An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5with Constant Nonnegative Scalar Curvature?

Bing TANG Ling YANG

Abstract Let M4be a closed minimal hypersurface in S5with constant nonnegative scalar curvature.Denote by f3the sum of the cubes of all principal curvatures,by g the number of distinct principal curvatures.It is proved that if both f3and g are constant,then M4is isoparametric.Moreover,the authors give all possible values for squared length of the second fundamental form of M4.This result provides another piece of supporting evidence to the Chern conjecture.

Keywords Chern conjecture,Isoparametric hypersurfaces,Scalar curvature,Minimal hypersurfaces in spheres

1 Introduction

More than 40 years ago,Chern[6–7]proposed the following problem in several places.

Problem 1.1Let Mnbe a closed minimal submanifold in Sn+mwith the second fundamental form of constant length.Denote by Anthe set of all the possible values for the squared length of the second fundamental form of Mn.Is Ana discrete set?

The affirmative hand of this question is usually called the Chern conjecture.

Denote by B the second fundamental form of Mnand letUsing the Gauss equations,one can easily deduce that

with R denoting the scalar curvature of Mn.It means that S is in fact an intrinsic geometric quantity,and the Chern conjecture is equivalent to claiming that the scalar curvature R has gap phenomena for closed minimal submanifolds in Euclidean spheres.

Up to now,it is far from a complete solution of this problem,even in the case that M is a hypersurface(see[15,Problem 105]).Moreover,because all known examples of closed minimal hypersurfaces in Sn+1with constant scalar curvature are all isoparametric hypersurfaces(the definition of isoparametric hypersurfaces will be introduced in Section 2),mathematicians turned the hypersurface case of Chern conjecture into the following new formulation(see[12,14]).

Conjecture 1.1Let Mnbe a closed minimal hypersurface in Sn+1with constant scalar curvature.Then M is an isoparametric hypersurface.

When n=2,this conjecture is trivial.For the case that n=3,Chang[4–5]gave a positive answer to the Chern conjecture.More precisely,it was shown that any closed minimal hypersurface M3in S4with constant scalar curvature has to be isoparametric,and A3={0,3,6}.

For n≥4,the Chern conjecture remains open,although some partial results exist for low dimensions and with additional conditions for the curvature functions,such as the following theorem.

Theorem 1.1(see[8]) Let M4be a closed minimal Willmore hypersurface in S5with constant nonnegative scalar curvature.Then M4is isoparametric.

Theorem 1.2(see[11])Let M6?S7be a closed hypersurface withconstant f4and R≥0.Then M6is isoparametric.

Here and in the sequel

with λ1,···,λnbeing the principal curvatures of M.

Note that in Theorem 1.1,the Willmore condition is equal to saying thatIt is natural to ask whether this conclusion holds whenis replaced by a weaker condition that f3≡const.In this paper,we give a partial positive answer to the above question and obtain the main theorem as follows.

Theorem 1.3Let M4be a closed minimal hypersurface in S5with constant nonnegative scalar curvature.If f3and the number g of distinct principal curvatures of M4are constant,then M4is isoparametric.

Finally,in conjunction with the theory of isoparametric hypersurfaces in Euclidean spheres,we arrive at a classification result(see Theorem 3.1),which gave a piece of supporting evidence to the Chern conjecture.

2 Isoparametric Minimal Hypersurfaces in S5

Let Mnbe an immersed hypersurface in Sn+1.If Mnhas constant principal curvatures,then Mnis said to be an isoparametric hypersurface.Each isoparametric hypersurface is an open subset of a level set of a so-called isoparametric function f.More precisely,there exists a smooth function f:Sn+1→R and c∈R,such thatandare both smooth functions of fandare respectively the gradient operator and Laplace-Beltrami operator on Sn+1),and f(p)=c for each p∈M.Conversely,given an isoparametric function f,the level sets of f consist of a smooth family of isoparametric hypersurfaces and 2 minimal submanifolds of higher codimension(called focal submanifolds).

The following theorem reveals some important geometric properties of isoparametric minimal hypersurfaces in Euclidean spheres(see[1–2,9–10]).

Theorem 2.1Let f:Sn+1→R be an isoparametric function.Then there exists a unique c0∈R,such that M:={x∈Sn+1:f(x)=c0}is an isoparametric minimal hypersurface.Let g be the number of distinct principal curvatures of M,λ1> ···> λgbe the distinct principal curvatures,whose multiplicities are m1,···,mg,respectively,and the denotation of S and R is the same as above.Then

(1)g=1,2,3,4 or 6.

(2)If g=1,then M has to be the totally geodesic great subsphere.

(3)If g=2,then M has to be a Clifford hypersurface,i.e.,

where 1≤r

(4)If g=3,then m1=m2=m3=2r(r=0,1,2 or 3).

(6)R≥0 and S=(g?1)n.

Cartan[3]constructed an example of minimal hypersurface in S5as follows.

Example 2.1Denote

A straightforward calculation shows thatis an isoparametric function andis a minimal isoparametric hypersurface with 4 distinct principal curvatures,which is usually called the Cartan minimal hypersurface.

3 Proof of the Main Theorem

Let M4be an immersed hypersurface in S5.If ν is a local unit normal vector field along M,then there exists a pointwise symmetric bilinear form h on TpM,such that

If{ω1,ω2,ω3,ω4}is a smooth orthonormal coframe field,then h can be written as

The covariant derivative?h with components hijkis given by

Here{ωij}is the connection forms of M4with respect to{ω1,ω2,ω3,ω4},which satisfy the following structure equations:

with Rijkldenoting the coefficients of the Riemannian curvature tensor on M4.

In this section,we shall give a proof of the main theorem in Section 1.

Proof of Theorem 1.3We shall consider this problem case by case,according to the value of g,i.e.,the number of distinct principal curvatures.

Case Ig=1.

In this case,all the principal curvatures are equal to 0 and hence M4is totally geodesic.

Case IIg=2.

Let λ and μ be distinct pricipal curvatures of M4with multiplicities m1=k,m2=4?k,respectively.We need to show that λ,μ are indeed constant functions.

we can solve m1,m2in terms of λ,μ and S,in other words,m1,m2can be seen as continuous functions of λ,μ and S.In conjunction with the fact that m1,m2take values in Z,both m1,m2are constant,so does k.Again from(3.3),we have

or

Thus λ and μ are both constant and M4is an isoparametric hypersurface.

Case IIIg=3.

Let λ,μ,σ be distinct principal curvatures of M4,with multiplicities p,q,r,respectively.Then

As in Case II,one can show that p,q,r are all constant integer-valued functions.Differentiating both sides of(3.6)gives

It follows that

where D:=(σ ?μ)(σ ?λ)(μ ?λ).Hence λ,μ and σ are all constant and M4is isoparametric.(In fact,Theorem 2.1 shows that there exists no isoparametric minimal hypersurface in S5with g=3,so this case cannot occur.)

Case IVg=4.

Let λ1< λ2< λ3< λ4be distinct principal curvatures of M4.We say that a coframe field(U,ω)is admissible(see[11])if

(1)U is an open subset of M4,

(2) ω :={ω1,ω2,ω3,ω4}is a smooth orthonormal coframe field on U,

(3)ω1∧ω2∧ω3∧ω4is the volume form of M4,

Denote by F:={e1,e2,e3,e4}the dual frame field of ω.Then it is easily-seen that,(U,ω)is admissible if and only if eiis a unit principal vector associated to λifor each 1 ≤ i≤ 4,and{e1,e2,e3,e4}is an oriented basis associated to the orientation of M4.Therefore,for every p∈ M,there exists an admissible coframe field(U,ω),such that p∈ U.

Now we introduce a 3-form on M4:For every admissible coframe field(U,ω),set

where?is the Hodge star operator.If(U,ω)andare both admissible coframe fields with,then onfor each 1≤i≤4,where αi=1 or?1 andDenote bythe connection form with respect to.Thenand hence

holds for any i

Now we compute the exterior differential of the form ψ.Due to the definition of the Hodge star operator,ψ can be written as

Combining(3.11)and(3.2)yields

Hence

where we have used Codazzi equations.A similar calculation shows

By the structure equations,

Combining(3.12)–(3.14)gives

Similarly,one can compute the exterior differential of each term of(3.10);taking the sum of these equations,we arrive at

where

Taking the exterior differential of

implies that

holds for each 1≤k≤4.Especially,letting k:=1 gives

Since λ1,λ2,λ3and λ4are distinct at every point,we can express hii1,i=2,3,4,in terms of

Let K:=deth be the Gauss-Kronecker curvature of M4and denote

Then

and hence

In a similar way,we have

Substituting(3.24)into(3.17),we deduce that

More precisely,

and

Observing that λ1< λ2< λ3< λ4,we can derive estimates as follows:

In the same way,I3≤0,I4≤0.

Note that M4is closed.Integrating both sides of(3.16)on M4and then using Stokes’s theorem gives

Combining Theorem 1.3 and Proposition 2.1 yields a classification theorem as follows.

AcknowledgementThe first author would like to thank his supervisor Professor Ling Yang for his constant encouragement and help.