Embedded constant mean curvature hypersurfaces on spheres

# Embedded constant mean curvature hypersurfaces on spheres

Oscar M. Perdomo Department of Mathematics
Central Connecticut State University
New Britain, CT 06050
July 13, 2019
###### Abstract.

Let and be any pair of integers. In this paper we prove that if is between the numbers and , then, there exists a non isoparametric, compact embedded hypersurface in with constant mean curvature that admits the group in their group of isometries, here is the set of orthogonal matrices and are the integers mod . When and is close to the boundary value , the hypersurfaces look like two very close -dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When and is close to , the hypersurfaces look like a necklets made out of spheres with catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when is close to the hypersurface is close to an isoparametric hypersurface with the same mean curvature. As a consequence of the expression of these bounds for , we have that every different from can be realized as the mean curvature of a non isoparametric CMC surface in . For hyperbolic spaces we prove that every non negative can be realized as the mean curvature of an embedded CMC hypersurface in , moreover we prove that when this hypersurface admits the group in its group of isometries. Here are the integer numbers. As a corollary of the properties proven for these hypersurfaces, for any , we construct non isoparametric compact minimal hypersurfaces in which cone in is stable. Also, we will prove that the stability index of every non isoparametric minimal hypersurface with two principal curvatures in is greater than .

###### 2000 Mathematics Subject Classification:
58E12, 58E20, 53C42, 53C43

## 1. Introduction

Minimal hypersurfaces on spheres with exactly two principal curvatures everywhere were studied by Otsuki in ([3]), he reduced the problem of classifying them all, to the problem of solving an ODE, and the problem of deciding about their compactness, to the problem of studying a real function given in term of an integral that related two periods of two functions involved in the immersions that he found. In this paper we changed the minimality condition for the constant mean curvature condition and using a slightly different point of view, we got similar results. As pointed out by Do Carmo and Dajczer in ([6]), these hypersurfaces are rotations of a plane profile curve. The existence of these hypersurfaces as immersions have been established in ([6]) and in ([7]). Partial result about the condition for small values of that guarantee embedding were found in ([8]). The main work in this paper consists in studying these profile curves and in deciding when they are embedded. Lemma (5.1) and its corollary (5.2) played an important role in the understanding of the period of these profile curves for they were responsible of getting explicit formulas for these two numbers

 an,m=cotπmandbn,m=(m2−2)√n−1n√m2−1

with the property that every between and can be realized as the mean curvature of a non isoparametric embedded CMC hypersurface in , such that its profile curve is invariant under the group of rotation by an angle , and therefore the hypersurface admits the group in its group of isometries.

One of the differences between the analysis of the profile curve in this work and Otsuki’s, is that Otsuki used the supporting function of this profile curve. We, instead, studied the radius and the angle separately, and we proved that the angle function is increasing, which helps us to decide when this curve is injective and consequently, when the immersion is an embedding. In order to understand this angle function we got some help from the understanding of three vector fields defined in section (3.3).

Since we can obtain a similar formula for the angle of the profile curve for CMC hypersurfaces in the hyperbolic space, we also extend the result in this case in order to explicitly exhibit embedded examples the hyperbolic space. Also, similar results are obtained in Euclidean spaces. These results on embedded hypersurfaces on hyperbolic spaces and Euclidean spaces where proven in ([1]) with different techniques.

As a consequence of the symmetries proven for all compact constant mean curvatures in with two principal curvatures everywhere, we proved that all these examples with , have stability index greater than . In this direction there is a conjecture that states that the only minimal hypersurfaces in with stability index are the isoparametric with two principal curvatures. Some partial results for this conjecture were proven in ([2]). Also, since it is not difficult to prove that these examples can be chosen to be as close as we want from the isoparametric examples, we proved that some of Otsuki’s minimal hypersurfaces are examples of non isoparametric compact stable minimal truncated cones in for .

The author would like to express his gratitude to Professor Bruce Solomon for discussing the hypersurfaces with him and pointing out the similarity between them and the Delaunay’s surfaces.

## 2. Preliminaries

Let be an n-dimensional hypersurface of the -dimensional unit sphere . Let be a Gauss map and the shape operator, notice that

 Ap(v)=−¯∇vνfor allv∈TpM

where is the Euclidean connection in . We will denote by the square of the norm of the shape operator.

If , and are vector fields on , represents the Levi-Civita connection on with respect to the metric induced by and represents the Lie bracket, then, the curvature tensor on is defined by

 (2.1) R(X,Y)Z=∇Y∇XZ−∇X∇YZ+∇[X,Y]Z

and the covariant derivative of is defined by

 (2.2) DA(X,Y,Z)=Z⟨A(X),Y⟩−⟨A(∇ZX),Y⟩−⟨A(X),∇ZY⟩

the Gauss equation is given by,

 (2.3) R(X,Y)Z=⟨X,Z⟩Y−⟨Y,Z⟩X+⟨A(X),Z⟩A(Y)−⟨A(Y),Z⟩X

and the Codazzi equations are given by,

 (2.4) DA(X,Y,Z)=DA(Z,Y,X)

Let us denote by the principal curvatures of and, by the mean curvature of . We will assume that has exactly two principal curvatures everywhere and that is a constant function on . More precisely, we will assume that

 κ1=⋯=κn−1=λ,κn=μand(n−1)λ+μ=nH

By changing by if necessary we can assume without loss of generality that . Let denotes a locally defined orthonormal frame such that

 (2.5) A(ei)=λeifor i=1,…,n−1 andA(en)=μen

The next Theorem is well known, see ([3]), for completeness sake and as part of preparation for the deduction of other formulas, we will show a proof here,

###### Theorem 2.1.

If is a CMC hypersurface with two principal curvatures and dimension greater than 2, and is a locally defined orthonormal frame such that (2.5) holds true, then,

 v(λ) = 0for any v∈Span{e1,…,en−1} ∇ven = en(λ)μ−λvfor any v∈Span{e1,…,en−1} ∇enen = 0 1+λμ = en(en(λ)λ−μ)−(en(λ)λ−μ)2
 [ei,ej]∈Span{e1,…,en−1}for any i,j∈{1,…,n−1}
###### Proof.

For any with (here we are using the fact that the dimension of is greater than 2) and any , we have that,

 DA(ei,ej,ek) = ek⟨A(ei),ej⟩−⟨A(∇ekei),ej⟩−⟨A(ei),∇ekej⟩ = ek(λ⟨ei,ej⟩)−⟨∇ekei,A(ej)⟩−λ⟨ei,∇ekej⟩ = ek(0)−λ⟨∇ekei,ej⟩−λ⟨ei,∇ekej⟩ = 0−λek(⟨ei,ej⟩) = 0

On the other hand,

 DA(ei,ei,ej) = ej⟨A(ei),ei⟩−⟨A(∇ejei),ei⟩−⟨A(ei),∇ejei⟩ = ej(λ)−λej(⟨ei,ei⟩) = ej(λ)

By Codazzi equation (2.4), we get that , for all , therefore for any . Now,

 DA(ei,en,ej) = ej⟨A(ei),en⟩−⟨A(∇ejei),en⟩−⟨A(ei),∇ejen⟩ = ej(λ⟨ei,en⟩)−⟨∇ejei,A(en)⟩−λ⟨ei,∇ejen⟩ = ej(0)−μ⟨∇ejei,en⟩−λ⟨ei,∇ejen⟩+(λ⟨∇ejei,en⟩−λ⟨∇ejei,en⟩) = (λ−μ)⟨∇ejei,en⟩−λ ej(⟨ei,en⟩) = (μ−λ)⟨ei,∇ejen⟩

Since , using Codazzi equations we get that,

 (2.6) ⟨ei,∇ejen⟩=0for % any i,j∈{1,…,n−1} with i≠j

Now, for any , using the same type of computations as above we can prove that,

 DA(ei,ei,en)=en(λ)=DA(ei,en,ei)=(μ−λ)⟨ei,∇eien⟩

and

 DA(en,en,ei)=ei(μ)=0=DA(ei,en,en)=(μ−λ)⟨ei,∇enen⟩

Therefore,

 (2.7) ⟨ei,∇eien⟩=en(λ)μ−λand⟨ei,∇enen⟩=0for any i∈{1,…,n−1}

Since is a unit vector field, we have that for any . From the equations (2.6 ) and (2.7 ) we conclude that

 ∇ven=en(λ)μ−λv%foranyv∈Span{e1,…,en−1}and∇enen=0

Notice that for any with , using equation (2.6), we get that

 ⟨[ei,ej],en⟩=⟨∇eiej−∇ejei,en⟩=⟨ei,∇ejen⟩−⟨ej,∇eien⟩=0

Therefore . Finally we will use Gauss equation to prove the differential equation on . First let us point out that, using the equation (2.7), we can prove that and therefore we have that . Now, by Gauss equation we get,

 1+λμ = ⟨R(en,e1)en,e1⟩ = ⟨∇e1∇enen−∇en∇e1en+∇[en,e1]en,e1⟩ = ⟨0−∇en(en(λ)μ−λe1)+en(λ)μ−λ[en,e1],e1⟩ = −en(en(λ)μ−λ)+en(λ)μ−λ⟨∇ene1−∇e1en,e1⟩ = −en(en(λ)μ−λ)−(en(λ)μ−λ)2 = en(en(λ)λ−μ)−(en(λ)λ−μ)2

## 3. Construction of the examples

We will maintain the notation of the previous section and we will prove a serious of identities and results that will allow us to easier state and prove the theorem that defines the examples at the end of this section.

### 3.1. The function w and its solution along a line of curvature

Since , we get that

 (3.1) λ−μ=λ−(nH−(n−1)λ)=n(λ−H)=nw−nwherew=(λ−H)−1n

Recall that we are assuming that is always positive, then, is a smooth differentiable function. By the definition of given in (3.1) we have that,

 (3.2) en(w)=−1n(λ−H)−n+1nen(λ)=−1nwn+1en(λ)=−wen(λ)λ−μ

The second order differential equation in Theorem (2.1) can be written using the function as,

 (3.3) en(en(w)w)+(en(w)w)2+1+λμ=0

and if we write and in terms of we get

 (3.4) en((en(w)w))+(en(w)w)2−(n−1)w2n−(n−2)Hwn+H2+1=0

Deriving the previous equation, we have used the following identities,

 (3.5) λ=w−n+Handμ=H−(n−1)w−n

From the Equation (3.2) we get that

 (3.6) en(λ)=−(λ−μ)en(w)w

The equation above allows us to write one of the equations in Theorem (2.1) as

 (3.7) ¯∇ven=en(w)wvfor any v∈Span{e1,…,en−1}

Notice that equation (3.4) reduces to

 (3.8) en(en(w))w−(n−1)w2n−(n−2)Hwn+H2+1=0

and therefore multiplying by we have that there exists a constant such that,

 (3.9) (en(w))2+w2−2n+(1+H2)w2+2Hw2−n=C

Let us denote by the position viewed as a map, and by the Euclidean connection on . Using the equations in Theorem (2.1) and the fact that , and , we get that

 (3.10) ¯∇enen = −x+μν (3.11) ¯∇enν = −μen (3.12) ¯∇enx = en

Let us fix a point , and let us denote by the only geodesic in such that and . Since vanishes, then . Notice that is also a line of curvature. Let us denote by . Equation (3.9) implies that

 (3.13) (g′)2+g2−2n+(1+H2)g2+2Hg2−n=C

or equivalently,

 (3.14) gn−1g′√Cg2n−2−1−(1+H2)g2n−2Hgn=±1

It is clear that the constant must be positive and moreover, in order to solve this equation we need to consider a constant such that the polynomial

 (3.15) ξ(s)=Cs2n−2−1−(1+H2)s2n−2Hsn

is positive on a interval with and . Notice that for every it is possible to pick such that is positive on an interval because is a polynomial of even degree with negative leading coefficient, , and if is big enough, this polynomial takes positive values for positive values of . Let us assume that and are as above and moreover let us assume that and are not zero, so that the function that we are about to define is well defined on . Let us consider

 G(s)=∫st1tn−1√Ct2n−2−1−(1+H2)t2n−2Htndtfort1≤u≤t2

and let . Since for , then we can consider the inverse of the function . Denoting by the inverse of , a direct verification shows that the -periodic function that satisfies

 g(u)=F(u)for0≤u≤T2andg(u)=F(T−u)forT2≤u≤T

is a solution of the equation (3.14)

### 3.2. The vector field η

Let us define the following normal vector field along

 η=−en(w)wen+λν−x

The vector field has the following properties

1. , this follows from Equation (3.9) and the definition of in terms of , (3.5).

2. . This is the equation that makes all the constructions work in this section. The equation follows from Equations (3.10), (3.11) and (3.12 ) and the first and second differential equations for the function , especially, Equation (3.3) and Equation (3.6).

3. For any , vanishes. The proof of this identity is similar, and additionally, uses the Equation (3.7).

4. .

### 3.3. Vector fields that lies on a plane

Now that we have computed the function , let us understand better the geodesic . The equations (3.10), (3.11) and (3.12 ) imply that

 X(u)=en(γ(u)),Y(u)=ν(γ(u))andZ(u)=γ(u)

satisfy an ordinary linear differential equation in the variable with periodic coefficients (notice that is a function of ). By the existence and uniqueness theorem of ordinary differential equations we get the solutions , and lies in the three dimensional space

 (3.16) Γp0=Span{en(p0),ν(p0),p0}

For the sake of simplification, we will consider the -periodic function defined by

 r(u)=g(u)√C

It is not difficult to check that the function satisfies the following equations

 (3.17) r′′r+1+λμ=0,(r′)2+r2(1+λ2)=1,λ′=−(λ−μ)r′r

In the previous equations we are abusing of the notation with the name of the functions and , in this case, and whenever is understood by the context, they will also denote the function and respectively. In the construction of the examples that we will be considering, the function needs to be positive. We can achieve this by assuming that because this condition will imply that , and therefore .

Let us define the following vector fields along

 B1(u) = η(γ(u))=−r′rX+λY−Z B2(u) = −rr′√1−r2X+r2λ√1−r2Y+√1−r2Z B3(u) = rλ√1−r2X+r′√1−r2Y

Using the equations in section (3.2), the Equations (3.10), (3.11) and (3.12 ) that give the derivative of the vector fields , and , and Equation (3.17), we can check the following properties.

1. , and lie on the three dimensional subspace

2.  B′1=−r′rB1
3.  ⟨B1,B2⟩=0,⟨B1,B3⟩=0and⟨B2,B3⟩=0
4.  ⟨B2,B2⟩=1,⟨B3,B3⟩=1and⟨B1,B1⟩=1r2
5. From the previous items we get that

 B′2=hB3 andB′3=−hB2 % for some function h:R→R

These equations hold true because

 ⟨B′2,B1⟩=−⟨B′1,B2⟩=r′r⟨B1,B2⟩=0likewise⟨B′3,B1⟩=0
6. From the previous item we get that the vectors and lie on a two dimensional subspace.

7. We have that

 ⟨B′3,Z⟩=−rλ√1−r2% and⟨B2,Z⟩=√1−r2

therefore the function in the previous item is given by . It follows that,

 B′2=rλ1−r2B3andB′3=−rλ1−r2B2

The fact that does not change sign when , in particular when , will help us prove that for some choices of the hypersurface is embedded.

8. If we assume without loss of generality that

 1|B1(0)|B1(0)=(0,…,1,0,0),B2(0)=(0,…,0,1,0)andB3(0)=(0,…,0,0,1)

then,

 B1(u) = 1r(0,…0,1,0,0) B2(u) = sin(θ(u))(0,…0,0,1)+cos(θ(u))(0,…,0,1,0) B3(u) = cos(θ(u))(0,…0,0,1)−sin(θ(u))(0,…,0,1,0)

where is a given by

 θ(u)=∫u0r(s)λ(s)1−r2(s)ds
9. If

 K=K(H,n,C)=θ(T)=∫T0r(s)λ(s)1−r2(s)ds=2∫T20r(s)λ(s)1−r2(s)ds

then, for any positive integer and any we have that

 θ(u)=mK+θ(u−mT)

This property is a consequence of the existence and uniqueness theorem for differential equation and will be used to prove the invariance of under some rotations.

10. If , then

 ⟨q,q⟩=1−r2and B2=q|q|i.eq=√1−r2B2

### 3.4. A classification of constant mean curvature hypersurfaces in spheres with two principal curvatures

We are ready to define the examples of constant mean curvature hypersurfaces on when . Here is the theorem:

###### Theorem 3.1.

Let be a positive integer greater than and let be a non negative real number.

1. Let be a -periodic solution of the equation (3.13) associated with this and a positive constant . If are defined by

 r=gC√C,λ=H+g−nCandθ(u)=∫u0r(s)λ(s)1−r2(s)ds

then, the map given by

 (3.18) ϕ(y,u)=(r(u)y,√1−r(u)2cos(θ(u)),√1−r(u)2sin(θ(u)))

is an immersion with constant mean curvature .

2. If for some positive integer , then, the image of the immersion is an embedded compact hypersurface in . In general, we have that if for a pair of integers, then, the image of the immersion is a compact hypersurface in .

3. Let be an integer greater than , and let be a connected compact hypersurface with two principal curvatures with multiplicity , and with multiplicity 1. If is positive and the mean curvature is non negative and constant, then, up to a rigid motion of the sphere, can be written as an immersion of the form (3.18). Moreover, contains in its group of isometries the group , is the positive integer such that , with and relative primes.

###### Proof.

Defining and as before we have that

 ϕ(y,u)=r(u)(y,0,0)+√1−r(u)2B2(u)

A direct verification shows that,

 ∂ϕ∂u=r′(y,0,0)−rr′√1−r2B2+λr√1−r2B3

We have that and that the tangent space of the immersion at is given by

 Tϕ(y,u)={(v,0,0)+s∂ϕ∂u:⟨v,y⟩=0ands∈R}

A direct verification shows that the map

 ν(y,u)=−r(u)λ(u)(y,0,0)+r2(u)λ(u)√1−r2(u)B2(u)+r′(u)√1−r2(u)B3(u)

satisfies that , and for any with we have that . It then follows that is a Gauss map of the immersion . The fact that the immersion has constant mean curvature follows because for any unit vector in perpendicular to , we have that

 β(t)=(rcos(t)y+rsin(t)v,0,0)+√1−r2B2=ϕ(cos(t)y+rsin(t)v,u)

satisfies that , and

 dν(β(t))dt∣∣t=0=dν(rv)=−rλv

Therefore, the tangent vectors of the form are principal directions with principal curvature and multiplicity . Now, since , we have that defines a principal direction, i.e. we must have that is a multiple of . A direct verification shows that if we define by , then,

 ⟨∂ν∂u,y⟩=−λ′r−λr′=(λ−μ)r′−λr′=−μr′=−(nH−(n−1)λ)r′

We also have that , therefore,

 ∂ν∂u=dν(∂ϕ∂u)=−μ∂ϕ∂u=−(nH−(n−1)λ)∂ϕ∂u

It follows that the other principal curvature is . Therefore defines an immersion with constant mean curvature , this proves the first item in the Theorem.

In order to prove the second item, we notice that if for some positive , then, we get that , this last fact makes the image of the immersion compact. This immersion is embedded because the immersion is one to one for values of between and as we can easily check using the fact that whenever , the function is strictly increasing. Recall that under these circumstances and . The prove of the other statement in this item is similar.

Let us prove the next item. For , let us now consider a minimal hypersurface with the properties of the statement. We will use the notation that we used in the preliminaries, in particular the function is defined by the relation , in particular we will assume that , and are chosen as before. By Theorem (2.1) we get that the distribution is completely integrable. Let us fix a point in and let us define the geodesic , and the functions as before and let us denote by the -dimensional integral submanifold of of this distribution that passes through . Let us define the vector field on as before. Recall that . Fixing a value , let us define the maps

 ρu,ζu:Mu→Rn+2byρu(x)=x+w2(x)Cηandζu(x)=ν(x)+λ(x)x

Using the equations in section (3.2) we get that the maps and