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 be a compact Riemannian manifold (without boundary) of dimension . Is there a metric g, conformal to , having constant scalar curvature?
In 1960, Yamabe  published an article with a proof of the statement. But in 1968, Trudinger  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 , Trudinger , 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 , 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 of solutions of the Yamabe problem on the product manifold, the existence of a bifurcation instant gives an entirely new sequence of solutions. They proved that if is a degenerate critical point of F, restricted to the conformal class (with unit volume), then is a candidate for a bifurcation instant. In fact, except for a finite number of instants s for which 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 with boundary. Here, we consider the following:
Let be a compact Riemannian manifold with boundary of dimension . The Yamabe problem in consists in finding a metric , conformal to , for which M has constant scalar curvature and 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 as , then 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
where K is constant, is the Laplacian operator of , is the scalar curvature of , is the mean curvature of relative to , and is the -unit normal field along pointing inside of M. Solving the above problem is equivalent to finding critical points for the Hilbert–Einstein functional , defined by
restricted to the set of metrics in the conformal class of 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 . We consider here the case of manifolds with boundary, which introduces new elements with respect to the theory developed in . Given a compact Riemannian manifold with and a compact Riemannian manifold with minimal boundary, both having constant scalar curvature, consider the product manifold , whose boundary is given by . Let and be the dimensions of and , respectively, and assume that . For each , define a family of metrics on M. Then, has constant scalar curvature and the mean curvature of relative to vanishes for all .
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 , for instance, proved that if there is a metric with constant scalar curvature and vanishing mean curvature in the -conformal class of an Einstein metric , then is Einstein. Moreover, if is not conformal to the round metric, then is unique, except for homothety. Here, we adapt the result proved by de Lima, Piccione, and Zedda , which states that every -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 , , is positive, then there are two sequences (one tending to zero and the other to ) of instants such that the corresponding metric 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  also remain valid in the case of manifolds with boundary.
2 General settings
2.1 Manifolds and conformal classes
Let be an m-dimensional oriented compact Riemannian manifold with boundary for . As the metric induces inner products and norms in all spaces of tensors on M and the Levi-Civita connection of induces a connection in all vector spaces of tensor fields on M, the space of -sections of the vector bundle of symmetric -tensors of class on M is a Banach space with the norm
and, therefore, it is a Banach manifold.
Given and , denote by the set of all Riemannian metrics of class on M, in the sense that the coefficients of the metrics are -functions on M. The set is an open cone of , so it is a Banach manifold itself, and for a metric .
Consider the open subset
of the Banach space . Now, for each , define the -conformal class of by
The -conformal class of a Riemannian metric is an embedded submanifold of .
Given , consider the injective map
whose differential is injective and has a left-inverse given by the linear bounded operator
Consequently,1 the image has a closed complement in and is an embedded submanifold of . ∎
In particular, is a Banach manifold with differential structure induced by and its tangent space is
which can be identified with .
For each , denote by the Ricci curvature and by the scalar curvature with respect to . Let be the unit (inward) vector field normal to and denote by the mean curvature of the boundary, induced by . These are -functions, and defined on M and defined on . Let be the volume form on M with respect to and let be the volume form induced on .
The volume function on is defined as
Observe that is smooth and that its differential is given by
for each and .
to be the subset of unit volume metrics in .
is a smooth embedded submanifold of .
Consider the smooth volume function defined on . For , set . Then, from (2.1) we get
that is, the differential is surjective. So, is the inverse image of a regular value. Moreover, the kernel (the tangent space )
has a closed complementary space, so the result follows. ∎
Observe that if , the conformal metric , for some , is not in , in general. Indeed,
So, for each , we define
which is an embedded submanifold of . The proof is similar to that of Proposition 2.2.
It is proved in  that is an embedded submanifold of .
The -conformal class of metrics of unit volume is an embedded submanifold of . Moreover,
Since we know that is a Banach manifold, we can define a smooth function for each by
with differential given by
for each .
Let be such that . Take to see that , that is, is surjective. Moreover, its kernel has complement in , which implies that is an embedded submanifold of .
Now, for , define the smooth maps and by
Observe that S is an injective immersion and T is a smooth left-inverse for S. Moreover, has a closed complement. Therefore, is an embedded submanifold of .
Finally, if we take and , we have
Since S is an immersion, we have . But since for all , , we obtain the desired expression for the tangent space, which can also be identified with
Note that . In , it is also proved that is transverse to , so is an embedded submanifold of .
Now, define the -normalized conformal class of a metric by
This is a non-empty set. Indeed, a result obtained by Escobar  assures that there is at least one metric with vanishing mean curvature in each conformal class, so given a conformal class , we can assume that .
The -normalized conformal class of can be identified with
which is a closed subset of .
Let . We denote with a tilde all quantities related with . Then, and
but and . Then,
Since and , it follows that if and only if . ∎
We want to show that the -normalized conformal class is a submanifold of the -conformal class . 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].
There exists a continuous linear map
such that, for , the following properties are satisfied:
vanishes on ;
Choose a finite set of local charts on M, , that satisfy the following properties:
is an open subset of M with for all r;
is an open neighborhood of ;
is a diffeomorphism from to carrying onto ;
for all .
Set , so that is an open cover of M, and let be a smooth partition of unity subordinated to such a cover. Given , consider, for all r, the function , which is of class . Let be defined by
where , , , and . Note that A straightforward calculation shows that . Let now . Clearly, Finally, define as
It is easy to see that satisfies the desired properties. ∎
The -normalized conformal class is an embedded submanifold of .
Given , let be the unit (inward) vector field normal to the boundary . Define
So, and the differential is given by
for all and . Now, by the last proposition, admits a bounded right-inverse for all . Therefore, the differential is surjective and its kernel, given by
has a closed complement in . It follows that is an embedded submanifold of . ∎
We can also combine both features of interest in the same conformal class, defining the -normalized conformal class consisting of metrics of unit volume as
This is an embedded submanifold of and an embedded submanifold of . For instance, we can express as . The corresponding tangent space is identified with
2.2 The Hilbert–Einstein functional
Consider the Hilbert–Einstein functional given by
It is well known that F is a smooth functional over and over . Let be a variation of a metric in the direction of , that is, and . Then, we can calculate the first variation of the functional F as
where the first variation of the scalar curvature is given by
with denoting the coordinates of the Ricci tensor of the metric . We have
Then, we have
and, by the divergence theorem,
where denote the coordinates, with respect to , of the inward unit normal vector field on the boundary. This expression can be written in a concise form as
and, after some further calculations, we get
For compact Riemannian manifolds (without boundary), it is well known that the critical points of F on are the Einstein metrics of unit volume on M and that if F is restricted to the -conformal classes of unit volume, then the critical points are those metrics conformal to 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 -normalized conformal classes.
The critical points of F on are those metrics conformal to which have unit volume, constant scalar curvature, and vanishing mean curvature. Indeed, when we take , the first variation becomes
So, by the fundamental lemma of the calculus of variations, we have for all with null integral if and only if is constant.
If is a critical point of F on , then the second variation of F is given by the quadratic form (see )
where and has vanishing integral, and is non-degenerate if or if is not an eigenvalue of with Neumann boundary conditions.
In fact, note that , as an operator from to , is Fredholm of index zero. Note also that this operator carries the subspace
into the subspace of consisting of functions with vanishing integral. It follows that the operator
is Fredholm of index zero. So, the quadratic form is non-degenerate if and only if the kernel . Indeed, the kernel is non-trivial if and only if is a non-zero eigenvalue of .
3 Local rigidity and bifurcation of solutions of the Yamabe problem
3.1 Local rigidity
In this section, we present a result obtained in , 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.
Let have constant scalar curvature in M. We say that is non-degenerate if either or if is not an eigenvalue of with Neumann boundary condition . In other words, is not a solution of the eigenvalue problem
Let be a non-degenerate constant scalar curvature metric. Then, there exists an open neighborhood U of in such that the set
is a smooth embedded submanifold of which is strongly transverse to the -normalized conformal classes.
The proof is a direct application of [11, Proposition 1]. ∎
Let be a non-degenerate metric on M with constant scalar curvature and vanishing mean curvature. Then, there is an open neighborhood U of in such that every -normalized conformal class of metrics in has at most one metric of constant scalar curvature and unit volume in U.
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 . Define a continuous path
of Riemannian metrics on M with constant scalar curvature and vanishing mean curvature for all .
An instant is called a bifurcation instant for the family if there exists a sequence and a sequence of Riemannian metrics on M satisfying the following:
but for all ;
has constant scalar curvature and vanishing mean curvature for all .
If is not a bifurcation instant, then the family is called locally rigid at .
An instant for which is a non-vanishing solution of the eigenvalue problem (3.1) is called a degeneracy instant for the family .
Let M be an m-dimensional compact manifold with boundary for and let
be a -path of Riemannian metrics on M having constant scalar curvature and vanishing mean curvature . Denote by the number of eigenvalues of the Laplace–Beltrami operator with Neumann boundary condition (counted with multiplicity) that are less than . Assume that if or , then or it is not an eigenvalue of , and . Then, there exists a bifurcation instant for the family .
The result follows from the non-equivariant bifurcation theorem [12, Theorem A.2]. ∎
Given a Riemannian manifold with minimal boundary , it is important to stress that, for all , the manifold also has minimal boundary, since . Moreover, we have , , and . This means that the spectrum of the operator with Neumann boundary condition is invariant by homothety of the metric. On the other hand, . When necessary, we will normalize metrics to have unit volume without change in the spectral theory of the operator with Neumann boundary condition.
4 Bifurcation of solutions of the Yamabe problem in product manifolds
Let be a compact Riemannian manifold with and constant scalar curvature, and let be a compact Riemannian manifold with minimal boundary and constant scalar curvature. Consider the product manifold whose boundary is given by . Let and be the dimensions of and , respectively, and assume that . For each , define a family of metrics on M. Then, .
The following statements, briefly justified, are valid.
has constant scalar curvature for all and its scalar curvature is given by
Since we can identify the tangent space of the product manifold with the direct sum for and , the interior vector field normal to can be written as
where is the interior vector field normal to .
The mean curvature of is zero, since we have
The Laplace–Beltrami operator with respect to is given by
Consider the family . For the purpose of studying bifurcation instants, this causes no loss of generality. In fact, is a bifurcation instant for the family if and only if is a bifurcation instant for the family on M. The same is valid for degeneracy instants.
Now, considering the remark at the end of the previous section, is a family of critical points of the functional
restricted to , 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 , we are interested in studying the spectrum of the operator
whose domain is
Denote by the sequence of all distinct eigenvalues of with geometric multiplicity , , and by the sequence of all distinct eigenvalues of subject to Neumann boundary condition, that is,
where and is the geometric multiplicity of , . Then, the spectrum of is given by
are the eigenvalues of with Neumann boundary condition on and with geometric multiplicity equal to the product .
We emphasize that the are not necessarily all distinct.
Let and be the smallest non-negative integers that satisfy
We say that the pair of metrics is degenerate if equalities hold in both cases, that is, if for all s. In this case, the operator is also called degenerate.
We can state that, if or if and , then is certainly non-degenerate. Observe also that if is degenerate, then zero is an eigenvalue of for all , otherwise there is only a discrete countable set of instants s for which the operator 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 as vary. At first glance, we can already draw some conclusions. For instance, if the function is not identically zero, for fixed , it has at most one zero in . Let us write as
so if and only if if and only if is a solution of (4.1). If for some s, then for all . Hence, σ is strictly monotone or constant, so if it is not identically null (constant), it certainly has at most one zero (strictly monotone).
Assume that the pair is non-degenerate and that with . Then, the functions satisfy the following properties:
For all , the map is strictly monotone in except for the maps which are constant and equal to when .
For and , the map admits a zero if and only if
either and , in which case is strictly increasing,
or if and , in which case is strictly decreasing.
If , then does not have zeros for any . If , then has a zero if and only if .
If , then does not have zeros for any . If , then has a zero if and only if .
The entire statement follows directly from a straightforward analysis of the expression
If is non-degenerate, then the set of instants s at which is singular, is countable and discrete. More precisely, it consists of a strictly decreasing sequence tending to 0 and a strictly increasing unbounded sequence . For all other values of s, is an isomorphism and, in particular, the family is locally rigid at these instants.
By Lemma 4.2, each function has at most one zero, thus there is only a countable number of degeneracy instants for . Let be the zero of . Then,
Now, we study the behavior of these zeros in the following two cases:
If and , then
If and , then
Therefore, all the zeros of the eigenvalues accumulate only at 0 and at . Let . Then, is an isomorphism, that is, or, equivalently, is not an eigenvalue of . So, is a non-degenerate metric. It follows from Proposition 3.2 that the family is locally rigid at . ∎
Note that is obviously different from zero, since we are considering only positive scalar curvature.
Let be a compact Riemannian manifold with positive constant scalar curvature and let be a compact manifold with boundary and with positive constant scalar curvature and minimal boundary . Assume that the pair is non-degenerate. For all , let be the metric on the product manifold with boundary . Then, there exists a sequence tending to 0 and a sequence tending to consisting of bifurcation instants for the family .
We prove that bifurcation instants consist of subsequences of the two sequences of instants where is singular, whose existence was proved above.
With the same notation used in Corollary 4.3, let be such that and for all . Then, there is for all such that the operator is an isomorphism on the intervals and , except for the instants and themselves.
As the zeros of the increasing eigenvalue functions accumulate at 0 and the zeros of the decreasing eigenvalue functions accumulate at , if is a non-increasing eigenvalue function, for all , , we have . So, if and only if . On the other hand, if we consider an increasing eigenvalue function , for all , we have , , , and the fact that ensures that there is no decreasing function that vanishes at . Hence, we can surely conclude that . By Theorem 3.5, it follows that the subsequence 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 and we can apply Theorem 3.5 to conclude that the subsequence 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 for all 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 and .
4.2 The case of non-positive scalar curvature
Now, consider the family on the product manifold in the case when one or both scalar curvatures or are non-positive, maintaining the Neumann boundary condition on . Observe that if and are both non-positive, then the pair is non-degenerate.
If and , then the pair is degenerate if and only if and as for all .
The following statements are true.
If and , then the family has no degeneracy instants and thus it is locally rigid at every .
If , , and the pair is non-degenerate, then the set of degeneracy instants for is a strictly decreasing sequence that converges to 0 as . Moreover, every degeneracy instant is a bifurcation instant for the family
Symmetrically, if , , and the pair is non-degenerate, then the set of degeneracy instants for is a strictly increasing unbounded sequence and every degeneracy instant is a bifurcation instant for the family .
The result follows from an analysis of the functions
For (i), it is straightforward to see that for all with , so has no vanishing eigenvalues and the result follows.
For (ii), the functions admit a zero only if and . If denotes the zero of such a function, then we have
as . Hence, there is a decreasing sequence of degeneracy instants that accumulates at 0. Since accumulates only at 0, for each , there exists such that the interval contains only as a degeneracy instant. Arguing as in the proof of Theorem 4.4, we have . The conclusion follows from Theorem 3.5.
Proceeding in a similar way, for (iii), the functions admit a zero only if and , and we have an unbounded increasing sequence of degeneracy instants, each of which is a bifurcation instant for the family . ∎
The authors gratefully acknowledge the supervision of Professor Paolo Piccione throughout the development of this work.
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About the article
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.