypersurface in unit sphere 𝑆^𝑛+1(1), we have the following.(1)When 𝐹^′≡0, 𝐹≡𝑐≠0 over (0,+∞), if 0≤𝜌≤𝑛, then 𝜌=0 or 𝜌=𝑛.(a)For 𝜌=0, then 𝐻=0, 𝜌=0, 𝑆=0, and 𝑀 is totally geodesic;(b)For 𝜌=𝑛, then 𝐻=0, 𝜌=𝑛, 𝑆=𝑛, and 𝑀=𝐶_{𝑚,𝑛−𝑚}=𝑆^𝑚(𝑚/𝑛)×𝑆^𝑛−𝑚((𝑛−𝑚)/𝑛) for some 𝑚 with 1≤𝑚≤𝑛−1.(2)When 𝑛𝐹−2𝑢𝐹^′>0,𝐹^′>0,𝐹^′′≥0 over (0,∞), if 0≤𝜌≤𝑛, then 𝜌=0 or 𝜌=𝑛.(a)For 𝜌=0, then 𝑀 is totally umbilical;(b)For 𝜌=𝑛, then 𝑛≡0(mod2), 𝐻=0, 𝜌=𝑛, 𝑆=𝑛, and 𝑀=𝐶_{𝑛/2,𝑛/2}=𝑆^𝑛/2(1/2)×𝑆^𝑛/2(1/2).(3)When 𝑛𝐹−2𝑢𝐹^′=0, 𝐹^′>0, 𝐹^′′≥0 over (0,+∞), if 0≤𝜌≤𝑛, then 𝐹(𝑢)=𝑐𝑢^𝑛/2, 𝑐>0, 𝜌=0, or 𝜌=𝑛.(a)For 𝜌=0, 𝑀 is totally umbilical;(b)For 𝜌=𝑛, 𝑀=𝑊_{𝑚,𝑛−𝑚}=𝑆^𝑚((𝑛−𝑚)/𝑛)×𝑆^𝑛−𝑚(𝑚/𝑛) for some 𝑚 with 1≤𝑚≤𝑛−1.(4)When 𝑛𝐹−2𝑢𝐹^′<0, 𝐹^′>0, 𝐹^′′≥0 over (0,+∞), if 𝐻=0, 0≤𝜌≤𝑛, then 𝐻=0, 𝜌=0 or 𝐻=0, 𝜌=𝑛.(a)For 𝐻=0, 𝜌=0, then 𝑀 is totally geodesic;(b)For 𝐻=0, 𝜌=𝑛, then 𝑛≡0(mod2), 𝐻=0, 𝜌=𝑛, 𝑆=𝑛, 𝑀=𝐶_{𝑛/2,𝑛/2}=𝑆^𝑛/2(1/2)×𝑆^𝑛/2(1/2).

Remark 6. When 𝐹(𝜌)=𝜌^𝑛/2,𝜌,𝜌^𝑟, Theorem 5 coincides with corresponding results in [6, 13, 15], respectively. Similar conclusions also hold for the other sign cases of 𝐹, 𝐹^′, and 𝐹^′′. The examples appeared in Theorem 5 would be constructed in Section 3.

2. Preliminaries

Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be an 𝑛-dimensional closed hypersurface in an (𝑛+1)-dimensional unit sphere and let Φ(⋅,⋅)∶𝑀×(−𝜖,𝜖)→𝑆^𝑛+1(1) be a variation of 𝜑; it means that 𝜑_𝑡=∶Φ(⋅,𝑡)∶𝑀×{𝑡}⟶𝑆^𝑛+1(1),∀𝑡∈(−𝜖,𝜖) (19) is still an isometric immersion with 𝜑₀≡𝜑.

Let 𝑠=(𝑠₁,…,𝑠_𝑛,𝑠_𝑛+1) and 𝜎=(𝜎¹,…,𝜎^𝑛,𝜎^𝑛+1) be the orthonormal local frames of 𝑇𝑆^𝑛+1(1) and 𝑇^∗𝑆^𝑛+1(1), respectively; then 𝑒=(𝑒₁,…,𝑒_𝑛,𝑒_𝑛+1)=𝜑_𝑡⁻¹𝑠 is the orthonormal local frames of the pullback vector bundle 𝜑_𝑡⁻¹𝑇𝑆^𝑛+1(1) over 𝑀×{𝑡}, such that {𝑒₁,…,𝑒_𝑛} are tangent to 𝑀 and {𝑒_𝑛+1} is normal to 𝑀 due to 𝜑_𝑡⁻¹𝑇𝑆^𝑛+1(1)=𝑇𝑀_𝑡⊕𝑇^⟂𝑀_𝑡. (20)

Use 𝜔 to denote the connection form over 𝑇𝑆^𝑛+1(1); by the pullback operation, we have the following decomposition: Φ^∗𝜎=𝜃+𝑉𝑑𝑡,Φ^∗𝜎^𝑖=𝜃^𝑖+𝑉^𝑖Φ𝑑𝑡,_∗𝜎^𝑛+1=𝑉^𝑛+1Φ𝑑𝑡=∶𝑓𝑑𝑡,_∗𝜔=𝜙+𝐿𝑑𝑡,Φ^∗𝜔^𝑗_𝑖=𝜙^𝑗_𝑖+𝐿^𝑗_𝑖Φ𝑑𝑡,_∗𝜔_𝑖^𝑛+1=𝜙_𝑖^𝑛+1+𝑏_𝑖𝑑𝑡. (21) Thus, one can derive that {𝜃^𝑖} are the orthonormal frames of 𝑇^∗𝑀 and 𝜃^𝑛+1≡0 when it is restricted over 𝑀, {𝑉^𝑖,𝑉^𝑛+1=∶𝑓} are the variation vector fields of Φ, {𝜙^𝑗_𝑖} is the connection form of 𝑇𝑀, and {𝜙_𝑖^𝑛+1=ℎ_𝑖𝑗𝜃^𝑗} is the second fundamental form. In particular, we have 𝜕𝜑𝜕𝑡^𝑡_𝑡=0=_𝑖𝑉^𝑖𝑒_𝑖+𝑓𝑒_𝑛+1. (22)

Use Ω and Ω^⊤ to denote curvature forms of 𝑇𝑆^𝑛+1(1) and 𝑇𝑀, respectively and write their components: Ω_𝐴𝐵=12𝑅_{𝐴𝐵𝐶𝐷}𝜎^𝐶∧𝜎^𝐷{},Ω,𝐴,𝐵,𝐶,𝐷∈1,…,𝑛+1_⊤𝑖𝑗=12𝑅_{𝑖𝑗𝑘𝑙}𝜃^𝑘∧𝜃^𝑙,𝑖,𝑗,𝑘,𝑙∈{1,2,…,𝑛}. (23) From any one standard differential geometry book, we know that the curvature of unit sphere is 𝑅_{𝐴𝐵𝐶𝐷}=−𝛿_𝐴𝐶𝛿_𝐵𝐷−𝛿_𝐴𝐷𝛿_𝐵𝐶. (24)

Using connection form {𝜙^𝑗_𝑖}, the covariant derivatives of ℎ_𝑖𝑗, 𝐻, 𝜌, 𝑉^𝑖, 𝑓, and 𝑏_𝑖 can be defined as ℎ_{𝑖𝑗,𝑘}𝜃^𝑘=𝑑_𝑀ℎ_𝑖𝑗−_𝑝ℎ_𝑝𝑗𝜙^𝑝_𝑖−_𝑝ℎ_𝑖𝑝𝜙^𝑝_𝑗,ℎ_{𝑖𝑗,𝑘𝑙}𝜃^𝑙=𝑑_𝑀ℎ_{𝑖𝑗,𝑘}−_𝑝ℎ_{𝑝𝑗,𝑘}𝜙^𝑝_𝑖−_𝑝ℎ_{𝑖𝑝,𝑘}𝜙^𝑝_𝑗−_𝑝ℎ_{𝑖𝑗,𝑝}𝜙^𝑝_𝑘,𝐻_,𝑖𝜃^𝑖=𝑑_𝑀𝐻,𝐻_,𝑖𝑗𝜃^𝑗=𝑑_𝑀𝐻_,𝑖−_𝑝𝐻_,𝑝𝜙^𝑝_𝑖,Δ𝐻=_𝑖𝐻_,𝑖𝑖,𝜌_,𝑖𝜃^𝑖=𝑑_𝑀𝜌,𝜌_,𝑖𝑗𝜃^𝑗=𝑑_𝑀𝜌_,𝑖−_𝑝𝜌_,𝑝𝜙^𝑝_𝑖,Δ𝜌=_𝑖𝜌_,𝑖𝑖,𝑉^𝑖_,𝑗𝜃^𝑗=𝑑_𝑀𝑉^𝑖+_𝑝𝑉^𝑝𝜙^𝑖_𝑝,𝑉^𝑖_,𝑗𝑘𝜃^𝑘=𝑑_𝑀𝑉^𝑖_,𝑗−_𝑝𝑉^𝑖_,𝑝𝜙^𝑝_𝑗+_𝑝𝑉^𝑝_,𝑗𝜙^𝑖_𝑝,𝑓_,𝑖𝜃^𝑖=𝑑_𝑀𝑓,𝑓_,𝑖𝑗𝜃^𝑗=𝑑_𝑀𝑓_,𝑖−_𝑝𝑓_,𝑝𝜙^𝑝_𝑖,Δ𝑓=𝑓_,𝑖𝑖,𝑏_𝑖,𝑗𝜃^𝑗=𝑑_𝑀𝑏_𝑖−_𝑝𝑏_𝑝𝜙^𝑝_𝑖,𝑏_{𝑖,𝑗𝑘}𝜃^𝑘=𝑑_𝑀𝑏_𝑖,𝑗−_𝑝𝑏_𝑝,𝑗𝜙^𝑝_𝑖−_𝑝𝑏_𝑖,𝑝𝜙^𝑝_𝑗. (25)

We use 𝑑_𝑀,𝑑_{𝑀×(−𝜖,𝜖)}=𝑑_𝑀+𝑑𝑡(𝜕/𝜕𝑡) and 𝑑 to denote the differential operators on 𝑀, 𝑀×(−𝜖,𝜖), and 𝑆^𝑛+1(1), respectively. By calculating directly and comparing both sides of the following equations: Ω=𝑑𝜔−𝜔∧𝜔,0=𝑑𝜎−𝜎∧𝜔,Φ^∗(Ω)=𝑑_{𝑀×(−𝜖,𝜖)}Φ^∗(𝜔)−Φ^∗(𝜔)∧Φ^∗(𝜔),0=𝑑_{𝑀×(−𝜖,𝜖)}Φ^∗(𝜎)−Φ^∗(𝜎)∧Φ^∗(𝜔), (26)(27)(28) we would derive out the lemmas below; also see article [10]. In fact, the right hand side of (27) is Φ^∗Ω^𝐵_𝐴=Φ^∗12𝑅_{𝐴𝐵𝐶𝐷}𝜎^𝐶∧𝜎^𝐷=12𝑅_{𝐴𝐵𝐶𝐷}Φ^∗𝜎^𝐶∧Φ^∗𝜎^𝐷=12𝑅_{𝐴𝐵𝐶𝐷}𝜃^𝐶∧𝜃^𝐷+𝑑𝑡∧𝑅_{𝐴𝐵𝐶𝐷}𝑉^𝐶𝜃^𝐷, (29) while the left hand side of (27) is 𝑑_{𝑀×(−𝜖,𝜖)}Φ^∗𝜔^𝐵_𝐴−Φ^∗𝜔^𝐶_𝐴∧Φ^∗𝜔^𝐵_𝐶=𝑑_𝑀𝜕+𝑑𝑡𝜙𝜕𝑡^𝐵𝐴+𝑑𝑡𝐿^𝐵_𝐴−𝜙^𝐶_𝐴+𝑑𝑡𝐿^𝐶_𝐴𝜙^𝐵_𝐶+𝑑𝑡𝐿^𝐵_𝐶=𝑑_𝑀𝜙^𝐵_𝐴−𝜙^𝐶_𝐴∧𝜙^𝐵_𝐶𝜕+𝑑𝑡𝜙𝜕𝑡^𝐵𝐴−𝑑_𝑀𝐿^𝐵_𝐴+𝐿^𝐶_𝐴𝜙^𝐵_𝐶−𝜙^𝐶_𝐴𝐿^𝐵_𝐶; (30) comparing (29) with (30), we obtain 𝑑_𝑀𝜙^𝐵_𝐴−𝜙^𝐶_𝐴∧𝜙^𝐵_𝐶=12𝑅_{𝐴𝐵𝐶𝐷}𝜃^𝐶∧𝜃^𝐷,𝜕𝜙𝜕𝑡^𝐵𝐴−𝑑_𝑀𝐿^𝐵_𝐴+𝐿^𝐶_𝐴𝜙^𝐵_𝐶−𝜙^𝐶_𝐴𝐿^𝐵_𝐶=𝑅_{𝐴𝐵𝐶𝐷}𝑉^𝐶𝜃^𝐷. (31) The equation of (28) is 0=𝑑_{𝑀×(−𝜖,𝜖)}Φ^∗(𝜎)−Φ^∗(𝜎)∧Φ^∗(𝜔)=𝑑_𝑀𝜕+𝑑𝑡()−()∧𝜕𝑡𝜃+𝑑𝑡𝑉𝜃+𝑑𝑡𝑉𝜙+𝑑𝑡𝐿=𝑑_𝑀𝜃−𝜃∧𝜙+𝑑𝑡𝜕𝜃𝜕𝑡−𝑑_𝑀𝑉−𝑉𝜙+𝜃𝐿. (32) Comparing both sides, we obtain 𝑑_𝑀𝜃−𝜃∧𝜙=0,𝜕𝜃𝜕𝑡−𝑑_𝑀𝑉−𝑉𝜙+𝜃𝐿=0. (33) From (31) and (33), we can easily conclude the following lemmas which are useful in the variation calculation and establishment of Simons’ type integral equalities.

Lemma 7. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be a hypersurface; one has structure equations: 𝑅_{𝑖𝑗𝑘𝑙}=−𝛿_𝑖𝑘𝛿_𝑗𝑙−𝛿_𝑖𝑙𝛿_𝑗𝑘−ℎ_𝑖𝑘ℎ_𝑗𝑙−ℎ_𝑖𝑙ℎ_𝑗𝑘,ℎ_{𝑖𝑗,𝑘}=ℎ_{𝑖𝑘,𝑗}. (34)

Lemma 8. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be a hypersurface; one has Ricci identity: ℎ_{𝑖𝑗,𝑘𝑙}−ℎ_{𝑖𝑗,𝑙𝑘}=−𝛿_𝑖𝑘ℎ_𝑗𝑙−𝛿_𝑖𝑙ℎ_𝑗𝑘−𝛿_𝑗𝑘ℎ_𝑖𝑙−𝛿_𝑗𝑙ℎ_𝑖𝑘−ℎ_𝑖𝑘ℎ_𝑝𝑙ℎ_𝑝𝑗−ℎ_𝑖𝑙ℎ_𝑝𝑘ℎ_𝑝𝑗−ℎ_𝑗𝑘ℎ_𝑝𝑙ℎ_𝑖𝑝−ℎ_𝑗𝑙ℎ_𝑝𝑘ℎ_𝑖𝑝. (35)

Lemma 9. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be a hypersurface and let 𝑉=𝑉^⊤+𝑉^⟂=𝑉^𝑖𝑒_𝑖+𝑓𝑒_𝑛+1 be a variation vector field; one has 𝜕𝜃^𝑖=𝑉𝜕𝑡^𝑖,𝑗−𝐿^𝑖_𝑗−ℎ_𝑖𝑗𝑓𝜃^𝑗,𝜕ℎ_𝑖𝑗𝜕𝑡=𝑓_,𝑖𝑗+_𝑝ℎ_{𝑖𝑗,𝑝}𝑉^𝑝+_𝑝ℎ_𝑝𝑗𝐿^𝑝_𝑖+_𝑝ℎ_𝑖𝑝𝐿^𝑝_𝑗+_𝑝ℎ_𝑖𝑝ℎ_𝑝𝑗𝑓+𝛿_𝑖𝑗𝑓. (36)

Lemma 10. With the same notations as above, one has _𝑖𝑗𝑘ℎ_𝑖𝑘ℎ_𝑘𝑗𝐿^𝑖_𝑗=0,_𝑖𝑗ℎ_𝑖𝑗𝐿^𝑖_𝑗=0. (37)

3. Variation Calculation and Examples

To calculate the first variation of 𝑊_(𝑛,𝐹)-Willmore functional, we need two lemmas.

Lemma 11. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be a hypersurface and let 𝑉=𝑉^⊤+𝑉^⟂=𝑉^𝑖𝑒_𝑖+𝑓𝑒_𝑛+1 be a variation vector field. Suppose that 𝑑𝑣=𝜃¹∧⋯∧𝜃^𝑛 denotes the volume element; one has 𝜕𝑑𝑣𝜕𝑡=_𝑖𝑉^𝑖_,𝑖−𝑛𝐻𝑓. (38)

Proof. By Lemma 9, we have (𝜕/𝜕𝑡)(𝜃^𝑖)=(𝑉^𝑖_,𝑗−𝐿^𝑖_𝑗−ℎ_𝑖𝑗𝑓)𝜃^𝑗. Hence, 𝜕𝑑𝑣𝜕𝑡=_𝑖𝜃¹𝜕∧⋯𝜃𝜕𝑡^𝑖⋯∧𝜃^𝑛=_𝑖𝑉^𝑖_,𝑖−𝑛𝐻𝑓. (39) With this, we complete the proof of Lemma 11.

Lemma 12. Let 𝜑∶𝑀^𝑛→𝑆^𝑛+1(1) be a hypersurface and let 𝑉=𝑉^⊤+𝑉^⟂=𝑉^𝑖𝑒_𝑖+𝑓𝑒_𝑛+1 be a variation vector field; one has 𝜕𝜌𝜕𝑡=_𝑖𝑗2ℎ_𝑖𝑗𝑓_,𝑖𝑗−2𝐻Δ𝑓+2𝑃₃𝑓−2𝑆𝐻𝑓+_𝑖𝜌_,𝑖𝑉^𝑖. (40)

Proof. By the definition of 𝜌,𝑆 and 𝐻, one has 𝜕𝜌=𝜕𝜕𝑡(𝑆)−𝜕𝜕𝑡𝜕𝑡𝑛𝐻². (41) It follows Lemmas 9 and 10 that 𝜕(𝑆)𝜕𝑡=2_𝑖𝑗ℎ_𝑖𝑗𝜕ℎ𝜕𝑡^𝑖𝑗=2_𝑖𝑗ℎ_𝑖𝑗𝑓_,𝑖𝑗+_𝑝ℎ_{𝑖𝑗,𝑝}𝑉^𝑝+_𝑝ℎ_𝑝𝑗𝐿^𝑝_𝑖+_𝑝ℎ_𝑖𝑃𝐿^𝑝_𝑗+_𝑝ℎ_𝑖𝑝ℎ_𝑝𝑗𝑓+𝛿_𝑖𝑗𝑓=2_𝑖𝑗ℎ_𝑖𝑗𝑓_,𝑖𝑗+_𝑖𝑆_,𝑖𝑉^𝑖+2𝑃₃𝑓+2𝑛𝐻𝑓. (42) By Lemmas 9 and 10 again, one has 𝜕𝜕𝑡𝑛𝐻²𝜕=2𝑛𝐻𝐻𝜕𝑡=2𝐻_𝑖𝑓_,𝑖𝑖+_𝑝ℎ_{𝑖𝑖,𝑝}𝑉^𝑝+_𝑝ℎ_𝑝𝑖𝐿^𝑝_𝑖+_𝑝ℎ_𝑝𝑖𝐿^𝑝_𝑖+_𝑝ℎ_𝑖𝑝ℎ_𝑝𝑖𝑓+𝛿_𝑖𝑖𝑓=2𝐻Δ𝑓+_𝑖𝑛𝐻²_,𝑖𝑉^𝑖+2𝐻𝑆𝑓+2𝑛𝐻𝑓. (43) Hence, 𝜕𝜌𝜕𝑡=_𝑖𝑗2ℎ_𝑖𝑗𝑓_,𝑖𝑗−2𝐻Δ𝑓+2𝑃₃𝑓−2𝐻𝑆𝑓+_𝑖𝜌_,𝑖𝑉^𝑖. (44) With this, we complete the proof of Lemma 12.

Proof of Theorem 1. By Lemmas 11 and 12, we can calculate 𝜕𝑊𝜕𝑡^(𝑛,𝐹)𝜑_𝑡=_𝑀𝑡𝐹^′𝜌𝜕𝜌𝜌𝜕𝑡+𝐹_𝑖𝑉^𝑖_,𝑖=−𝑛𝐻𝑓𝑑𝑣_𝑀𝑡𝐹^′𝜌_𝑖𝑗2ℎ_𝑖𝑗𝑓_,𝑖𝑗−2𝐻Δ𝑓+2𝑃₃𝑓−2𝑆𝐻𝑓+_𝑖𝜌_,𝑖𝑉^𝑖+𝐹𝜌_𝑖𝑉^𝑖_,𝑖=−𝑛𝐻𝑓𝑑𝑣_𝑀𝑡2_𝑖𝑗𝐹^′𝜌ℎ_{𝑖𝑗,𝑗𝑖}−Δ𝐹^′𝜌𝐻+𝑃₃−𝑆𝐻𝐹^′𝜌−𝑛2𝐻𝐹𝜌𝑓𝑑𝑣. (45) With this, we complete the proof of Theorem 1.

From Theorem 1, we can find a few examples in unit sphere 𝑆^𝑛+1(1), in particular, the isoparametric hypersurfaces. Since all principal curvatures 𝑘₁,…,𝑘_𝑛 are constant, then 𝜌, 𝐻, 𝑆 are constant and 𝐻=(1/𝑛)𝑃₁, 𝑆=𝑃₂, 𝜌=𝑃₂−(1/𝑛)(𝑃₁)². Thus, 𝑊_(𝑛,𝐹)-Willmore hypersurface equation becomes 𝑃₃−1𝑛𝑃₁𝑃₂𝐹^′𝜌−12𝑃₁𝐹𝜌=0. (46)

Example 13. Obviously, totally geodesic hypersurfaces are 𝑊_(𝑛,𝐹)-Willmore for any abstract function 𝐹, totally umbilical but not totally geodesic hypersurfaces are 𝑊_(𝑛,𝐹)-Willmore if and only if 𝐹(0)=0.

Example 14. For a particular hypersurface with 𝑛≡0(mod2), 𝐶_{𝑛/2,𝑛/2}=𝑆^𝑛/21√2×𝑆^𝑛/21√2⟶𝑆^𝑛+1(1). (47) All principal curvatures are 𝑘₁=⋯=𝑘_𝑛/2=1,𝑘_(𝑛/2)+1=⋯=𝑘_𝑛=−1. (48) We can derive the quantities 𝑃₁,𝑃₂,𝑃₃,𝜌, respectively, 𝑃₁=0,𝑃₂=𝑛,𝑃₃=0,𝜌=𝑛. Obviously 𝐶_{𝑛/2,𝑛/2} always is a 𝑊_(𝑛,𝐹)-Willmore hypersurface of 𝑆^𝑛+1(1) for any function 𝐹.

Example 15. For a family hypersurfaces with parameters 𝜆,𝜇,0<𝜆, 𝜇<1, 𝜆²+𝜇²=1. 𝑆^𝑚(𝜆)×𝑆^𝑛−𝑚𝜇⟶𝑆^𝑛+1(1),1≤𝑚≤𝑛−1. (49) Obviously, all principal curvatures are 𝑘₁=⋯=𝑘_𝑚=𝜇𝜆,𝑘_𝑚+1=⋯=𝑘_𝑛𝜆=−𝜇. (50) Then the quantities 𝑃₁, 𝑃₂, 𝑃₃, 𝜌 are, respectively, 𝑃₁𝜇=𝑚𝜆−()𝜆𝑛−𝑚𝜇,𝑃₂𝜇=𝑚²𝜆²+()𝜆𝑛−𝑚²𝜇²,𝑃₃𝜇=𝑚³𝜆³−()𝜆𝑛−𝑚³𝜇³,𝑚()𝜌=𝑛−𝑚𝑛𝜇²𝜆²+𝜆²𝜇²+2. (51) Setting 𝜇/𝜆=𝑥>0, 𝑊_(𝑛,𝐹)-Willmore hypersurface equation becomes 2𝑚(𝑛−𝑚)𝐹^′𝜌𝑥⁶−𝑚𝑛𝐹𝜌−2(𝑛−𝑚)𝐹^′𝜌𝑥⁴+(𝑛−𝑚)𝑛𝐹𝜌−2𝑚𝐹^′𝜌𝑥²−2𝑚(𝑛−𝑚)𝐹^′𝜌=0. (52)

Example 16 (Clifford torus [16]; 𝐹(𝜌)≡1). Consider that 𝐶_{𝑚,𝑛−𝑚}=𝑆^𝑚𝑚𝑛×𝑆^𝑛−𝑚𝑛−𝑚𝑛,1≤𝑚≤𝑛−1 (53) are minimal hypersurfaces with 𝐻≡0,𝑆≡𝑛,𝜌≡𝑛.

Example 17 (classic Willmore torus [6]; 𝐹(𝜌)=𝜌^𝑛/2). Consider that 𝑊_{𝑚,𝑛−𝑚}=𝑆^𝑚𝑛−𝑚𝑛×𝑆^𝑛−𝑚𝑚𝑛,1≤𝑚≤𝑛−1 (54) are 𝑊_{(𝑛,𝑛/2)}-Willmore hypersurfaces with 𝜌=𝑛. When some 𝑊_{𝑚,𝑛−𝑚} is minimal, then 𝑛≡0(mod2), 𝑚=𝑛/2.

4. Simons’ Type Equalities and Gap Phenomenon

Some lemmas are needed for the establishment of Simons’ type integral equalities and the discussion of gap phenomenon.

Lemma 18. For 𝜌, one has Δ𝜌=_𝑖𝑗2𝑛ℎ_𝑖𝑗𝐻_,𝑖𝑗−2𝑛𝐻Δ𝐻+2|∇ℎ|²−2𝑛|∇𝐻|²+2𝑛𝜌+2𝑛𝐻𝑃₃−2𝑆², (55) where |∇ℎ|²=∑_𝑖𝑗𝑘ℎ²_{𝑖𝑗,𝑘}.

Proof. By the definition of 𝜌 and Laplacian operator, we have Δ𝜌=_𝑘𝑖𝑗ℎ²_𝑖𝑗−𝑛𝐻²_,𝑘𝑘=_𝑘𝑖𝑗2ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘}−2𝑛𝐻𝐻_,𝑘,𝑘=_𝑘𝑖𝑗2ℎ²_{𝑖𝑗,𝑘}+_𝑖𝑗2ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘𝑘}−2𝑛𝐻²_,𝑘−2𝑛𝐻𝐻_,𝑘𝑘=2|∇ℎ|²−2𝑛𝐻Δ𝐻−2𝑛|∇𝐻|²+2_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘𝑘}=2|∇ℎ|²−𝑛𝐻Δ𝐻−𝑛|∇𝐻|²+𝑇_{1𝑖𝑗𝑘}ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘𝑘}, (56) for term 𝑇₁=∑_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘𝑘}; by Lemmas 7 and 8, easily we obtain 𝑇₁=_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑗,𝑘𝑘}=_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑘,𝑗𝑘}=_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑘,𝑗𝑘}−ℎ_{𝑖𝑘,𝑘𝑗}+ℎ_{𝑖𝑘,𝑘𝑗}=_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑘𝑘,𝑖𝑗}+_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑘,𝑗𝑘}−ℎ_{𝑖𝑘,𝑘𝑗}=_𝑖𝑗𝑛ℎ_𝑖𝑗𝐻_,𝑖𝑗+𝑇_{2𝑖𝑗𝑘}ℎ_𝑖𝑗ℎ_{𝑖𝑘,𝑗𝑘}−ℎ_{𝑖𝑘,𝑘𝑗}, (57) for term 𝑇₂=∑_𝑖𝑗𝑘ℎ_𝑖𝑗(ℎ_{𝑖𝑘,𝑗𝑘}−ℎ_{𝑖𝑘,𝑘𝑗}); by Lemma 8, we have 𝑇₂=_𝑖𝑗𝑘ℎ_𝑖𝑗ℎ_{𝑖𝑘,𝑗𝑘}−ℎ_{𝑖𝑘,𝑘𝑗}=_𝑖𝑗𝑘ℎ_𝑖𝑗−𝛿_𝑗𝑘ℎ_𝑖𝑘−𝛿_𝑘𝑘ℎ_𝑖𝑗−ℎ_𝑗𝑘ℎ_𝑖𝑝ℎ_𝑝𝑘−ℎ_𝑘𝑘ℎ_𝑖𝑝ℎ_𝑝𝑗+−𝛿_𝑖𝑗ℎ_𝑘𝑘−𝛿_𝑖𝑘ℎ_𝑗𝑘−ℎ_𝑖𝑗ℎ_𝑘𝑝ℎ_𝑝𝑘−ℎ_𝑖𝑘ℎ_𝑘𝑝ℎ_𝑝𝑗=−𝑆+𝑛𝑆−_{𝑖𝑗𝑘𝑝}ℎ_𝑖𝑗ℎ_𝑗𝑘ℎ_𝑖𝑝ℎ_𝑝𝑘+𝑛𝐻𝑃₃−𝑛²𝐻²+𝑆−𝑆²+_{𝑖𝑗𝑘𝑝}ℎ_𝑖𝑗ℎ_𝑖𝑘ℎ_𝑘𝑝ℎ_𝑝𝑗=𝑛𝜌−𝑆²+𝑛𝐻𝑃₃. (58) Substituting (58) into (57), we have 𝑇₁=_𝑖𝑗𝑛ℎ_𝑖𝑗𝐻_,𝑖𝑗+𝑛𝜌−𝑆²+𝑛𝐻𝑃₃. (59) Substituting (59) into (56), we have Δ𝜌=_𝑖𝑗2𝑛ℎ_𝑖𝑗𝐻_,𝑖𝑗−2𝑛𝐻Δ𝐻+2|∇ℎ|²−2𝑛|∇𝐻|²+2𝑛𝜌+2𝑛𝐻𝑃₃−2𝑆². (60) With this, we complete the proof of Lemma 18.

Lemma 19 (Huisken’s estimate [17]). Consider that |∇ℎ|²≥3𝑛²||𝑛+2∇𝐻²≥𝑛|∇𝐻|², (61) and |∇ℎ|²=𝑛|∇𝐻|² if and only if ∇ℎ=0.

Proof. We decompose the tensor ∇ℎ as follows: ℎ_{𝑖𝑗,𝑘}=𝐸_𝑖𝑗𝑘+𝐹_𝑖𝑗𝑘, (62) where 𝐸_𝑖𝑗𝑘=𝑛𝐻𝑛+2^,𝑖𝛿_𝑗𝑘+𝐻_,𝑗𝛿_𝑖𝑘+𝐻_,𝑘𝛿_𝑖𝑗,𝐹_𝑖𝑗𝑘=ℎ_{𝑖𝑗,𝑘}−𝐸_𝑖𝑗𝑘. (63) Through a computation, we obtain |𝐸|²=3𝑛²||𝑛+2∇𝐻²,⟨𝐸,𝐹⟩=0. (64) Finally, by the triangle inequality, we have |∇ℎ|²≥|𝐸|²=3𝑛²||𝑛+2∇𝐻²≥𝑛|∇𝐻|². (65) If |∇ℎ|²=𝑛|∇𝐻|², then all inequalities become equalities, 𝐹_𝑖𝑗𝑘=0, ∇𝐻=0, and 𝐸_𝑖𝑗𝑘=0; thus, ∇ℎ=0. With this, we complete the proof of Lemma 19.

Lemma 20. For 𝐹(𝜌), one has Δ𝐹𝜌=𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²−2𝜌𝜌−𝑛𝐹^′𝜌+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌+2𝑛_𝑖𝑗𝐹^′𝜌ℎ_𝑖𝑗𝐻_,𝑖𝑗−𝐹^′𝜌+𝑃𝐻Δ𝐻₃−𝑆𝐻𝐹^′𝜌𝑛𝐻−2𝐹𝜌𝐻². (66)

Proof. We know that Δ𝐹𝜌=𝐹^′′𝜌∇𝜌²+𝐹^′𝜌Δ𝜌. (67) By Lemmas 18 and 19, we can easily obtain the identity.

Proof of Theorem 3. Integrating the equality in Lemma 20 over 𝑀, we have 0=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²−2𝜌𝜌−𝑛𝐹^′𝜌+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌+2𝑛_𝑖𝑗𝐹^′𝜌ℎ_𝑖𝑗𝐻_,𝑖𝑗−𝐹^′𝜌+𝑃𝐻Δ𝐻₃−𝑆𝐻𝐹^′𝜌𝑛𝐻−2𝐹𝜌𝐻². (68) By Theorem 1, we know that _𝑖𝑗𝐹^′𝜌ℎ_{𝑖𝑗,𝑗𝑖}−Δ𝐹^′𝜌𝐻+𝑃₃−𝑆𝐻𝐹^′𝜌−𝑛2𝐹𝜌𝐻=0. (69) Using Stokes theorem, we obtain 0=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²−2𝜌𝜌−𝑛𝐹^′𝜌+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌. (70) We complete the proof of Theorem 3.

In order to prove Theorem 5, we need two very important conclusions which are treated as a lemma and a main Theorem in Chern et al.’s article [16]. For a hypersurface, we choose frame fields in such a way that ℎ_𝑖𝑗=0,for all𝑖≠𝑗,ℎ_𝑖=ℎ_𝑖𝑖.

Lemma 21 (Lemma 3 of [16]). Let 𝑥∶𝑀^𝑛→𝑆^𝑛+1(1) be a compact hypersurface with ∇ℎ≡0; then there are two cases.(1)ℎ₁=⋯=ℎ_𝑛=𝜆=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, and 𝑀 is a totally umbilic (𝜆>0) or totally geodesic (𝜆=0);(2)ℎ₁=⋯ℎ_𝑚=𝜆=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡>0, ℎ_𝑚+1=⋯=ℎ_𝑛=−(1/𝜆),1≤𝑚≤𝑛−1, and 𝑀 is a Riemannian product of 𝑀₁×𝑀₂, where 𝑀₁=𝑆^𝑚(1/1+𝜆²),𝑀₂=𝑆^𝑛−𝑚(𝜆/1+𝜆²).

Lemma 22 (main Theorem of [16]). Clifford torus 𝐶_{𝑚,𝑛−𝑚},1 ≤ 𝑚 ≤ 𝑛−1 are the only compact minimal (𝐻=0) hypersurfaces of dimension 𝑛 in unit sphere 𝑆^𝑛+1(1) satisfying 𝑆=𝑛.

Proof of Theorem 5. Obviously, case (1) of Theorem 5 is a corollary of Lemma 22; thus, we just focus cases (2) and (3); the rest can be proved by similar argument.
(2) When 𝑛𝐹−2𝑢𝐹^′>0,𝐹^′>0,𝐹^′′≥0 over (0,+∞), by the case (4) of Theorem 3, we have 2𝜌𝜌−𝑛𝐹_𝑀^′𝜌=_𝑀𝐹^′′𝜌∇𝜌²+2𝐹^′𝜌|∇ℎ|²−𝑛|∇𝐻|²+𝑛𝐻²𝑛𝐹𝜌−2𝜌𝐹^′𝜌≥0. (71) If 0≤𝜌≤𝑛, the right hand side of identity (71) is nonpositive, while the left hand side of (71) is nonnegative; thus, we can conclude that 𝜌=0 or 𝜌=𝑛. For 𝜌=0, 𝑀 is totally umbilical; for 𝜌=𝑛, substituting it into the above equality together with Lemma 19, we can derive ∇ℎ=0, 𝐻=0, 𝜌=𝑛, and 𝑆=𝑛, by Lemma 22, we know that 𝑀=𝐶_{𝑚,𝑛−𝑚}; moreover, by 𝑊_(𝑛,𝐹)-Willmore equation, we conclude that 𝑛≡0(mod2), 𝐻=0, 𝜌=𝑛, 𝑆=𝑛, and 𝑀=𝐶_{𝑛/2,𝑛/2}.
(3) When 𝑛𝐹−2𝑢𝐹^′=0, 𝐹^′>0, 𝐹^′′≥0(0,+∞), we know that 𝐹(𝑢)=𝑐𝑢^𝑛/2, 𝑐=const>0. By the case (4) of Theorem 3, we have _𝑀𝜌^𝑛/2𝜌−𝑛=_𝑀𝑛−24∇𝜌²+𝜌^(𝑛/2)−1|∇ℎ|²−𝑛|∇𝐻|²≥0. (72) Moreover, if 0≤𝜌≤𝑛, then 𝜌=0 or 𝜌=𝑛. For 𝜌=0, 𝑀 is totally umbilical; for 𝜌=𝑛, substituting it into the above equality together with Lemma 19, we obtain ∇ℎ=0; by Lemma 22, we know that 𝑀 is a torus; then, by Example 17, we can conclude that 𝑀 is a classic Willmore torus 𝑊_{𝑚,𝑛−𝑚} for some 𝑚.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The second author would like to express thanks to Professor H. Z. Li for his encouragement and help.