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Advances in Nonlinear Analysis

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On the bifurcation of solutions of the Yamabe problem in product manifolds with minimal boundary

Elkin Dario Cárdenas Diaz / Ana Cláudia da Silva Moreira
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  • Instituto de Matemática e Estatística, Universidade de São Paulo, 05508-090 São Paulo, Brazil
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Published Online: 2016-01-19 | DOI: https://doi.org/10.1515/anona-2015-0170

Abstract

In this paper, we study the multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

Keywords: Yamabe problem; bifurcation theory

MSC 2010: 53C20; 58E11; 58J32

1 Introduction

Geometers have been interested in finding canonical metrics on a Riemannian manifold for years. Probably, the best known problem on this topic is the uniformization problem for compact surfaces, which assures that there exists a metric of constant Gaussian curvature in each conformal class. For higher dimensions, a Japanese mathematician named Hidehiko Yamabe proposed the following question, that later came to be known as the Yamabe problem:

Let (M,g0) be a compact Riemannian manifold (without boundary) of dimension m3. Is there a metric g, conformal to g0, having constant scalar curvature?

In 1960, Yamabe [17] published an article with a proof of the statement. But in 1968, Trudinger [16] found an error in Yamabe’s proof and gave a complete argument for the existence of a solution to the problem for non-positive scalar curvature. Only in 1989, the combined work of Aubin [2], Trudinger [16], and Schoen [13, 14] lead to a complete proof of the existence of a solution for the Yamabe problem in its full generality. It is known that the critical points of the Hilbert–Einstein functional F, restricted to the set of metrics on M having unit volume, are the Einstein metrics of unit volume on M, and the critical points of F, restricted to the metrics conformal to g with unit volume, are the constant scalar curvature metrics of unit volume.

From then on, many questions have been raised on the uniqueness and the multiplicity of solutions, on the compactness and the non-compactness of the set of solutions, and so on. Moreover, variations of the problem have been proposed by several authors. Nowadays, a great number of results on the Yamabe problem and its generalizations can be found in the literature. Our starting point originates from an article of de Lima, Piccione, and Zedda [12], where the authors studied the local rigidity and the multiplicity of constant scalar curvature metrics on compact product manifolds using a bifurcation result. Given a family {gs}s[a,b] of solutions of the Yamabe problem on the product manifold, the existence of a bifurcation instant s* gives an entirely new sequence (gn)n>0 of solutions. They proved that if gs* is a degenerate critical point of F, restricted to the conformal class (with unit volume), then s* is a candidate for a bifurcation instant. In fact, except for a finite number of instants s for which gs is degenerate, all the rest are bifurcation instants.

Bifurcation techniques for the Yamabe problem have been used in the case of collapsing Riemannian manifolds and in the case of homogeneous metrics on spheres, see [4, 3]. There are several possible formulations of the Yamabe problem on manifolds (M,g¯) with boundary. Here, we consider the following:

Let (M,g¯) be a compact Riemannian manifold with boundary of dimension m3. The Yamabe problem in (M,g¯) consists in finding a metric g~, conformal to g¯, for which M has constant scalar curvature and M has vanishing mean curvature.

The question of existence and some other aspects of solutions for the above problem have been studied, for instance, in [1, 5, 8, 7, 9]. Writing a metric conformal to g¯ as g~=u4n-2g¯, then g~ is a solution with constant scalar curvature and vanishing mean curvature of the Yamabe problem on manifolds with boundary if and only if the function u is a solution of the Neumann problem

{4(m-1)m-2Δg¯u+Rg¯u-Kum+2m-2=0in M,(m-1)ηg¯u+m-22Hg¯u=0on M,

where K is constant, Δg¯ is the Laplacian operator of g¯, Rg¯ is the scalar curvature of g¯, Hg¯ is the mean curvature of M relative to g¯, and ηg¯ is the g¯-unit normal field along M pointing inside of M. Solving the above problem is equivalent to finding critical points for the Hilbert–Einstein functional F:k,α(M), defined by

F(g)=MRgωg,

restricted to the set of metrics in the conformal class of g¯ having unit volume.

The aim of this paper is to determine the multiplicity of solutions of the Yamabe problem in manifolds obtained as a product of a compact manifold (without boundary) and a compact manifold with boundary, using bifurcation theory. We consider a setup similar to the problem studied in [12]. We consider here the case of manifolds with boundary, which introduces new elements with respect to the theory developed in [12]. Given a compact Riemannian manifold (M1,g(1)) with M1= and a compact Riemannian manifold (M2,g¯(2)) with minimal boundary, both having constant scalar curvature, consider the product manifold M=M1×M2, whose boundary is given by M=M1×M2. Let m1 and m2 be the dimensions of M1 and M2, respectively, and assume that dim(M)=m=m1+m23. For each s(0,+), define a family g¯s=g(1)sg¯(2) of metrics on M. Then, (M,g¯s) has constant scalar curvature and the mean curvature of M relative to g¯s vanishes for all s>0.

The main difficulty faced in the transition from the case without boundary to the case with boundary was to find a result on uniqueness (to ensure rigidity) that can be well adapted to the case with boundary. Several results on uniqueness are known. Escobar [6], for instance, proved that if there is a metric g~ with constant scalar curvature and vanishing mean curvature in the 𝒞k,α-conformal class of an Einstein metric g¯, then g~ is Einstein. Moreover, if g¯ is not conformal to the round metric, then g~ is unique, except for homothety. Here, we adapt the result proved by de Lima, Piccione, and Zedda [11], which states that every 𝒞k,α-conformal class of metrics of unit volume has at most one metric of unit volume and constant scalar curvature in a neighborhood of a non-degenerate metric with constant scalar curvature.

The main results in this paper, Theorem 4.4 and Theorem 4.5, state that when the scalar curvature of each factor (Mi,g(i)), i=1,2, is positive, then there are two sequences (one tending to zero and the other to +) of instants s*(0,+) such that the corresponding metric g¯s* is the limit of a sequence of metrics of distinct solutions of the Yamabe problem. Precise definitions of these bifurcating branches of solutions will be given below. In particular, we have a multiplicity of solutions of the Yamabe problem on the normalized conformal classes of these new metrics. For all other values of s, the family is locally rigid, which means that, locally, the metrics of the family are the unique solutions of the Yamabe problem, up to homotheties.

The paper is organized as follows. In Section 2, we study the variational framework used for the bifurcation result. This framework is given by considering the Hilbert–Einstein functional restricted to the normalized conformal classes. In Section 3, we discuss some results about rigidity and bifurcation that are used to obtain the conclusions of this work. Finally, in Section 4, we verify that essentially the same results obtained by de Lima, Piccione, and Zedda [12] also remain valid in the case of manifolds with boundary.

2 General settings

2.1 Manifolds and conformal classes

Let (M,g¯) be an m-dimensional oriented compact Riemannian manifold with boundary M for m3. As the metric g¯ induces inner products and norms in all spaces of tensors on M and the Levi-Civita connection ¯ of g¯ induces a connection in all vector spaces of tensor fields on M, the space Γk(T*MT*M) of 𝒞k-sections of the vector bundle T*MT*M of symmetric (0,2)-tensors of class 𝒞k on M is a Banach space with the norm

τ𝒞k=maxj=0,,k[maxpM¯(j)τ(p)g¯]

and, therefore, it is a Banach manifold.

Given k3 and α(0,1], denote by k,α(M) the set of all Riemannian metrics of class 𝒞k,α on M, in the sense that the coefficients of the metrics are 𝒞k,α-functions on M. The set k,α(M) is an open cone of Γk,α(T*MT*M), so it is a Banach manifold itself, and Tg¯k,α(M)=Γk,α(T*MT*M) for a metric g¯k,α(M).

Consider the open subset

𝒞+k,α(M)={ϕ𝒞k,α(M):ϕ>0}

of the Banach space 𝒞k,α(M). Now, for each g¯k,α(M), define the 𝒞k,α-conformal class of g¯ by

[g¯]:={ϕg¯:ϕ𝒞+k,α(M)}.

Proposition 2.1.

The Ck,α-conformal class of a Riemannian metric g¯Mk,α(M) is an embedded submanifold of Mk,α(M).

Proof.

Given g¯k,α(M), consider the injective map

g¯:𝒞+k,α(M)k,α(M),ϕϕg¯,

whose differential (dg¯)ϕ:𝒞k,α(M)Γk,α(T*MT*M) is injective and has a left-inverse given by the linear bounded operator

𝒥g¯:Γk,α(T*MT*M)𝒞k,α(M),h1mtrg¯h.

Consequently,1 the image Imdg¯ has a closed complement in Γk,α(T*MT*M) and Img¯=[g¯] is an embedded submanifold of k,α(M). ∎

In particular, [g¯] is a Banach manifold with differential structure induced by 𝒞k,α(M) and its tangent space is

Tg¯[g¯]={ψg¯:ψ𝒞k,α(M)},

which can be identified with 𝒞k,α(M).

For each g¯k,α(M), denote by Ricg¯ the Ricci curvature and by Rg¯ the scalar curvature with respect to g¯. Let ηg¯ be the unit (inward) vector field normal to M and denote by Hg¯ the mean curvature of the boundary, induced by g¯. These are 𝒞k-2,α-functions, Ricg¯ and Rg¯ defined on M and Hg¯ defined on M. Let ωg¯ be the volume form on M with respect to g¯ and let σg¯ be the volume form induced on M.

The volume function on k,α(M) is defined as

𝒱(g¯):=Mωg¯.

Observe that 𝒱 is smooth and that its differential is given by

(d𝒱)g¯(h)=12Mtrg¯(h)ωg¯(2.1)

for each g¯k,α(M) and hTg¯k,α(M).

We define

k,α(M)1:={g¯k,α(M):𝒱(g¯)=1}

to be the subset of unit volume metrics in k,α(M).

Proposition 2.2.

k,α(M)1 is a smooth embedded submanifold of Mk,α(M).

Proof.

Consider the smooth volume function 𝒱 defined on k,α(M). For g¯k,α(M)1, set h=g¯. Then, from (2.1) we get

(d𝒱)g¯(g¯)=12Mtrg¯(g¯)ωg¯=m2𝒱(g¯)=m20,

that is, the differential is surjective. So, k,α(M)1=𝒱-1(1) is the inverse image of a regular value. Moreover, the kernel (the tangent space Tg¯k,α(M)1)

ker{(d𝒱)g¯}={hTg¯k,α(M):Mtrg¯(h)ωg¯=0}

has a closed complementary space, so the result follows. ∎

Observe that if g¯k,α(M)1, the conformal metric ϕg¯, for some ϕ𝒞+k,α(M), is not in k,α(M)1, in general. Indeed,

𝒱(ϕg¯)=Mωϕg¯=Mϕm2ωg¯.

So, for each g¯k,α(M)1, we define

[g¯]1={ϕg¯:ϕ𝒞+k,α(M),Mϕm2ωg¯=1},

which is an embedded submanifold of [g¯]. The proof is similar to that of Proposition 2.2.

It is proved in [11] that [g¯]1 is an embedded submanifold of k,α(M)1.

Proposition 2.3.

The Ck,α-conformal class of metrics [g¯]1 of unit volume is an embedded submanifold of Mk,α(M)1. Moreover,

Tg¯[g¯]1={ψg¯:ψ𝒞k,α(M),Mψωg¯=0}.

Proof.

Since we know that 𝒞+k,α(M) is a Banach manifold, we can define a smooth function 𝒱g¯:𝒞+k,α(M) for each g¯k,α(M)1 by

𝒱g¯(ϕ):=𝒱(ϕg¯)

with differential given by

(d𝒱g¯)ϕ(ψ)=m2Mϕm2-1ψωg¯

for each ψTϕ𝒞+k,α(M)=𝒞k,α(M).

Let ϕ𝒞+k,α(M) be such that 𝒱g¯(ϕ)=1. Take ψ=ϕ to see that (d𝒱g¯)ϕ(ϕ)=m20, that is, (d𝒱g¯)ϕ is surjective. Moreover, its kernel has complement in 𝒞k,α(M), which implies that 𝒱g¯-1(1) is an embedded submanifold of 𝒞+k,α(M).

Now, for g¯(M)k,α(M)1, define the smooth maps S:𝒱g¯-1(1)k,α(M)1 and T:k,α(M)1𝒞k,α(M) by

S(ϕ)=ϕg¯andT(g~)=1mtrg¯(g~).

Observe that S is an injective immersion and T is a smooth left-inverse for S. Moreover, Im(dS)ϕ has a closed complement. Therefore, [g¯]1=ImS is an embedded submanifold of k,α(M)1.

Finally, if we take ϕ=1 and S(1)=g¯, we have

T1𝒱g¯-1(1)={ψ𝒞k,α(M):Mψωg¯=0}.

Since S is an immersion, we have Tg¯[g¯]1=Im(dS)1. But since (dS)ϕ(ψ)=ψg¯ for all ϕ𝒱g¯-1(1), ψTϕ𝒱g¯-1(1), we obtain the desired expression for the tangent space, which can also be identified with

Tg¯[g¯]1={ψ𝒞k,α(M):Mψωg¯=0}.

Note that [g¯]1=k,α(M)1[g¯]. In [11], it is also proved that k,α(M)1 is transverse to [g¯], so [g¯]1 is an embedded submanifold of k,α(M).

Now, define the 𝒞k,α-normalized conformal class of a metric g¯k,α(M) by

[g¯]0={g~[g¯]:Hg~=0}.

This is a non-empty set. Indeed, a result obtained by Escobar [8] assures that there is at least one metric with vanishing mean curvature in each conformal class, so given a conformal class [g¯], we can assume that Hg¯=0.

Proposition 2.4.

The Ck,α-normalized conformal class of g¯ can be identified with

𝒞+k,α(M)0={ϕ𝒞+k,α(M):ηg¯ϕ=0 on M},

which is a closed subset of Ck,α(M).

Proof.

Let g~=ϕ4m-2g¯[g¯]0. We denote with a tilde all quantities related with g~. Then, η~=ϕ-2m-2ηg¯ and

H~=g~ijg~(η~,~ij)=g~ijg~(η~,Γ~ijrr),

but Γ~ijr=Γijr+2m-2ϕ-1(δjriϕ+δirjϕ-g¯ijrϕ) and g~ij=ϕ-4m-2g¯ij. Then,

H~=ϕ-2m-2(Hg¯+2(m-1)m-2ϕ-1ηg¯ϕ)=ϕ-mm-2(ϕHg¯+2(m-1)m-2ηg¯ϕ).

Since Hg¯=0 and ϕ>0, it follows that H~=0 if and only if ηg¯ϕ=0. ∎

We want to show that the 𝒞k,α-normalized conformal class [g¯]0 is a submanifold of the 𝒞k,α-conformal class [g¯]. To this aim, we need the following proposition which is an elementary version of a more general result that can be found in [15, Chapter IV, Theorem 4].

Proposition 2.5.

There exists a continuous linear map

:𝒞k,α(M)𝒞k+1,α(M)

such that, for ξCk,α(M), the following properties are satisfied:

  • (i)

    (ξ) vanishes on M ;

  • (ii)

    η(ξ)=ξ.

Proof.

Choose a finite set of local charts (Ur,φr) on M, r=1,,n, that satisfy the following properties:

  • (i)

    Ur is an open subset of M with UrM for all r;

  • (ii)

    U=r=1nUr is an open neighborhood of M;

  • (iii)

    φr is a diffeomorphism from Ur to m-1×[0,+) carrying UrM onto m-1×{0};

  • (iv)

    (dφr)p(η(p))=xm for all pUrM.

Set U0=MM, so that (Ur)r=0,,n is an open cover of M, and let (ρr)r=0n be a smooth partition of unity subordinated to such a cover. Given ξ𝒞k,α(M), consider, for all r, the function ξr=ξφr-1:m-1, which is of class 𝒞k,α. Let Fξr:m be defined by

Fξr(x)=1xmm-1Q(x)ξr(z)𝑑z,

where x=(x1,,xm-1,xm), Q(x)=i=1m-1[xi-12xm,xi+12xm], z=(z1,,zm-1), and xm0. Note that Fξr(x1,,xm-1,0)=0. A straightforward calculation shows that Fξr𝒞k+1,α(m). Let now r=Fξrφr. Clearly, r𝒞k+1,α(Ur). Finally, define (ξ):M as

(ξ):=r=1nρrr.

It is easy to see that (ξ) satisfies the desired properties. ∎

Proposition 2.6.

The Ck,α-normalized conformal class [g¯]0 is an embedded submanifold of [g¯].

Proof.

Given g¯k,α(M), let ηg¯ be the unit (inward) vector field normal to the boundary M. Define

𝒩g¯:[g¯]𝒞k-1,α(M),ϕg¯ηg¯ϕ.

So, 𝒩g¯-1({0})=[g¯]0 and the differential (d𝒩g¯)ϕg¯:𝒞k,α(M)𝒞k-1,α(M) is given by

(d𝒩g¯)ϕg¯(ψ)=ηg¯ψ

for all ϕg¯[g¯] and ψ𝒞k,α(M). Now, by the last proposition, (d𝒩g¯)ϕg¯ admits a bounded right-inverse for all ϕg¯[g¯]. Therefore, the differential is surjective and its kernel, given by

ker(d𝒩g¯)ϕg¯={ψ𝒞k,α(M):ηg¯ψ=0},

has a closed complement in 𝒞k,α(M). It follows that [g¯]0 is an embedded submanifold of [g¯]. ∎

We can also combine both features of interest in the same conformal class, defining the 𝒞k,α-normalized conformal class consisting of metrics of unit volume as

[g¯]10={ϕg¯:ϕ𝒞+k,α(M),ηg¯ϕ=0,Mϕm2ωg¯=1}.

This is an embedded submanifold of k,α(M)1 and an embedded submanifold of [g¯]. For instance, we can express [g¯]10 as [g¯]0k,α(M)1. The corresponding tangent space is identified with

Tg¯[g¯]10={ψ𝒞k,α(M)0:Mψωg¯=0}.

2.2 The Hilbert–Einstein functional

Consider the Hilbert–Einstein functional F:k,α(M)1 given by

F(g¯)=MRg¯ωg¯.(2.2)

It is well known that F is a smooth functional over k,α(M) and over [g¯]. Let g¯(t) be a variation of a metric g¯k,α(M) in the direction of hTg¯k,α(M), that is, g¯(0)=g¯ and ddtg¯(t)|t=0=h. Then, we can calculate the first variation of the functional F as

δF(g¯)h:=ddt|t=0F(g¯(t))=MδRg¯(t)ωg¯+Rg¯δωg¯(t),

where the first variation of the scalar curvature is given by

δRg¯(t)=δ(g¯(t)ijRij(t))=δg¯(t)ijRij(t)+g¯(t)ijδRij(t)=-hijRij(t)+gijδRij(t)

with Rij(t) denoting the coordinates of the Ricci tensor of the metric g¯(t). We have

gijδRij(t)=¯i(¯jhij-¯i(g¯lmhlm))

and

δωg¯(t)=12g¯ijhijωg¯.

Then, we have

δF(g¯)h=M(-hijRij+12g¯ijhijRg¯)ωg¯+M¯i(¯jhij-¯i(g¯lmhlm))ωg¯

and, by the divergence theorem,

δF(g¯)h=-M(hijRij-12g¯ijhijRg¯)ωg¯-M(¯jhij-¯i(g¯lmhlm))ηiσg¯,

where ηi denote the coordinates, with respect to g¯, of the inward unit normal vector field on the boundary. This expression can be written in a concise form as

δF(g¯)h=-MRicg¯-12Rg¯g¯,hg¯ωg¯-M(¯jhij)ηiσg¯+M¯trg¯h,ηg¯σg¯

and, after some further calculations, we get

δF(g¯)h=-MRicg¯-12Rg¯g¯,hg¯ωg¯-2M(δHg¯+12IIg¯,h)σg¯.

For compact Riemannian manifolds (without boundary), it is well known that the critical points of F on k(M)1 are the Einstein metrics of unit volume on M and that if F is restricted to the 𝒞k,α-conformal classes [g¯]1 of unit volume, then the critical points are those metrics conformal to g¯ which have unit volume and constant scalar curvature. Since we are dealing with manifolds with boundary, we are interested in critical points for the restriction to the 𝒞k,α-normalized conformal classes.

The critical points of F on [g¯]10 are those metrics conformal to g¯ which have unit volume, constant scalar curvature, and vanishing mean curvature. Indeed, when we take h=ψg¯Tg¯[g¯]10, the first variation becomes

δF(g¯)ψg¯=-MRicg¯-12Rg¯g¯,ψg¯g¯ωg¯=-M(ψRg¯-12mψRg¯)ωg¯=m-22MψRg¯ωg¯.

So, by the fundamental lemma of the calculus of variations, we have δF(g¯)ψg¯=0 for all ψ𝒞k,α(M)0 with null integral if and only if Rg¯ is constant.

If g¯k,α(M)1 is a critical point of F on [g¯]10, then the second variation of F is given by the quadratic form (see [10])

δ2F(g¯)(ψ)=m-22M((m-1)Δg¯ψ-Rg¯ψ)ψωg¯,

where ψ𝒞k,α(M)0 and has vanishing integral, and g¯ is non-degenerate if Rg¯=0 or if Rg¯m-1 is not an eigenvalue of Δg¯ with Neumann boundary conditions.

In fact, note that (m-1)Δg-Rg, as an operator from 𝒞k,α(M) to 𝒞k-2,α, is Fredholm of index zero. Note also that this operator carries the subspace

{ψ𝒞k,α(M):ηg¯ψ=0,Mψ=0}

into the subspace of 𝒞k-2,α consisting of functions with vanishing integral. It follows that the operator

(m-1)Δg¯-Rg¯:Tg¯[g¯]1𝒞k-2,α(M)

is Fredholm of index zero. So, the quadratic form δ2F(g¯)(ψg¯,ψg¯) is non-degenerate if and only if the kernel ker((m-1)Δg-Rg)={0}. Indeed, the kernel is non-trivial if and only if Rg¯m-1 is a non-zero eigenvalue of Δg¯.

3 Local rigidity and bifurcation of solutions of the Yamabe problem

3.1 Local rigidity

In this section, we present a result obtained in [11], but written in a slightly different way to better suit our context. We refer to [11, Proposition 3 and Corollary 4] for details. The proofs are essentially identical. We begin with the following definition.

Definition 3.1.

Let g¯k,α(M)1 have constant scalar curvature Rg¯ in M. We say that g¯ is non-degenerate if either Rg¯=0 or if Rg¯m-1 is not an eigenvalue of Δg¯ with Neumann boundary condition ηg¯f=0. In other words, Rg¯m-1 is not a solution of the eigenvalue problem

{Δg¯f=λfin M,ηg¯f=0on M.(3.1)

Proposition 3.2.

Let g¯*Mk,α(M)1 be a non-degenerate constant scalar curvature metric. Then, there exists an open neighborhood U of g¯* in Mk,α(M)1 such that the set

S={g¯U:Rg¯ is constant}

is a smooth embedded submanifold of Mk,α(M)1 which is strongly transverse to the Ck,α-normalized conformal classes.

Proof.

The proof is a direct application of [11, Proposition 1]. ∎

Corollary 3.3.

Let g¯*Mk,α(M)1 be a non-degenerate metric on M with constant scalar curvature and vanishing mean curvature. Then, there is an open neighborhood U of g¯* in Mk,α(M)1 such that every Ck,α-normalized conformal class of metrics in Mk,α(M)1 has at most one metric of constant scalar curvature and unit volume in U.

Proof.

The fact that the manifold S is transverse to the normalized conformal class guarantees the local uniqueness of intersections. ∎

3.2 Bifurcation of solutions

Let M be an m-dimensional compact Riemannian manifold with boundary for m3. Define a continuous path

[a,b]k,α(M)1,k3,sg¯s,

of Riemannian metrics on M with constant scalar curvature Rg¯s and vanishing mean curvature Hg¯s for all s[a,b].

Definition 3.4.

An instant s*[a,b] is called a bifurcation instant for the family {g¯s}s[a,b] if there exists a sequence (sn)n1[a,b] and a sequence (g¯n)n1k,α(M)1 of Riemannian metrics on M satisfying the following:

  • (i)

    limnsn=s* and limng¯n=g¯s*k,α(M)1;

  • (ii)

    g¯n[g¯sn] but g¯ng¯sn for all n1;

  • (iii)

    g¯n has constant scalar curvature and vanishing mean curvature for all n1.

If s*[a,b] is not a bifurcation instant, then the family {g¯s}s[a,b] is called locally rigid at s*.

An instant s[a,b] for which Rg¯sm-1 is a non-vanishing solution of the eigenvalue problem (3.1) is called a degeneracy instant for the family {gs}s[a,b].

Theorem 3.5.

Let M be an m-dimensional compact manifold with boundary M for m3 and let

[a,b]k,α(M)1,k3,sg¯s,

be a C1-path of Riemannian metrics on M having constant scalar curvature Rg¯s and vanishing mean curvature Hg¯s. Denote by ns the number of eigenvalues of the Laplace–Beltrami operator Δg¯s with Neumann boundary condition (counted with multiplicity) that are less than Rg¯sm-1. Assume that if s=a or s=b, then Rg¯sm-1=0 or it is not an eigenvalue of Δg¯s, and nanb. Then, there exists a bifurcation instant s*(a,b) for the family {g¯s}s[a,b].

Proof.

The result follows from the non-equivariant bifurcation theorem [12, Theorem A.2]. ∎

Remark 3.6.

Given a Riemannian manifold (M,g¯) with minimal boundary M, it is important to stress that, for all s+, the manifold (M,sg¯) also has minimal boundary, since Hsg¯=1sHg¯. Moreover, we have Δsg¯=1sΔg¯, Rsg¯=1sRg¯, and ηsg¯=1sη. This means that the spectrum of the operator Δg¯-Rg¯m-1 with Neumann boundary condition is invariant by homothety of the metric. On the other hand, ωsg¯=sm2ωg¯. When necessary, we will normalize metrics to have unit volume without change in the spectral theory of the operator Δg¯-Rg¯m-1 with Neumann boundary condition.

4 Bifurcation of solutions of the Yamabe problem in product manifolds

Let (M1,g(1)) be a compact Riemannian manifold with M1= and constant scalar curvature, and let (M2,g¯(2)) be a compact Riemannian manifold with minimal boundary and constant scalar curvature. Consider the product manifold M=M1×M2 whose boundary is given by M=M1×M2. Let m1 and m2 be the dimensions of M1 and M2, respectively, and assume that dim(M)=m=m1+m23. For each s(0,+), define a family g¯s=g(1)sg¯(2) of metrics on M. Then, {g¯s}sk,α(M).

The following statements, briefly justified, are valid.

  • (i)

    (M,g¯s) has constant scalar curvature for all s>0 and its scalar curvature is given by

    Rg¯s=Rg(1)+Rsg¯(2)=Rg(1)+1sRg¯(2).

  • (ii)

    Since we can identify the tangent space of the product manifold T(p,q)M with the direct sum TpM1TqM2 for pM1 and qM2, the interior vector field ηs normal to M can be written as

    ηs=0+1sη2,

    where η2 is the interior vector field normal to M2.

  • (iii)

    The mean curvature of M is zero, since we have

    Hg¯s=Hsg¯(2)=1sHg¯(2).

  • (iv)

    The Laplace–Beltrami operator with respect to g¯s is given by

    Δg¯s=(Δg(1)I)+1s(IΔg¯(2)).

  • (v)

    Consider the family {g¯s}s>0. For the purpose of studying bifurcation instants, this causes no loss of generality. In fact, s* is a bifurcation instant for the family {g(1)sg¯(2)}s>0 if and only if s* is a bifurcation instant for the family {1sg(1)g¯(2)}s>0 on M. The same is valid for degeneracy instants.

Now, considering the remark at the end of the previous section, {g¯s}s>0 is a family of critical points of the functional

F:k,α(M)1,

restricted to [g¯]10, so it is a family of solutions with minimal boundary of the Yamabe problem in product manifolds.

In order to investigate the existence of bifurcation instants for the family {g¯}s>0, we are interested in studying the spectrum of the operator

𝒥s=Δg¯s-Rg¯sm-1,

whose domain is

{ψ𝒞k,α(M)0:Mψωg¯s=0}.

Denote by 0=ρ0(1)<ρ1(1)<ρ2(1)< the sequence of all distinct eigenvalues of Δg(1) with geometric multiplicity μi(1), i0, and by 0=ρ0(2)<ρ1(2)<ρ2(2)< the sequence of all distinct eigenvalues of Δg¯(2) subject to Neumann boundary condition, that is,

{Δg¯(2)f(2)=ρj(2)f(2)in M,η2f(2)=0on M,(4.1)

where j0 and μj(2) is the geometric multiplicity of ρj(2), j0. Then, the spectrum of 𝒥s is given by

Σ(𝒥s)={σi,j:i,j0,i+j>0},

where

σi,j(s)=ρi(1)+1sρj(2)-1m-1(Rg(1)+1sRg¯(2))

are the eigenvalues of Js with Neumann boundary condition on M and with geometric multiplicity equal to the product μi(1)μj(2).

We emphasize that the σi,j are not necessarily all distinct.

Definition 4.1.

Let i* and j* be the smallest non-negative integers that satisfy

ρi*(1)Rg(1)m-1andρj*(2)Rg¯(2)m-1.

We say that the pair of metrics (g(1),g¯(2)) is degenerate if equalities hold in both cases, that is, if σi*,j*(s)=0 for all s. In this case, the operator 𝒥s is also called degenerate.

We can state that, if Rg(1)<0 or if Rg¯(2)<0 and Hg¯(2)=0, then (g(1),g¯(2)) is certainly non-degenerate. Observe also that if (g(1),g¯(2)) is degenerate, then zero is an eigenvalue of 𝒥s for all s(0,+), otherwise there is only a discrete countable set S(0,+) of instants s for which the operator 𝒥s is singular. First, we consider the case when both scalar curvatures are positive.

4.1 The case of positive scalar curvature

We are interested in studying the zeros of the function sσi,j(s) as i,j vary. At first glance, we can already draw some conclusions. For instance, if the function σi,j is not identically zero, for fixed i,j, it has at most one zero in (0,+). Let us write σi,j as

σi,j(s)=ρi(1)-Rg(1)m-1+1s(ρj(2)-Rg¯(2)m-1).

Derive

σi,j(s)=-1s2(ρj(2)-Rg¯(2)m-1),

so σi,j(s)=0 if and only if ρj(2)=Rg¯(2)m-1 if and only if Rg¯(2) is a solution of (4.1). If σi,j(s)=0 for some s, then σi,j(s)=0 for all s(0,+). Hence, σ is strictly monotone or constant, so if it is not identically null (constant), it certainly has at most one zero (strictly monotone).

Lemma 4.2.

Assume that the pair (g(1),g¯(2)) is non-degenerate and that Rg(1),Rg¯(2)>0 with Hg¯(2)=0. Then, the functions σi,j(s) satisfy the following properties:

  • (i)

    For all i,j0 , the map sσi,j(s) is strictly monotone in (0,+) except for the maps σi,j* which are constant and equal to ρi(1)-Rg(1)m-1 when ρj*(2)=Rg¯(2)m-1.

  • (ii)

    For ii* and jj* , the map σi,j(s) admits a zero if and only if

    • either j<j* and i>i* , in which case σi,j is strictly increasing,

    • or if j>j* and i<i* , in which case σi,j is strictly decreasing.

  • (iii)

    If ρi*(1)=Rg(1)m-1 , then σi*,j does not have zeros for any j(0,+) . If ρi*(1)>Rg(1)m-1 , then σi*,j has a zero if and only if j<j*.

  • (iv)

    If ρj*(2)=Rg¯(2)m-1 , then σi,j* does not have zeros for any i(0,+) . If ρj*(2)>Rg¯(2)m-1 , then σi,j* has a zero if and only if i<i*.

Proof.

The entire statement follows directly from a straightforward analysis of the expression

σi,j(s)=(ρi(1)-Rg(1)m-1)+1s(ρj(2)-Rg¯(2)m-1).

Corollary 4.3.

If (g(1),g¯(2)) is non-degenerate, then the set S(0,+) of instants s at which Js is singular, is countable and discrete. More precisely, it consists of a strictly decreasing sequence (sn(1))n tending to 0 and a strictly increasing unbounded sequence (sn(2))n. For all other values of s, Js is an isomorphism and, in particular, the family {g¯s}s>0 is locally rigid at these instants.

Proof.

By Lemma 4.2, each function σi,j has at most one zero, thus there is only a countable number of degeneracy instants for 𝒥s. Let sij be the zero of σij. Then,

0<sij=-ρj(2)-Rg¯(2)m-1ρi(1)-Rg(1)m-1.

Now, we study the behavior of these zeros in the following two cases:

  • If j>j* and i<i*, then

    sij=ρj(2)-Rg¯(2)m-1Rg(1)m-1-ρi(1)(ρj(2)-Rg¯(2)m-1)1Rg(1)m-1-ρi*-1(1)+

    as j+.

  • If i>i* and j<j*, then

    sij=Rg¯(2)m-1-ρj(2)ρi(1)-Rg(1)m-11ρi(1)-Rg(1)m-1(Rg¯(2)m-1-ρ1(2))0

    as i+.

Therefore, all the zeros of the eigenvalues σi,j accumulate only at 0 and at +. Let s*(0,+)S. Then, 𝒥s* is an isomorphism, that is, 0Σ(Js*) or, equivalently, Rg¯s*m-1 is not an eigenvalue of Δg¯s*. So, g¯s*{g¯s}s>0 is a non-degenerate metric. It follows from Proposition 3.2 that the family {g¯s}s>0 is locally rigid at s*. ∎

Note that Rg¯ is obviously different from zero, since we are considering only positive scalar curvature.

Theorem 4.4.

Let (M1,g(1)) be a compact Riemannian manifold with positive constant scalar curvature and let (M2,g¯(2)) be a compact manifold with boundary and with positive constant scalar curvature and minimal boundary M2. Assume that the pair (g(1),g¯(2)) is non-degenerate. For all s(0,+), let g¯s=g(1)sg¯(2) be the metric on the product manifold with boundary M=M1×M2. Then, there exists a sequence tending to 0 and a sequence tending to + consisting of bifurcation instants for the family {g¯s}s>0.

Proof.

We prove that bifurcation instants consist of subsequences of the two sequences of instants where 𝒥s is singular, whose existence was proved above.

With the same notation used in Corollary 4.3, let n0>0 be such that sn(1)<s1(2) and s1(1)<sn(2) for all n>n0. Then, there is ε>0 for all n>n0 such that the operator 𝒥() is an isomorphism on the intervals [sn(1)-ε,sn(1)+ε] and [sn(2)-ε,sn(2)+ε], except for the instants sn(1) and sn(2) themselves.

As the zeros of the increasing eigenvalue functions accumulate at 0 and the zeros of the decreasing eigenvalue functions accumulate at +, if σp,q is a non-increasing eigenvalue function, for all s(0,sn(1)+ε], n>n0, we have σp,q(s)0. So, σp,q(sn(1)-ε)<0 if and only if σp,q(sn(1)+ε)<0. On the other hand, if we consider an increasing eigenvalue function σi,j, for all n>n0, we have σi,j(sn(1))=0, σi,j(sn(1)-ε)<0, σi,j(sn(1)+ε)>0, and the fact that sn(1)<s1(2) ensures that there is no decreasing function that vanishes at sn(1). Hence, we can surely conclude that nsn(1)-εnsn(1)+ε. By Theorem 3.5, it follows that the subsequence (sn(1))n>n0 is the sought after sequence of bifurcation instants tending to 0.

Now, analyzing the non-decreasing eigenvalue function and the decreasing eigenvalue function in a similar way, we obtain nsn(2)-εnsn(2)+ε and we can apply Theorem 3.5 to conclude that the subsequence (sn(2))n>n0 is the sought after sequence of bifurcation instants tending to +. ∎

Note that the case of degenerate pairs cannot be treated with Theorem 3.5, because in this case the zero is present in the spectrum of the operator 𝒥s for all s(0,+) and thus the hypotheses of the theorem are never satisfied. Another interesting observation is that we do not know, by Theorem 4.4, if bifurcation occurs at degeneracy instants s between s1(1) and s1(2).

4.2 The case of non-positive scalar curvature

Now, consider the family {g¯s}s>0 on the product manifold M1×M2 in the case when one or both scalar curvatures Rg(1) or Rg¯(2) are non-positive, maintaining the Neumann boundary condition on M2. Observe that if Rg(1) and Rg¯(2) are both non-positive, then the pair (g(1),g¯(2)) is non-degenerate.

If Rg(1)0 and Rg¯(2)>0, then the pair (g(1),g¯(2)) is degenerate if and only if Rg(1)=0 and ρj*(2)=Rg¯(2)m-1 as ρi(1)0 for all i{0,1,2,}.

Theorem 4.5.

The following statements are true.

  • (i)

    If Rg(1)0 and Rg¯(2)0 , then the family {g¯s}s>0 has no degeneracy instants and thus it is locally rigid at every s(0,+).

  • (ii)

    If Rg(1)0, Rg¯(2)>0 , and the pair (g(1),g¯(2)) is non-degenerate, then the set of degeneracy instants for 𝒥s is a strictly decreasing sequence (sn)n that converges to 0 as n . Moreover, every degeneracy instant is a bifurcation instant for the family {g¯s}s>0

  • (iii)

    Symmetrically, if Rg(1)>0, Rg¯(2)0 , and the pair (g(1),g¯(2)) is non-degenerate, then the set of degeneracy instants for 𝒥s is a strictly increasing unbounded sequence (sn)n and every degeneracy instant is a bifurcation instant for the family {g¯s}s>0.

Proof.

The result follows from an analysis of the functions

σi,j(s)=ρi(1)+1sρj(2)-1m-1(Rg(1)+1sRg¯(2)).

For (i), it is straightforward to see that σi,j(s)>0 for all i,j=0,1,2, with i+j>0, so 𝒥s has no vanishing eigenvalues and the result follows.

For (ii), the functions σi,j admit a zero only if i0 and j<j*. If si,j denotes the zero of such a function, then we have

0<si,j=|ρj(2)-Rg¯(2)m-1ρi(1)-Rg(1)m-1|Rg¯(2)ρi(1)-Rg(1)m-10

as i+. Hence, there is a decreasing sequence (sn)n of degeneracy instants that accumulates at 0. Since (sn)n accumulates only at 0, for each n, there exists ε>0 such that the interval [sn-ε,sn+ε] contains only sn as a degeneracy instant. Arguing as in the proof of Theorem 4.4, we have nsn-εnsn+ε. The conclusion follows from Theorem 3.5.

Proceeding in a similar way, for (iii), the functions σi,j admit a zero only if i<i* and j0, and we have an unbounded increasing sequence of degeneracy instants, each of which is a bifurcation instant for the family {g¯s}s>0. ∎

Acknowledgements

The authors gratefully acknowledge the supervision of Professor Paolo Piccione throughout the development of this work.

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Footnotes

  • 1

    Let T:EF be a linear bounded operator between Banach spaces. If T admits a continuous right-inverse S:FE, then kerT admits a closed complement. Moreover, this complement is ImS. 

About the article

Received: 2015-12-09

Accepted: 2015-12-12

Published Online: 2016-01-19

Published in Print: 2018-02-01


The first author is partially sponsored by CAPES-Brazil via the Instituto de Matemática e Estatística, Universidade de São Paulo.


Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 1, Pages 1–14, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2015-0170.

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