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

Authors: Yanqi Zhu; Jin Liu; Guohua Wu
Journal: The Scientific World Journal
 (2014)

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## Abstract

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


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## Body

## 1. Introduction

Letφ:Mn→Sn+1(1) be an n-dimensional compact without boundary hypersurface in unit sphere. Choose an orthonormal frames field {e1,…,en,en+1} along M such that {e1,…,en} are tangent to M and {en+1} is normal to M. Their dual frames are {θ1,…,θn} and {θn+1}, respectively; obviously, θn+1=0 when it is restricted over M. Let h denote the second fundamental form of the immersion φ:M→Sn+1(1). Write
(1)h=hijθi⊗θj
and define
(2)H=1n∑i=1nhii,S=∑ij‍(hij)2,ρ=S-nH2,Pk=tr(hk).
One calls H, S, Pk to be mean curvature, square of the length, and kth power polynomial of second fundamental form, respectively. It is well known that hypersurface φ:Mn→Sn+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(3)W(n,n/2)(φ)=∫Mρn/2dv.
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):
(4)W(2,p)(φ)=∫M(k1-k2)pdv,
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 Sn+1(1):
(5)W(n,1)(φ)=∫Mρdv.
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 Sn+1(1):
(6)W(n,r)(φ)=∫Mρrdv.
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 calledW(n,F)-Willmore functional:
(7)W(n,F)(φ)=∫MF(ρ)dv,
where we always assume that the abstract function satisfies
(8)F:u⟶F(u),F∈C3([0,+∞)⟶R).
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.
Letφ:Mn→Sn+1(1) be an n-dimensional hypersurface in an (n+1)-dimensional unit sphere Sn+1(1); then M is a W(n,F)-Willmore hypersurface if and only if
(9)∑ij(F′(ρ)hij),ji-Δ(F′(ρ)H)+(P3-SH)F′(ρ)-n2F(ρ)H=0.
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
(10)(P3-SH)F′(ρ)-n2F(ρ)H=0.Remark 2.
WhenF(ρ)=ρn/2,ρ,ρr, 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 ifM is an n-dimensional compact minimal hypersurface in (n+1)-dimensional unit sphere Sn+1(1), then
(11)∫MS(n-S)dv≤0.
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 Sn+1(1), then
(12)∫Mρn/2(n-ρ)dv≤0.
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.
LetM be an n-dimensional compact W(n,F)-Willmore hypersurface in sphere Sn+1(1); then we have Simons’ type equality and can give a discussion according to the sign of F, F′, and F′′:
(13)0=∫MF′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)-2ρ(ρ-n)F′(ρ)+nH2(nF(ρ)-2ρF′(ρ))dv.(1) When ρ≡n, there holds
(14)0=∫M2F′(n)(|∇h|2-n|∇H|2)+nH2(nF(n)-2nF′(n))dv.(2) When ρ≡0, there holds
(15)0=∫M2F′(0)(|∇h|2-n|∇H|2)+n2H2F(0)dv.(3) When F′≡0, F≡c≠0, there holds
(16)0=∫Mcn2H2dv.(4) When nF-2uF′≥0, F′≥0, F′′≥0, there holds
(17)∫M2ρ(ρ-n)F′(ρ)=∫MF′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)+nH2(nF(ρ)-2ρF′(ρ))dv.(5) When nF-2uF′≤0, F′≥0, F′′≥0, there holds
(18)∫M2ρ(ρ-n)F′(ρ)-nH2(nF(ρ)-2ρF′(ρ))=∫MF′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)dv.Remark 4.
WhenF(ρ)=ρn/2,ρ,ρr, 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 Theorem3, together with two famous Chern et al. results in [16], we can obtain some conclusions of gap phenomenon.Theorem 5.
LetM be an n-dimensional closed W(n,F)-Willmore hypersurface in unit sphere Sn+1(1), we have the following.(1)
WhenF′≡0, F≡c≠0 over (0,+∞), if 0≤ρ≤n, then ρ=0 or ρ=n.
(a)
Forρ=0, then H=0, ρ=0, S=0, and M is totally geodesic;(b)
Forρ=n, then H=0, ρ=n, S=n, and M=Cm,n-m=Sm(m/n)×Sn-m((n-m)/n) for some m with 1≤m≤n-1.(2)
WhennF-2uF′>0,F′>0,F′′≥0 over (0,∞), if 0≤ρ≤n, then ρ=0 or ρ=n.
(a)
Forρ=0, then M is totally umbilical;(b)
Forρ=n, then n≡0(mod⁡2), H=0, ρ=n, S=n, and M=Cn/2,n/2=Sn/2(1/2)×Sn/2(1/2).(3)
WhennF-2uF′=0, F′>0, F′′≥0 over (0,+∞), if 0≤ρ≤n, then F(u)=cun/2, c>0, ρ=0, or ρ=n.
(a)
Forρ=0, M is totally umbilical;(b)
Forρ=n, M=Wm,n-m=Sm((n-m)/n)×Sn-m(m/n) for some m with 1≤m≤n-1.(4)
WhennF-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)
ForH=0, ρ=0, then M is totally geodesic;(b)
ForH=0, ρ=n, then n≡0(mod⁡2), H=0, ρ=n, S=n, M=Cn/2,n/2=Sn/2(1/2)×Sn/2(1/2).Remark 6.
WhenF(ρ)=ρn/2,ρ,ρr, 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

Letφ:Mn→Sn+1(1) be an n-dimensional closed hypersurface in an (n+1)-dimensional unit sphere and let Φ(·,·):M×(-ϵ,ϵ)→Sn+1(1) be a variation of φ; it means that
(19)φt=:Φ(·,t):M×{t}⟶Sn+1(1),∀t∈(-ϵ,ϵ)
is still an isometric immersion with φ0≡φ.Lets=(s1,…,sn,sn+1) and σ=(σ1,…,σn,σn+1) be the orthonormal local frames of TSn+1(1) and T*Sn+1(1), respectively; then e=(e1,…,en,en+1)=φt-1s is the orthonormal local frames of the pullback vector bundle φt-1TSn+1(1) over M×{t}, such that {e1,…,en} are tangent to M and {en+1} is normal to M due to
(20)φt-1TSn+1(1)=TMt⊕T⊥Mt.Useω to denote the connection form over TSn+1(1); by the pullback operation, we have the following decomposition:
(21)Φ*σ=θ+Vdt,Φ*σi=θi+Vidt,Φ*σn+1=Vn+1dt=:fdt,Φ*ω=ϕ+Ldt,Φ*ωij=ϕij+Lijdt,Φ*ωin+1=ϕin+1+bidt.
Thus, one can derive that {θi} are the orthonormal frames of T*M and θn+1≡0 when it is restricted over M, {Vi,Vn+1=:f} are the variation vector fields of Φ, {ϕij} is the connection form of TM, and {ϕin+1=hijθj} is the second fundamental form. In particular, we have
(22)∂∂tφt|t=0=∑iViei+fen+1.UseΩ and Ω⊤ to denote curvature forms of TSn+1(1) and TM, respectively and write their components:
(23)ΩAB=12R¯ABCDσC∧σD,A,B,C,D∈{1,…,n+1},Ωij⊤=12Rijklθk∧θl,i,j,k,l∈{1,2,…,n}.
From any one standard differential geometry book, we know that the curvature of unit sphere is
(24)R¯ABCD=-(δACδBD-δADδBC).Using connection form{ϕij}, the covariant derivatives of hij, H, ρ, Vi, f, and bi can be defined as
(25)hij,kθk=dMhij-∑phpjϕip-∑phipϕjp,hij,klθl=dMhij,k-∑phpj,kϕip-∑phip,kϕjp-∑phij,pϕkp,H,iθi=dMH,H,ijθj=dMH,i-∑pH,pϕip,ΔH=∑i‍H,ii,ρ,iθi=dMρ,ρ,ijθj=dMρ,i-∑pρ,pϕip,Δρ=∑iρ,ii,V,jiθj=dMVi+∑pVpϕpi,V,jkiθk=dMV,ji-∑pV,piϕjp+∑pV,jpϕpi,f,iθi=dMf,f,ijθj=dMf,i-∑pf,pϕip,Δf=∑f,ii,bi,jθj=dMbi-∑pbpϕip,bi,jkθk=dMbi,j-∑pbp,jϕip-∑pbi,pϕjp.We usedM,dM×(-ϵ,ϵ)=dM+dt(∂/∂t) and d to denote the differential operators on M, M×(-ϵ,ϵ), and Sn+1(1), respectively. By calculating directly and comparing both sides of the following equations:
(26)Ω=dω-ω∧ω,0=dσ-σ∧ω,(27)Φ*(Ω)=dM×(-ϵ,ϵ)Φ*(ω)-Φ*(ω)∧Φ*(ω),(28)0=dM×(-ϵ,ϵ)Φ*(σ)-Φ*(σ)∧Φ*(ω),
we would derive out the lemmas below; also see article [10]. In fact, the right hand side of (27) is
(29)Φ*(ΩAB)=Φ*(12R¯ABCDσC∧σD)=12R¯ABCDΦ*(σC)∧Φ*(σD)=12R¯ABCDθC∧θD+dt∧(R¯ABCDVCθD),
while the left hand side of (27) is
(30)dM×(-ϵ,ϵ)Φ*(ωAB)-Φ*(ωAC)∧Φ*(ωCB)=(dM+dt∂∂t)(ϕAB+dtLAB)-(ϕAC+dtLAC)(ϕCB+dtLCB)=(dMϕAB-ϕAC∧ϕCB)+dt(∂∂tϕAB-(dMLAB+LACϕCB-ϕACLCB));
comparing (29) with (30), we obtain
(31)(dMϕAB-ϕAC∧ϕCB)=12R¯ABCDθC∧θD,∂∂tϕAB-(dMLAB+LACϕCB-ϕACLCB)=R¯ABCDVCθD.
The equation of (28) is
(32)0=dM×(-ϵ,ϵ)Φ*(σ)-Φ*(σ)∧Φ*(ω)=(dM+dt∂∂t)(θ+dtV)-(θ+dtV)∧(ϕ+dtL)=dMθ-θ∧ϕ+dt(∂θ∂t-dMV-Vϕ+θL).
Comparing both sides, we obtain
(33)dMθ-θ∧ϕ=0,∂θ∂t-dMV-Vϕ+θL=0.
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.
Letφ:Mn→Sn+1(1) be a hypersurface; one has structure equations:
(34)Rijkl=-(δikδjl-δilδjk)-(hikhjl-hilhjk),hij,k=hik,j.Lemma 8.
Letφ:Mn→Sn+1(1) be a hypersurface; one has Ricci identity:
(35)hij,kl-hij,lk=-(δikhjl-δilhjk)-(δjkhil-δjlhik)-(hikhplhpj-hilhpkhpj)-(hjkhplhip-hjlhpkhip).Lemma 9.
Letφ:Mn→Sn+1(1) be a hypersurface and let V=V⊤+V⊥=Viei+fen+1 be a variation vector field; one has
(36)∂θi∂t=(V,ji-Lji-hijf)θj,∂hij∂t=f,ij+∑phij,pVp+∑phpjLip+∑phipLjp+∑p‍hiphpjf+δijf.Lemma 10.
With the same notations as above, one has(37)∑ijkhikhkjLji=0,∑ijhijLji=0.


## 3. Variation Calculation and Examples

To calculate the first variation ofW(n,F)-Willmore functional, we need two lemmas.Lemma 11.
Letφ:Mn→Sn+1(1) be a hypersurface and let V=V⊤+V⊥=Viei+fen+1 be a variation vector field. Suppose that dv=θ1∧⋯∧θn denotes the volume element; one has
(38)∂dv∂t=∑iV,ii-nHf.Proof.
By Lemma9, we have (∂/∂t)(θi)=(V,ji-Lji-hijf)θj. Hence,
(39)∂dv∂t=∑iθ1∧⋯∂∂t(θi)⋯∧θn=∑iV,ii-nHf.
With this, we complete the proof of Lemma 11.Lemma 12.
Letφ:Mn→Sn+1(1) be a hypersurface and let V=V⊤+V⊥=Viei+fen+1 be a variation vector field; one has
(40)∂ρ∂t=∑ij2hijf,ij-2HΔf+2P3f-2SHf+∑iρ,iVi.Proof.
By the definition ofρ,S and H, one has
(41)∂∂t(ρ)=∂∂t(S)-∂∂t(nH2).
It follows Lemmas 9 and 10 that
(42)∂∂t(S)=2∑ij‍hij∂∂t(hij)=2∑ijhij[f,ij+∑phij,pVp+∑phpjLip+∑phiPLjp+∑phiphpjf+δijf]=2∑ij‍hijf,ij+∑i‍S,iVi+2P3f+2nHf.
By Lemmas 9 and 10 again, one has
(43)∂∂t(nH2)=2nH∂∂tH=2H∑i[f,ii+∑p‍hii,pVp+∑p‍hpiLip+∑p‍hpiLip+∑p‍hiphpif+δiif]=2HΔf+∑i‍(nH2),iVi+2HSf+2nHf.
Hence,
(44)∂ρ∂t=∑ij2hijf,ij-2HΔf+2P3f-2HSf+∑iρ,iVi.
With this, we complete the proof of Lemma 12.Proof of Theorem1.
By Lemmas11 and 12, we can calculate
(45)∂∂tW(n,F)(φt)=∫MtF′(ρ)∂∂t(ρ)+F(ρ)(∑iV,ii-nHf)dv=∫MtF′(ρ)(∑ij2hijf,ij-2HΔf+2P3f-2SHf+∑iρ,iVi)+F(ρ)(∑i‍V,ii-nHf)dv=∫Mt‍2[∑ij(F′(ρ)hij),ji-Δ(F′(ρ)H)+(P3-SH)F′(ρ)-n2HF(ρ)∑ij]fdv.
With this, we complete the proof of Theorem 1.From Theorem1, we can find a few examples in unit sphere Sn+1(1), in particular, the isoparametric hypersurfaces. Since all principal curvatures k1,…,kn 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
(46)(P3-1nP1P2)F′(ρ)-12P1F(ρ)=0.Example 13.
Obviously, totally geodesic hypersurfaces areW(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 withn≡0(mod⁡2),
(47)Cn/2,n/2=Sn/2(12)×Sn/2(12)⟶Sn+1(1).
All principal curvatures are
(48)k1=⋯=kn/2=1,k(n/2)+1=⋯=kn=-1.
We can derive the quantities P1,P2,P3,ρ, respectively, P1=0,P2=n,P3=0,ρ=n. Obviously Cn/2,n/2 always is a W(n,F)-Willmore hypersurface of Sn+1(1) for any function F.Example 15.
For a family hypersurfaces with parametersλ,μ,0<λ, μ<1, λ2+μ2=1. 							(49)Sm(λ)×Sn-m(μ)⟶Sn+1(1),1≤m≤n-1.
Obviously, all principal curvatures are
(50)k1=⋯=km=μλ,km+1=⋯=kn=-λμ.
Then the quantities P1, P2, P3, ρ are, respectively,
(51)P1=mμλ-(n-m)λμ,P2=mμ2λ2+(n-m)λ2μ2,P3=mμ3λ3-(n-m)λ3μ3,ρ=m(n-m)n[μ2λ2+λ2μ2+2].
Setting μ/λ=x>0, W(n,F)-Willmore hypersurface equation becomes
(52)2m(n-m)F′(ρ)x6-m[nF(ρ)-2(n-m)F′(ρ)]x4+(n-m)[nF(ρ)-2mF′(ρ)]x2-2m(n-m)F′(ρ)=0.Example 16 (Clifford torus [16]; F(ρ)≡1).
Consider that(53)Cm,n-m=Sm(mn)×Sn-m(n-mn),1≤m≤n-1
are minimal hypersurfaces with H≡0,S≡n,ρ≡n.Example 17 (classic Willmore torus [6]; F(ρ)=ρn/2).
Consider that(54)Wm,n-m=Sm(n-mn)×Sn-m(mn),1≤m≤n-1
are W(n,n/2)-Willmore hypersurfaces with ρ=n. When some Wm,n-m is minimal, then n≡0(mod⁡2), 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.
Forρ, one has
(55)Δρ=∑ij2nhijH,ij-2nHΔH+2|∇h|2-2n|∇H|2+2nρ+2nHP3-2S2,
where |∇h|2=∑ijk‍hij,k2.Proof.
By the definition ofρ and Laplacian operator, we have
(56)Δρ=∑k(∑ijhij2-nH2),kk=∑k(∑ij‍2hijhij,k-2nHH,k),k=∑k(∑ij2hij,k2+∑ij2hijhij,kk-2nH,k2-2nHH,kk)=2|∇h|2-2nHΔH-2n|∇H|2+2∑ijkhijhij,kk=2(|∇h|2-nHΔH-n|∇H|2+(T1)∑ijkhijhij,kk),
for term T1=∑ijk‍hijhij,kk; by Lemmas 7 and 8, easily we obtain
(57)T1=∑ijkhijhij,kk=∑ijkhijhik,jk=∑ijkhij(hik,jk-hik,kj+hik,kj)=∑ijkhijhkk,ij+∑ijkhij(hik,jk-hik,kj)=∑ijnhijH,ij+(T2)∑ijkhij(hik,jk-hik,kj),
for term T2=∑ijk‍hij(hik,jk-hik,kj); by Lemma 8, we have
(58)T2=∑ijk‍hij(hik,jk-hik,kj)=∑ijkhij{[-(δjkhik-δkkhij)-(hjkhiphpk-hkkhiphpj)]+[-(δijhkk-δikhjk)-(hijhkphpk-hikhkphpj)]}=-S+nS-∑ijkphijhjkhiphpk+nHP3-n2H2+S-S2+∑ijkp‍hijhikhkphpj=nρ-S2+nHP3.
Substituting (58) into (57), we have
(59)T1=∑ijnhijH,ij+nρ-S2+nHP3.
Substituting (59) into (56), we have
(60)Δρ=∑ij2nhijH,ij-2nHΔH+2|∇h|2-2n|∇H|2+2nρ+2nHP3-2S2.
With this, we complete the proof of Lemma 18.Lemma 19 (Huisken’s estimate [17]).
Consider that(61)|∇h|2≥3n2n+2|∇H|2≥n|∇H|2,
and |∇h|2=n|∇H|2 if and only if ∇h=0.Proof.
We decompose the tensor∇h as follows:
(62)hij,k=Eijk+Fijk,
where
(63)Eijk=nn+2(H,iδjk+H,jδik+H,kδij),Fijk=hij,k-Eijk.
Through a computation, we obtain
(64)|E|2=3n2n+2|∇H|2,〈E,F〉=0.
Finally, by the triangle inequality, we have
(65)|∇h|2≥|E|2=3n2n+2|∇H|2≥n|∇H|2.
If |∇h|2=n|∇H|2, then all inequalities become equalities, Fijk=0, ∇H=0, and Eijk=0; thus, ∇h=0. With this, we complete the proof of Lemma 19.Lemma 20.
ForF(ρ), one has
(66)ΔF(ρ)=F′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)-2ρ(ρ-n)F′(ρ)+nH2(nF(ρ)-2ρF′(ρ))+2n[∑ijF′(ρ)hijH,ij-F′(ρ)HΔH+(P3-SH)F′(ρ)H-n2F(ρ)H2∑ij].Proof.
We know that(67)ΔF(ρ)=F′′(ρ)|∇ρ|2+F′(ρ)Δρ.
By Lemmas 18 and 19, we can easily obtain the identity.Proof of Theorem3.
Integrating the equality in Lemma20 over M, we have
(68)0=∫M‍F′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)-2ρ(ρ-n)F′(ρ)+nH2(nF(ρ)-2ρF′(ρ))+2n[∑ijF′(ρ)hijH,ij-F′(ρ)HΔH+(P3-SH)F′(ρ)H-n2F(ρ)H2∑ij].
By Theorem 1, we know that
(69)∑ij(F′(ρ)hij),ji-Δ(F′(ρ)H)+(P3-SH)F′(ρ)-n2F(ρ)H=0.
Using Stokes theorem, we obtain
(70)0=∫M‍F′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)-2ρ(ρ-n)F′(ρ)+nH2(nF(ρ)-2ρF′(ρ)).
We complete the proof of Theorem 3.In order to prove Theorem5, 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 hij=0,for alli≠j,hi=hii.Lemma 21 (Lemma 3 of [16]).
Letx:Mn→Sn+1(1) be a compact hypersurface with ∇h≡0; then there are two cases.(1)
h1=⋯=hn=λ=constant, and M is a totally umbilic (λ>0) or totally geodesic (λ=0);(2)
h1=⋯hm=λ=constant>0, hm+1=⋯=hn=-(1/λ),1≤m≤n-1, and M is a Riemannian product of M1×M2, where M1=Sm(1/1+λ2),M2=Sn-m(λ/1+λ2).Lemma 22 (main Theorem of [16]).
Clifford torusCm,n-m,1 ≤ m ≤ n-1 are the only compact minimal (H=0) hypersurfaces of dimension n in unit sphere Sn+1(1) satisfying S=n.Proof of Theorem5.
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
(71)∫M‍2ρ(ρ-n)F′(ρ)=∫MF′′(ρ)|∇ρ|2+2F′(ρ)(|∇h|2-n|∇H|2)+nH2(nF(ρ)-2ρF′(ρ))≥0.
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=Cm,n-m; moreover, by W(n,F)-Willmore equation, we conclude that n≡0(mod⁡2), H=0, ρ=n, S=n, and M=Cn/2,n/2.
(
3
) When nF-2uF′=0, F′>0, F′′≥0(0,+∞), we know that F(u)=cun/2, c=const>0. By the case (4) of Theorem 3, we have
(72)∫Mρn/2(ρ-n)=∫M‍n-24|∇ρ|2+ρ(n/2)-1(|∇h|2-n|∇H|2)≥0.
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 Wm,n-m for some m.



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*Source: 697132-2014-06-05.xml*