About curvature, conformal metrics and warped products
ABOUT CURVATURE, CONFORMAL METRICS AND
WARPED PRODUCTS
FERNANDO DOBARRO
BÜLENT ÜNAL
Abstract. We consider the curvature of a family of warped prod-
ucts of two pseduo-Riemannian manifolds (B, gB) and (F, gF ) furnished
with metrics of the form c2gB ⊕ w
2gF and, in particular, of the type
gB ⊕w
gF , where c, w : B → (0,∞) are smooth functions and µ is a
real parameter. We obtain suitable expressions for the Ricci tensor and
scalar curvature of such products that allow us to establish results about
the existence of Einstein or constant scalar curvature structures in these
categories. If (B, gB) is Riemannian, the latter question involves nonlin-
ear elliptic partial differential equations with concave-convex nonlinear-
ities and singular partial differential equations of the Lichnerowicz-York
type among others.
1. Introduction
The main concern of this paper is the curvature of a special family of
warped pseudo-metrics on product manifolds. We introduce a suitable form
for the relations among the involved curvatures in such metrics and apply
them to the existence and/or construction of Einstein and constant scalar
curvature metrics in this family.
Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Riemannian manifolds
of dimensions m ≥ 1 and k ≥ 0, respectively and also let B × F be the
usual product manifold of B and F . For a given smooth function w ∈
C∞>0(B) = {v ∈ C∞(B) : v(x) > 0, ∀x ∈ B}, the warped product B ×w F =
((B ×w F )m+k, g = gB +w2gF ) was defined by Bishop and O’Neill in [19] in
order to study manifolds of negative curvature.
Date: November 4, 2018.
1991 Mathematics Subject Classification. Primary: 53C21, 53C25, 53C50
Secondary: 35Q75, 53C80, 83E15, 83E30.
Key words and phrases. Warped products, conformal metrics, Ricci curvature, scalar
curvature, semilinear equations, positive solutions, Lichnerowicz-York equation, concave-
convex nonlinearities, Kaluza-Klein theory, string theory.
http://arxiv.org/abs/0704.0595v1
2 FERNANDO DOBARRO & BÜLENT ÜNAL
In this article, we deal with a particular class of warped products, i.e.
when the pseudo-metric in the base is affected by a conformal change. Pre-
cisely, for given smooth functions c, w ∈ C∞>0(B) we will call ((B × F )m+k, g =
c2gB+w
2gF ) as a [c, w]-base conformal warped product (briefly [c, w]-bcwp),
denoted by B ×[c,w] F . We will concentrate our attention on a special sub-
class of this structure, namely when there is a relation between the conformal
factor c and the warping function w of the form c = wµ, where µ is a real
parameter and we will call the [ψµ, ψ]-bcwp as a (ψ, µ)-bcwp. Note that we
generically called the latter case as special base conformal warped products,
briefly sbcwp in [29].
As we will explain in §2, metrics of this type play a relevant role in several
topics of differential geometry and theoretical physics (see also [29]). This
article concerns curvature related questions of these metrics which are of
interest not only in the applications, but also from the points of view of dif-
ferential geometry and the type of the involved nonlinear partial differential
equations (PDE), such as those with concave-convex nonlinearities and the
Lichnerowicz-York equations.
The article is organized in the following way: in §2 after a brief description
of several fields where pseudo-metrics described as above are applied, we
formulate the curvature problems that we deal within the next sections and
give the statements of the main results. In §3, we state Theorems 2.2 and
2.3 in order to express the Ricci tensor and scalar curvature of a (ψ, µ)-bcwp
and sketch their proofs (see [29, Section 3] for detailed computations). In §4
and 5, we establish our main results about the existence of (ψ, µ)-bcwp’s of
constant scalar curvature with compact Riemannian base.
2. Motivations and Main results
As we announced in the introduction, we firstly want to mention some of
the major fields of differential geometry and theoretical physics where base
conformal warped products are applied.
i: In the construction of a large class of non trivial static anti de Sitter
vacuum space-times
• In the Schwarzschild solutions of the Einstein equations (see
[10, 18, 41, 59, 69, 74]).
• In the Riemannian Schwarzschild metric, namely (see [10]).
• In the “generalized Riemannian anti de Sitter T2 black hole
metrics” (see §3.2 of [10] for details).
• In the Bañados-Teitelboim-Zanelli (BTZ) and de Sitter (dS)
black holes (see [1, 15, 16, 28, 45] for details).
Indeed, all of them can be generated by an approach of the fol-
lowing type: let (F2, gF ) be a pseudo-Riemannian manifold and g be
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 3
a pseudo-metric on R+ × R× F2 defined by
(2.1) g =
u2(r)
dr2 ± u2(r)dt2 + r2gF .
After the change of variables s = r2, y =
t, there results ds2 =
4r2dr2 and dy2 =
dt2. Then (2.1) is equivalent to
ds2 ± 4
s)dy2
+ sgF
2 )2(−
2 ))2(−1)ds2 ± (2s
2 ))2dy2
2 )2gF .
(2.2)
Note that roughly speaking, g is a nested application of two (ψ, µ)-
bcwp’s. That is, on R+ × R and taking
(2.3) ψ1(s) = 2s
2 ) and µ1 = −1,
the metric inside the brackets in the last member of (2.2) is a (ψ1, µ1)-
bcwp, while the metric g on (R+ × R)× F2 is a (ψ2, µ2)-bcwp with
(2.4) ψ2(s, y) = s
2 and µ2 = −
In the last section of [29], through the application of Theorems 2.2
and 2.3 below and several standard computations, we generalized
the latter approach to the case of an Einstein fiber (Fk, gF ) with
dimension k ≥ 2.
ii: In the study of the equivariant isometric embeddings of space-time
slices in Minkowski spaces (see [39, 38]).
iii: In the Kaluza-Klein theory (see [76, §7.6, Particle Physics and Ge-
ometry], [60] and [77]) and in the Randall-Sundrum theory [30, 40,
63, 64, 65, 71] with µ as a free parameter. For example, in [46] the
following metric is considered
(2.5) e2A(y)gijdx
idxj + e2B(y)dy2,
with the notation {xi}, i = 0, 1, 2, 3 for the coordinates in the 4-
dimensional space-time and x5 = y for the fifth coordinate on an
extra dimension. In particular, Ito takes the ansatz
(2.6) B = αA,
which corresponds exactly to our sbcwp metrics, considering gB =
dy2, gF = gijdx
idxj , ψ(y) = e
α = eA(y) and µ = α.
iii: In String and Supergravity theories, for instance, in the Maldacena
conjecture about the duality between compactifications of M/string
4 FERNANDO DOBARRO & BÜLENT ÜNAL
theory on various Anti-de Sitter space-times and various confor-
mal field theories (see [55, 62]) and in warped compactifications
(see [40, 72] and references therein). Besides all of these, there are
also frequent occurrences of this type of metrics in string topics (see
[33, 34, 35, 36, 37, 53, 61, 71] and also [1, 12, 67] for some reviews
about these topics).
iv: In the derivation of effective theories for warped compactification
of supergravity and the Hor̆ava-Witten model (see [50, 51]). For in-
stance, in [51] the ansatz ds2 = hαds2(X4) + h
βds2(Y ) is considered
where X4 is a four-dimensional space-time with coordinates x
is a Calabi-Yau manifold (the so called internal space) and h de-
pends on the four-dimensional coordinates xµ, in order to study the
dynamics of the four-dimensional effective theory. We note that in
those articles, the structure of the expressions of the Ricci tensor
and scalar curvature of the involved metrics result particularly use-
ful. We observe that they correspond to very particular cases of the
expressions obtained by us in [29], see also Theorems 2.2 and 2.3 and
Proposition 2.4 stated below.
v: In the discussion of Birkhoff-type theorems (generally speaking these
are the theorems in which the gravitational vacuum solutions admit
more symmetry than the inserted metric ansatz, (see [41, page 372]
and [17, Chapter 3]) for rigorous statements), especially in Equation
6.1 of [66] where, H-J. Schmidt considers a special form of a bcwp and
basically shows that if a bcwp of this form is Einstein, then it admits
one Killing vector more than the fiber. In order to achieve that, the
author considers for a specific value of µ, namely µ = (1 − k)/2, in
the following problem:
Does there exist a smooth function ψ ∈ C∞>0(B) such that
the corresponding (ψ, µ)-bcwp (B2 × Fk, ψ2µgB + ψ2gF ) is
an Einstein manifold? (see also (Pb-Eins.) below.)
vi: In the study of bi-conformal transformations, bi-conformal vector
fields and their applications (see [32, Remark in Section 7] and [31,
Sections 7 and 8]).
vii: In the study of the spectrum of the Laplace-Beltrami operator for
p−forms. For instance in Equation (1.1) of [11], the author considers
the structure that follows: let M be an n-dimensional compact, Rie-
mannian manifold with boundary, and let y be a boundary-defining
function; she endows the interior M of M with a Riemannian metric
ds2 such that in a small tubular neighborhood of ∂M inM , ds2 takes
the form
(2.7) ds2 = e−2(a+1)tdt2 + e−2btdθ2∂M ,
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 5
where t := − log y ∈ (c,+∞) and dθ2
is the Riemannian metric on
∂M (see [11, 56] and references therein for details).
Notation 2.1. From now on, we will use the Einstein summation convention
over repeated indices and consider only connected manifolds. Furthermore,
we will denote the Laplace-Beltrami operator on a pseudo-Riemannian man-
ifold (N,h) by ∆N (·), i.e., ∆N (·) = ∇N
i∇Ni(·). Note that ∆N is elliptic if
(N,h) is Riemannian and it is hyperbolic when (N,h) is Lorentzian. If (N,h)
is neither Riemannian nor Lorentzian, then the operator is ultra-hyperbolic.
Furthermore, we will consider the Hessian of a function v ∈ C∞(N), denoted
by Hvh or H
N , so that the second covariant differential of v is given by
Hvh = ∇(∇v). Recall that the Hessian is a symmetric (0, 2) tensor field
satisfying
(2.8) Hvh(X,Y ) = XY v − (∇XY )v = h(∇X(grad v), Y ),
for any smooth vector fields X,Y on N.
For a given pseudo-Riemannian manifold N = (N,h) we will denote its
Riemann curvature tensor, Ricci tensor and scalar curvature by RN , RicN
and SN , respectively.
We will denote the set of all lifts of all vector fields of B by L(B). Note that
the lift of a vector field X on B denoted by X̃ is the vector field on B × F
given by dπ(X̃) = X where π : B × F → B is the usual projection map.
In Section 3, we will sketch the proofs of the following two theorems related
to the Ricci tensor and the scalar curvature of a generic (ψ, µ)-bcwp.
Theorem 2.2. Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Rieman-
nian manifolds with m ≥ 3 and k ≥ 1, respectively and also let µ ∈ R \
{0, 1, µ, µ±} be a real number with
µ := − k
m− 2 and µ± := µ±
µ2 − µ.
Suppose ψ ∈ C∞>0(B). Then the Ricci curvature tensor of the corresponding
(ψ, µ)-bcwp, denoted by Ric verifies the relation
Ric = RicB + β
− β∆ 1
α∆ gB on L(B)× L(B),
Ric = 0 on L(B)× L(F ),
Ric = RicF −
ψ2(µ−1)
α∆ gF on L(F )× L(F ),
(2.9)
6 FERNANDO DOBARRO & BÜLENT ÜNAL
where
(2.10)
(m− 2)µ+ k ,
(m− 2)µ+ k ,
−[(m− 2)µ + k]
µ[(m− 2)µ+ k] + k(µ− 1) ,
[(m− 2)µ + k]2
µ[(m− 2)µ+ k] + k(µ− 1) .
Theorem 2.3. Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Rieman-
nian manifolds of dimensions m ≥ 2 and k ≥ 0, respectively. Suppose that
SB and SF denote the scalar curvatures of B = (Bm, gB) and F = (Fk, gF ),
respectively. If µ ∈ R and ψ ∈ C∞>0(B), then the scalar curvature S of the
corresponding (ψ, µ)-bcwp verifies,
(i) If µ 6= − k
m− 1 , then
(2.11) − β∆Bu+ SBu = Su2µα+1 − SFu2(µ−1)α+1
where
(2.12) α =
2[k + (m− 1)µ]
{[k + (m− 1)µ] + (1− µ)}k + (m− 2)µ[k + (m− 1)µ] ,
(2.13) β = α2[k + (m− 1)µ] > 0
and ψ = uα > 0.
(ii) If µ = − k
m− 1 , then
(2.14) − k
|∇Bψ|2B
m−1 [S − SFψ−2]− SB.
From the mathematical and physical points of view, there are several
interesting questions about (ψ, µ)-bcwp’s. In [29] we began the study of
existence and/or construction of Einstein (ψ, µ)-bcwp’s and those of constant
scalar curvature. These questions are closely connected to Theorems 2.2 and
In [29], by applying Theorem 2.2, we give suitable conditions that allow
us to study some particular cases of the problem:
(Pb-Eins.) Given µ ∈ R, does there exist a smooth function
ψ ∈ C∞>0(B) such that the corresponding (ψ, µ)-bcwp is an
Einstein manifold?
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 7
In particular, we obtain the following result as an immediate corollary of
Theorem 2.2.
Proposition 2.4. Let us assume the hypothesis of Theorem 2.2. Then
the corresponding (ψ, µ)-bcwp is an Einstein manifold with λ if and only
if (F, gF ) is Einstein with ν constant and the system that follows is verified
λψ2µgB = RicB + β
− β∆ 1
α∆ gB on L(B)× L(B)
λψ2 = ν − 1
ψ2(µ−1)
(2.15)
where the coefficients are given by (2.10).
Compare the system (2.15) with the well known one for a classical warped
product in [18, 49, 59]. By studying (2.15), we have obtained the generaliza-
tion of the construction exposed in the above motivational examples in i and
v, among other related results. We suggest the interested reader consider
the results about the problem (Pb-Eins.) stated in [29].
Now, we focus on the problems which we will deal in §4. Let B = (Bm, gB)
and F = (Fk, gF ) be pseudo-Riemannian manifolds.
There is an extensive number of publications about the well known Yamabe
problem namely:
(Ya) [79, 75, 68, 13] Does there exist a function ϕ ∈ C∞>0(B)
such that (Bm, ϕ
m−2 gB) has constant scalar curvature?
Analogously, in several articles the following problem has been studied:
(cscwp) [27] Is there a function w ∈ C∞>0(B) such that the
warped product B ×w F has constant scalar curvature?
In the sequel we will suppose that B = (Bm, gB) is a Riemannian manifold.
Thus, both problems bring to the study of the existence of positive solutions
for nonlinear elliptic equations on Riemannian manifolds. The involved non-
linearities are powers with Sobolev critical exponent for the Yamabe problem
and sub-linear (linear if the dimension k of the fiber is 3) for the problem of
constant scalar curvature of a warped product.
In Section 4, we deal with a mixed problem between (Ya) and (cscwp)
which is already proposed in [29], namely:
8 FERNANDO DOBARRO & BÜLENT ÜNAL
(Pb-sc) Given µ ∈ R, does there exist a function ψ ∈
C∞>0(B) such that the corresponding (ψ, µ)-bcwp has constant
scalar curvature?
Note that when µ = 0, (Pb-sc) corresponds to the problem (cscwp),
whereas when the dimension of the fiber k = 0 and µ = 1, then (Pb-sc)
corresponds to (Ya) for the base manifold. Finally (Pb-sc) corresponds to
(Ya) for the usual product metric with a conformal factor in C∞>0(B) when
µ = 1.
Under the hypothesis of Theorem 2.3 i, the analysis of the problem (Pb-
sc) brings to the study of the existence and multiplicity of solutions u ∈
C∞>0(B) of
(2.16) − β∆Bu+ SBu = λu2µα+1 − SFu2(µ−1)α+1,
where all the components of the equation are like in Theorem 2.3 i and λ
(the conjectured constant scalar curvature of the corresponding (ψ, µ)-bcwp)
is a real parameter. We observe that an easy argument of separation of
variables, like in [24, §2] and [27], shows that there exists a positive solution
of (2.16) only if the scalar curvature of the fiber SH is constant. Thus this
will be a natural assumption in the study of (Pb-sc).
Furthermore, note that the involved nonlinearities in the right hand side of
(2.16) dramatically change with the choice of the parameters, an exhaustive
analysis of these changes is the subject matter of [29, §6].
There are several partial results about semi-linear elliptic equations like
(2.16) with different boundary conditions, see for instance [2, 5, 6, 9, 21,
23, 26, 73, 78] and references in [29].
In this article we will state our first results about the problem (Pb-sc) when
the base B is a compact Riemannian manifold of dimension m ≥ 3 and the
fiber F has non-positive constant scalar curvature SF .
For brevity of our study, it will be useful to introduce the following notation:
µsc := µsc(m,k) = −
m− 1 and µpY = µpY (m,k) := −
k + 1
m− 2 (sc as scalar
curvature and Y as Yamabe). Notice that µpY < µsc < 0.
We plan to study the case of µ = µsc in a preceding project, therefore the
related results are not going to be presented here.
We can synthesize our results about (Pb-sc) in the case of non-positive SF
as follow.
• The case of scalar flat fiber, i.e. SF = 0.
Theorem 2.5. If µ ∈ (µpY , µsc) ∪ (µsc,+∞) the answer to (Pb-sc)
is affirmative.
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 9
By assuming some additional restrictions on the scalar curvature of
the base SB , we obtain existence results for the range µ ≤ µpY .
• The case of fiber with negative constant scalar curvature, i.e. SF < 0.
In order to describe the µ−ranges of validity of the results, we will
apply the notations introduced in [29, §5] (see Appendix A for a brief
introduction of these notations).
Theorem 2.6. If “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD and
µ ∈ (0, 1) ∩ (µ−, µ+)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩ C[µ−, µ+]”,
then the answer to (Pb-sc) is affirmative.
Remark 2.7. The first two cases in Theorem 2.6 will be studied by
adapting the ideas in [5] and the last case by applying the results in
[73, p. 99]. In the former - Theorem 4.15, the involved nonlinearities
correspond to the so called concave-convex whereas in the latter -
Theorem 4.16, they are singular as in the Lichnerowicz-York equation
about the constraints for the Einstein equations (see [22], [43], [58],
[57, p. 542-543] and [73, Chp.18]).
Similarly to the case of SF = 0, we obtain existence results for some
remaining µ−ranges by assuming some additional restrictions for the
scalar curvature of the base SB .
Naturally the study of (Pb-sc) allows us to obtain partial results of the
related question:
Given µ ∈ R and λ ∈ R does there exist a function ψ ∈
C∞>0(B) such that the corresponding (ψ, µ)-bcwp has constant
scalar curvature λ?
These are stated in the several theorems and propositions in §4.
3. The curvature relations - Sketch of the proofs
The proofs of Theorems 2.2 and 2.3 require long and yet standard com-
putations of the Riemann and Ricci tensors and the scalar curvature of a
general base conformal warped product. Here, we reproduce the results for
the Ricci tensor and the scalar curvature, and we also suggest the reader see
[29, §3] for the complete computations.
Theorem 3.1. The Ricci tensor of [c, w]-bcwp, denoted by Ric satisfies
(1) Ric = RicB −
(m− 2)1
HcB + k
+2(m− 2) 1
dc⊗ dc+ k 1
[dc⊗ dw + dw ⊗ dc]
(m− 3)gB(∇
Bc,∇Bc)
gB(∇Bw,∇Bc)
on L(B)× L(B),
10 FERNANDO DOBARRO & BÜLENT ÜNAL
(2) Ric = 0 on L(B)× L(F ),
(3) Ric = RicF −
(m− 2)gB(∇
Bw,∇Bc)
+(k − 1)gB(∇
Bw,∇Bw)
gF on L(F )× L(F ).
Theorem 3.2. The scalar curvature S of a [c, w]-bcwp is given by
c2S = SB + SF
− 2(m− 1)∆Bc
− 2k∆Bw
− (m− 4)(m− 1)gB(∇
Bc,∇Bc)
− 2k(m− 2)gB(∇
Bw,∇Bc)
− k(k − 1)gB(∇
Bw,∇Bw)
The following two lemmas (3.3 and 3.7) play a central role in the proof
of Theorems 2.2 and 2.3. Indeed, it is sufficient to apply them in a suitable
mode and make use of Theorems 3.1 and 3.2 several times, the reader can
find all the details in [29, §2 and 4].
LetN = (Nn, h) be a pseudo-Riemannian manifold of dimension n, |∇(·)|2 =
|∇N (·)|2N = h(∇N (·),∇N (·)) and ∆h = ∆N .
Lemma 3.3. Let Lh be a differential operator on C
>0(N) defined by
(3.1) Lhv =
where ri, ai ∈ R and ζ :=
riai, η :=
i . Then,
(3.2) Lhv = (η − ζ)
‖grad hv‖2h
(ii) If ζ 6= 0 and η 6= 0, for α = ζ
and β =
, then we have
(3.3) Lhv = β
Remark 3.4. We also applied the latter lemma in the study of curvature of
multiply warped products (see [28]).
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 11
Corollary 3.5. Let Lh be a differential operator defined by
(3.4) Lhv = r1
for v ∈ C∞>0(N),
where r1a1 + r2a2 6= 0 and r1a21 + r2a22 6= 0. Then, by changing the variables
v = uα with 0 < u ∈ C∞(N), α = r1a1 + r2a2
1 + r2a
and β =
(r1a1 + r2a2)
1 + r2a
α(r1a1 + r2a2) there results
(3.5) Lhv = β
Remark 3.6. By the change of variables as in Corollary 3.5 equations of the
(3.6) Lhv = r1
= H(v, x, s),
transform into
(3.7) β∆hu = uH(u
α, x, s).
Lemma 3.7. Let Hh be a differential operator on C∞>0(N) defined by
(3.8) Hhv =
riai and η :=
i , where the indices extend from 1 to l ∈ N and
any ri, ai ∈ R. Hence,
(3.9) Hhv = (η − ζ)
dv ⊗ dv + ζ 1
where ⊗ is the usual tensorial product. If furthermore, ζ 6= 0 and η 6= 0,
(3.10) Hhv = β
where α =
and β =
4. The problem (Pb-sc) - Existence of solutions
Throughout this section, we will assume that B is not only a Riemannian
manifold of dimension m ≥ 3, but also “compact” and connected. We further
assume that F is a pseudo-Riemannian manifold of dimension k ≥ 0 with
constant scalar curvature SF ≤ 0. Moreover, we will assume that µ 6= µsc.
Hence, we will concentrate our attention on the relations (2.11), (2.12) and
(2.13) by applying Theorem 2.3 (i).
12 FERNANDO DOBARRO & BÜLENT ÜNAL
Let λ1 denote the principal eigenvalue of the operator
(4.1) L(·) = −β∆B(·) + SB(·),
and u1 ∈ C∞>0(B) be the corresponding positive eigenfunction with ‖u1‖∞ =
1, where β is as in Theorem 2.3.
First of all, we will state some results about uniqueness and non-existence
of positive solutions for Equation (2.16) under the latter hypothesis.
About the former, we adapt Lemma 3.3 in [5, p. 525] to our situation (for
a detailed proof see [5], [20, Method II, p. 103] and also [70]).
Lemma 4.1. Let f ∈ C0(R>0) such that t−1f(t) is decreasing. If v and w
satisfy
(4.2)
−β∆Bv + SBv ≤ f(v),
v ∈ C∞>0(B),
(4.3)
−β∆Bw + SBw ≥ f(w),
w ∈ C∞>0(B),
then w ≥ v on B.
Proof. Let θ(t) be a smooth nondecreasing function such that θ(t) ≡ 0 for
t ≤ 0 and θ(t) ≡ 1 for t ≥ 1. Thus for all ǫ > 0,
θǫ(t) := θ
is smooth, nondecreasing, nonnegative and θ(t) ≡ 0 for t ≤ 0 and θ(t) ≡ 1
for t ≥ ǫ. Furthermore γǫ(t) :=
sθ′ǫ(s)ds satisfies 0 ≤ γǫ(t) ≤ ǫ, for any
t ∈ R.
On the other hand, since (B, gB) is a compact Riemannian manifold without
boundary and β > 0, like in [5, Lemma 3.3, p. 526] there results
(4.4)
[−vβ∆Bw + wβ∆Bv]θǫ(v − w)dvgB ≤
[−β∆Bv]γǫ(v − w)dvgB .
Hence, by the above considerations about θǫ and γǫ, (4.4) implies that
(4.5)
[−vβ∆Bw + wβ∆Bv]θǫ(v − w)dvgB ≤ ǫ
[−β∆Bv≥0]
[−β∆Bv]dvgB .
Now, by applying (4.2) and (4.3) there results
(4.6)
− vβ∆Bw+wβ∆Bv = vLw−wLv ≥ vf(w)−wf(v) = vw
− f(v)
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 13
Thus by combining (4.6) and (4.5), as ǫ→ 0+ we led to
(4.7)
[v>w]
− f(v)
dvgB ≤ 0
and conclude the proof like in [5, Lemma 3.3, p. 526-527]. But
on [v > w] and hence meas[v > w] = 0; thus v ≤ w. 1 �
Corollary 4.2. Let f ∈ C0(R>0) such that t−1f(t) is decreasing. Then
(4.8)
−β∆Bv + SBv = f(v),
v ∈ C∞>0(B)
has at most one solution.
Proof. Assume that v and w are two solutions of (4.8). Then by applying
Lemma 4.1 firstly with v and w, and conversely with w and v, the conclusion
is proved. �
Remark 4.3. Notice that Lemma 4.1 and Corollary 4.2 allow the function
f ∈ C0(R>0) to be singular at 0.
Related to the non-existence of smooth positive solutions for Equation
(2.16), we will state an easy result under the general hypothesis of this
section.
Proposition 4.4. If either maxB SB ≤ infu∈R>0 u2µα(λ− SFu−2α) or
minB SB ≥ supu∈R>0 u
2µα(λ − SFu−2α), then (2.16) has no solution in
C∞>0(B).
Proof. It is sufficient to apply the maximum principle with some easy ad-
justments to the particular involved coefficients. �
• The case of scalar flat fiber, i.e. SF = 0.
In this case, the term containing the nonlinearity u2(µ−1)α+1 becomes non-
influent in (2.16), thus (Pb-sc) equivalently results to the study of existence
of solutions for the problem:
(4.9)
−β∆Bu+ SBu = λu2µα+1,
u ∈ C∞>0(B),
where λ is a real parameter (i.e., it is the searched constant scalar curvature)
and ψ = uα.
1meas denotes the usual gB−measure on the compact Riemannian manifold (Bm, gB)
14 FERNANDO DOBARRO & BÜLENT ÜNAL
Remark 4.5. 2 Let p ∈ R\{1} and (λ0, u0) ∈ (R\{0})×C∞>0(B) be a solution
(4.10)
−β∆Bu+ SBu = λup,
u ∈ C∞>0(B).
Hence, by the difference of homogeneity between both members of (4.9), it
is easy to show that if λ ∈ R satisfies sign(λ) = sign(λ0), then (λ, uλ) is a
solution of (4.10), where uλ = tλu0 and tλ =
Thus by (4.9), we obtain geometrically: if the parameter µ is given in a
way that p := 2µα + 1 6= 1 and B ×[ψµ0 ,ψ0] F has constant scalar curvature
λ0 6= 0, then for any λ ∈ R verifying sign(λ) = sign(λ0), there results that
B×[ψµ
F is of scalar curvature λ, where ψλ = t
ψ0 and tλ given as above.
Theorem 4.6. (Case : µ = 0) The scalar curvature of a (ψ, 0)-bcwp of base
B and fiber F (i.e., a singly warped product B ×ψ F ) is a constant λ if and
only if λ = λ1 and ψ is a positive multiple of u
1 (i.e., ψ = tu
1 for some
t ∈ R>0).
Proof. First of all note that µ = 0 implies α =
k + 1
. On the other hand, in
this case, the problem (4.9) is linear, so it is sufficient to apply the well known
results about the principal eigenvalue and its associated eigenfunctions of
operators like (4.1) in a suitable setting. �
Theorem 4.7. (Case : µsc < µ < 0) The scalar curvature of a (ψ, µ)-bcwp
of base B and fiber F is a constant λ, only if sign(λ) = sign(λ1). Further-
more,
(1) if λ = 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has
constant scalar curvature 0 if and only if λ1 = 0. Moreover, such
ψ’s are the positive multiples of uα1 , i.e. tu
1 , t ∈ R>0.
(2) if λ > 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has
constant scalar curvature λ if and only if λ1 > 0. In this case, the
solution ψ is unique.
(3) if λ < 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has
constant scalar curvature λ when λ1 < 0 and is close enough to 0.
Proof. The condition µsc < µ < 0 implies that 0 < p := 2µα + 1 < 1,
i.e., the problem (4.9) is sublinear. Thus, to prove the theorem one can use
variational arguments as in [24] (alternatively, degree theoretic arguments
as in [7] or bifurcation theory as in [27]).
2Along this article we consider the sign function defined by sign = χ(0,+∞) − χ(−∞,0),
where χA is the characteristic function of the set A.
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 15
We observe that in order to obtain the positivity of the solutions required in
(4.9), one may apply the maximum principle for the case of λ > 0 and the
antimaximum principle for the case of λ < 0.
The uniqueness for λ > 0 is a consequence of Corollary 4.2. �
Remark 4.8. In order to consider the next case we introduce the following
notation. For a given p such that 1 < p ≤ pY , let
(4.11) κp := inf
|∇Bv|2 + SB
dvgB ,
where
Hp :=
v ∈ H1(B) :
|v|p+1dvgB = 1
Now, we consider the following two cases.
(1 < p < pY ): In this case by adapting [42, Theorem 1.3], there ex-
ists up ∈ C∞>0(B) such that (βκp, up) is a solution of (4.10) and∫
up+1p dvgB = 1.
(p = pY ): For this specific and important value, analogously to [42, §2],
we distinguish three subcases along the study of our problem (4.10),
in correspondence with the sign(κpY ).
κpY = 0: in this case, there exists upY ∈ C∞>0(B) such that (0, upY )
is a solution of (4.10) and
upY +1pY dvgB = 1.
κpY < 0: here there exists upY ∈ C∞>0(B) such that (βκpY , upY ) is
a solution of (4.10) and
upY +1pY dvgB = 1.
κpY > 0: this is a more difficult case, let Km be the sharp Eu-
clidean Sobolev constant
(4.12) Km =
m(m− 2)ω
where ωm is the volume of the unit m−sphere. Thus, if
(4.13) κpY <
then there exists upY ∈ C∞>0(B) such that (βκpY , upY ) is a solu-
tion of (4.10) and
upY +1pY dvgB = 1. Furthermore, the condi-
(4.14) κpY ≤
16 FERNANDO DOBARRO & BÜLENT ÜNAL
is sharp by [42], so that this is independent of the underlying
manifold and the potential considered.
The equality case in (4.14) is discussed in [44].
This results allow to establish the following two theorems.
Theorem 4.9. (Cases : µpY < µ < µsc or 0 < µ) There exists ψ ∈ C∞>0(B)
such that the scalar curvature of B ×[ψµ,ψ] F is a constant λ if and only if
sign(λ) = sign(κp) where p := 2µα+1 and κp is given by (4.11). Furthermore
if λ < 0, then the solution ψ is unique.
Proof. The conditions (µpY < µ < µsc or 0 < µ) imply that 1 < p := 2µα+
1 < pY , i.e. the problem (4.9) is superlinear but subcritical with respect to
the Sobolev immersion theorem (see [29, Remark 5.5]). By recalling that
ψ = uα, it is sufficient to prove that follows.
Let up be defined as in the case of (1 < p < pY ) in Remark 4.8. If (λ, u)
is a solution of (4.9), then multiplying (4.9) by up and integrating by parts
there results
(4.15) βκp
upudvgB = λ
pdvgB .
Thus sign(λ) = sign(κp) since β, up and u are all positive.
Conversely, if λ is a real constant such that sign(λ) = sign(κp) 6= 0,
then by Remark 4.5, (λ, uλ) is a solution of (4.9), where uλ = tλup and
On the other side, if λ = κp = 0, then (0, up) is a solution of (4.9).
Since 1 < p, the uniqueness for λ < 0 is a consequence of Corollary 4.2. �
Theorem 4.10. (Cases : µ = µpY ) If there exists ψ ∈ C∞>0(B) such that the
scalar curvature of B ×[ψµpY ,ψ] F is a constant λ, then sign(λ) = sign(κpY ).
Furthermore, if λ ∈ R verifying sign(λ) = sign(κpY ) and (4.13), then there
exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµpY ,ψ] F is λ.
Besides, if λ ∈ R is negative, then there exists at most one ψ ∈ C∞>0(B) such
that the scalar curvature of B ×[ψµpY ,ψ] F is λ.
Proof. The proof is similar to that of Theorem 4.9, but follows from the
application of the case of (p = pY ) in Remark 4.8. Like above, the uniqueness
of λ < 0 is a consequence of Corollary 4.2. �
In the next proposition including the supercritical case, we will apply the
following result (see also [73, p.99]).
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 17
Lemma 4.11. Let (Nn, gN ) be a compact connected Riemannian manifold
without boundary of dimension n ≥ 2 and ∆gN be the corresponding Laplace-
Beltrami operator. Consider the equation of the form
(4.16)
−∆gNu = f(·, u),
u ∈ C∞>0(N)
where f ∈ C∞(N × R>0). If there exist a0 and a1 ∈ R>0 such that
(4.17)
u < a0 ⇒ f(·, u) > 0
u > a1 ⇒ f(·, u) < 0,
then (4.16) has a solution satisfying a0 ≤ u ≤ a1.
Proposition 4.12. (Cases : −∞ < µ < µsc or 0 < µ) If maxSB < 0, then
for all λ < 0 there exists ψ ∈ C∞>0(B) such that the scalar curvature of
B ×[ψµ,ψ] F is the constant λ. Furthermore, the solution ψ is unique.
Proof. The conditions (−∞ < µ < µsc or 0 < µ) imply that 1 < p :=
2µα+ 1.
On the other hand, since B is compact, by taking
f(., u) = −SB(·)u+ λup = (−SB + λup−1)u,
we obtain that limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus
(4.17) is verified.
Hence, the proposition is proved by applying Lemma 4.11 on (Bm, gB). No-
tice that a0 can take positive values and eventually gets close enough to 0
due to the condition of limu−→0+ f(·, u), and consequently the corresponding
solution results positive.
Again, since λ < 0 and 1 < p the uniqueness is a consequence of Corollary
4.2. �
Proof. (of Theorem 2.5) This is an immediate consequence of the above
results. �
• The case of a fiber with negative constant scalar curvature, i.e. SF < 0.
Here, the (Pb-sc) becomes equivalent to the study of the existence for
the problem
(4.18)
−β∆Bu+ SBu = λup − SFuq,
u ∈ C∞>0(B),
where λ is a real parameter (i.e., the searched constant scalar curvature),
ψ = uα, p = 2µα+ 1 and q = 2(µ − 1)α + 1.
Remark 4.13. Let u be a solution of (4.18).
18 FERNANDO DOBARRO & BÜLENT ÜNAL
(i) If λ1 ≤ 0, then λ < 0. Indeed, multiplying the equation in (4.18) by
u1 and integrating by parts there results:
(4.19) λ1
u1udvgB + SF
qdvgB = λ
pdvgB ,
where u1 and u are positive.
(ii) If λ = 0, then λ1 > 0.
(iii) If µ = 0 (the warped product case), then λ < λ1. These cases have
been studied in [27, 24].
(iv) If µ = 1 (the Yamabe problem for the usual product with conformal
factor in C∞>0(B)), there results sign(λ) = sign(λ1 + SF ).
An immediate consequence of Remark 4.13 is the following lemma.
Lemma 4.14. Let B and F be given like in Theorem 2.3(i). Suppose further
that B is a compact connected Riemannian manifold and F is a pseudo-
Riemannian manifold of constant scalar curvature SF < 0. If λ ≥ 0 and
λ1 ≤ 0 (for instance when SB ≤ 0 on B), then there is no ψ ∈ C∞>0(B) such
that the scalar curvature of B ×[ψµ,ψ] F is λ.
Theorem 4.15. [29, Rows 6 and 8 in Table 4] Under the hypothesis of
Theorem 2.3(i), let B be a compact connected Riemannian manifold and
F be a pseudo-Riemannian manifold of constant scalar curvature SF < 0.
Suppose that “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩
C[µ−, µ+]”.
(1) If λ1 ≤ 0, then λ ∈ R is the scalar curvature of a B ×[ψµ,ψ] F if and
only if λ < 0.
(2) If λ1 > 0, then there exists Λ ∈ R>0 such that λ ∈ R \ {Λ} is the
scalar curvature of a B ×[ψµ,ψ] F if and only if λ < Λ.
Furthermore if λ ≤ 0, then there exists at most one ψ ∈ C∞>0(B) such that
B ×[ψµ,ψ] F has scalar curvature λ.
Proof. The proof of this theorem is the subject matter of §5. �
Once again we make use of Lemma 4.11 for the next theorem about the
singular case and the following propositions.
Theorem 4.16. [29, Row 7 Table 4] Under the hypothesis of Theorem 2.3(i),
let B be a compact connected Riemannian manifold and F be a pseudo-
Riemannian manifold of constant scalar curvature SF < 0. Suppose that
“(m,k) ∈ CD and µ ∈ (0, 1) ∩ (µ−, µ+)”, then for any λ < 0 there exists
ψ ∈ C∞>0(B) such that the scalar curvature of B×[ψµ,ψ] F is λ. Furthermore
the solution ψ is unique.
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 19
Proof. First of all note that the conditions “(m,k) ∈ CD and µ ∈ (0, 1) ∩
(µ−, µ+)” imply that q < 0 and 1 < p , i.e. the problem (4.18) is superlinear
in p but singular in q.
On the other hand, since B is compact, taking
f(., u) = −SB(·)u+ λup − SFuq = [(−SB(·) + λup−1)u1−q − SF ]uq,
there result limu−→0+ f(·, u) = +∞ and limu−→+∞ f(·, u) = −∞. Thus
(4.17) is verified.
Thus by an application of Lemma 4.11 for (Bm, gB), we conclude the proof
for the existence part.
The uniqueness part just follows from Corollary 4.2. �
Remark 4.17. We observe that the arguments applied in the proof of The-
orem 4.16 can be adjusted to the case of a compact connected Riemannian
manifold B with 0 ≤ q < 1 < p, λ < 0 and SF < 0, so that some of
the situations included in Theorem 4.15. However, both argumentations are
compatible but different.
Proof. (of Theorem 2.6) This is an immediate consequence of the above
results. �
The approach in the next propositions is similar to Proposition 4.12 and
Theorem 4.16.
Proposition 4.18. [29, Row 10 Table 4] Let 1 < µ < +∞. If maxSB < 0,
then for all λ < 0 there exists ψ ∈ C∞>0(B) such that the scalar curvature of
B ×[ψµ,ψ] F is the constant λ.
Proof. The condition 1 < µ < +∞ implies that 1 < q < p.
On the other hand, since B is compact, taking
f(., u) = −SB(·)u+ λup − SFuq = [−SB(·) + (λup−q − SF )uq−1]u,
there result limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus (4.17)
is satisfied.
Thus an elementary application of Lemma 4.11 for (Bm, gB) proves the
proposition. �
Proposition 4.19. [29, Rows 2, 4 and 3 in Table 4] Let either “(m,k) ∈
D and µ ∈ (µsc, 0)” or “(m,k) ∈ CD and µ ∈ (µsc, 0) ∩ C[µ−, µ+]” or
“(m,k) ∈ CD and µ ∈ (µsc, 0)∩ (µ−, µ+)”. If minSB > 0, then for all λ ≤ 0
there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of
B ×[ψµ,ψ] F is the constant λ.
Proof. If either “(m,k) ∈ D and µ ∈ (µsc, 0)” or “(m,k) ∈ CD and µ ∈
(µsc, 0) ∩ C[µ−, µ+]”, then 0 < q < p < 1.
20 FERNANDO DOBARRO & BÜLENT ÜNAL
On the other hand, since B is compact, taking
f(., u) = −SB(·)u+ λup − SFuq = [−SB(·)u1−q + λup−q − SF ]uq,
there result limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus (4.17)
is verified and again we can apply Lemma 4.11 for (Bm, gB).
If “(m,k) ∈ CD and µ ∈ (µsc, 0) ∩ (µ−, µ+)”, then q < 0 < p < 1. Con-
sidering the limits as above, limu−→0+ f(·, u) = +∞ and limu−→+∞ f(·, u) =
−∞. So, an application of Lemma 4.11 concludes the proof. �
Remark 4.20. Notice that in Theorems 4.15 and 4.16 we do not assume
hypothesis related to the sign of SB(·), unlike in Propositions 4.12, 4.18 and
4.19.
Proposition 4.21. [29, Rows 5 and 9 in Table 4] Let (m,k) ∈ CD be.
(1) If either “µ ∈
m− 1 , 0
∩ {µ−, µ+} and minSB > 0” or “µ ∈
(0, 1) ∩ {µ−, µ+}”, then for all λ < 0 there exists a smooth function
ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is the
constant λ. In the second case, ψ is also unique .
(2) If either “µ ∈
m− 1 , 0
∩ {µ−, µ+}” or “µ ∈ (0, 1) ∩ {µ−, µ+}”
and furthermore λ1 > 0, then there exists a smooth function ψ ∈
C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is 0.
Proof. In both cases q = 0, so by considering
f(., u) = −SB(·)u+ λup − SF ,
the proof of (1) follows as in the latter propositions, while that of (2) is a
consequence of the linear theory and the maximum principle. �
Remark 4.22. Finally, we observe a particular result about the cases studied
in [27]. If µ = 0, then p = 1 and q = 1−2α = k − 3
k + 1
. When the dimension of
the fiber is k = 2, the exponent q = −1
. So, writing the involved equation
∆Bu = f(., u) = −SB(·)u + λu− SFu−
and by applying Lemma 4.11 as above, we obtain that if λ < minSB, then
there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of
B ×ψ F is the constant λ. Furthermore, by Corollary 4.2 such ψ is unique
(see [27, 24] and [25]).
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 21
5. Proof of the Theorem 4.15
The subject matter of this section is the proof of the Theorem 4.15, so we
naturally assume its hypothesis.
Most of the time, we need to specify the dependence of λ of (4.18), we will
do that by writing (4.18)λ. Furthermore, we will denote the right hand side
of (4.18)λ by fλ(t) := λt
p − SF tq.
The conditions either “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD
and µ ∈ (0, 1) ∩ C[µ−, µ+]”, imply that 0 < q < 1 < p. But the type of
nonlinearity in the right hand side of (4.18)λ changes with the signλ, i.e. it
is purely concave for λ < 0 and concave-convex for λ > 0.
The uniqueness for λ ≤ 0 is again a consequence of Corollary 4.2.
In order to prove the existence of a solution for (4.18)λ with signλ 6= 0, we
adapt the approach of sub and upper solutions in [5].
Thus, the proof of Theorem 4.15 will be an immediate consequence of the
results that follows.
Lemma 5.1. (4.18)0 has a solution if and only if λ1 > 0.
Proof. This situation is included in the results of the second case of Theorem
4.7 by replacing −SF with λ (see [24, Proposition 3.1]). �
Lemma 5.2. Let us assume that {λ : (4.18)λ has a solution} is non-empty
and define
(5.1) Λ = sup{λ : (4.18)λ has a solution}.
(i) If λ1 ≤ 0, then Λ ≤ 0.
(ii) If λ1 > 0, then there exists λ > 0 finite such that Λ ≤ λ.
Proof.
(i) It is sufficient to observe Remark 4.13 i.
(ii) Like in [5], let λ > 0 such that
(5.2) λ1t < λt
p − SF tq,∀t ∈ R, t > 0.
Thus, if (λ, u) is a solution of (4.18)λ, then
p − SF
λ1u1u < λ
p − SF
so λ < λ.
Lemma 5.3. Let
(5.3) Λ = sup{λ : (4.18)λ has a solution}.
22 FERNANDO DOBARRO & BÜLENT ÜNAL
Figure 1. The nonlinearity fλ in Lemma 5.3, i.e. 0 < q <
1 < p, SF < 0, λ1 > 0, λ > 0.
(i) Let E ∈ R>0. There exist 0 < λ0 = λ0(E) and 0 < M = M(E,λ0)
such that ∀λ : 0 < λ ≤ λ0, so we have
(5.4) 0 < E
fλ(EM)
(ii) If λ1 > 0, then {λ > 0 : (4.18)λ has a solution} 6= ∅. As a conse-
quence of that, Λ is finite.
(iii) If λ1 > 0, then for all 0 < λ < Λ there exists a solution of the
problem (4.18)λ.
Proof.
(i) For any 0 < λ < λ0
0 < gλ(r) := E
fλ(Er)
= Erq−1(λEp−1rp−q − SFEq−1)
< Erq−1(λ0E
p−1rp−q − SFEq−1).
It is easy to see that
q − 1
p−q 1
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 23
is a minimum point for gλ0 and
gλ0(r0) = E
q − 1
) q−1
q − 1
p− 1 − 1
→ 0+, as λ0 → 0+.
Hence there exist 0 < λ0 = λ0(E) and 0 < M =M(E,λ0) such that
(5.4) is verified.
(ii) Since λ1 > 0, by the maximum principle, there exists a solution
e ∈ C∞>0(B) of
(5.5) LB(e) = −β∆Be+ SBe = 1.
Then, applying item (i) above with E = ‖e‖∞ there exists 0 < λ0 =
λ0(‖e‖∞) and 0 < M = M(‖e‖∞, λ0) such that ∀λ with 0 < λ ≤ λ0
we have that
(5.6) LB(Me) =M ≥ fλ(Me),
hence Me is a supersolution of (4.18)λ.
On the other hand, since ǔ1 := inf u1 > 0, for all λ > 0
(5.7) ǫ−1fλ(ǫǔ1) = ǫ
q−1[λǫp−qǔ
1 − SF ǔ
1] → +∞, as ǫ→ 0+.
Furthermore, note that fλ is nondecreasing when λ > 0. Hence for
any 0 < λ there exists a small enough 0 < ǫ verifying
(5.8) LB(ǫu1) = ǫλ1u1 ≤ ǫλ1‖u1‖∞ ≤ fλ(ǫǔ1) ≤ fλ(ǫu1),
thus ǫu1 is a subsolution of (4.18)λ.
Then for any 0 < λ < λ0, (taking eventually 0 < ǫ smaller if
necessary), we have that the above constructed couple sub super
solution satisfies
(5.9) ǫu1 < Me.
Now, by applying the monotone iteration scheme, we have that
{λ > 0 : (4.18)λ has a solution} 6= ∅. Furthermore by Lemma 5.2
(ii) there results Λ is finite.
(iii) The proof of this item is completely analogous to Lemma 3.2 in [5].
We will rewrite this to be self contained.
Given λ < Λ, let uν be a solution of (4.18)ν with λ < ν < Λ.
Then uν is a supersolution of (4.18)λ and for small enough 0 < ǫ, the
subsolution ǫu1 of (4.18)λ verifies ǫu1 < uν , then as above (4.18)λ
has a solution.
Lemma 5.4. For any λ < 0, there exists γλ > 0 such that ‖u‖∞ ≤ γλ for
any solution u of (4.18)λ. Furthermore if SB is nonnegative, then positive
zero of fλ can be choose as γλ.
24 FERNANDO DOBARRO & BÜLENT ÜNAL
fΛHtL
fΛHtL+Νt
Figure 2. The nonlinearity in Lemma 5.5 , i.e. 0 < q < 1 <
p, SF < 0, λ1 > 0, λ < 0.
Proof. Define ŠB := minSB (recall that B is compact). There are two
different situations, namely.
• 0 ≤ ŠB: since there exists x1 ∈ B such that u(x1) = ‖u‖∞ and
0 ≤ −β∆Bu(x1) = −SB(x1)‖u‖∞ + λ‖u‖p∞ − SF‖u‖q∞, there results
‖u‖∞ ≤ γλ, where γλ is the strictly positive zero of fλ.
• ŠB < 0: we consider f̃λ(t) := λtp − SF tq − ŠBt. Now, our problem
(4.18)λ is equivalent to
−β∆Bu+ (SB − ŠB)u = f̃λ(u),
u ∈ C∞>0(B).
But here the potential of (SB− ŠB) is non negative and the function
f̃λ has the same behavior of fλ with a positive zero γ̃λ on the right
side of the positive zero γλ of fλ. Thus, repeating the argument for
the case of ŠB ≥ 0, we proved ‖u‖∞ ≤ γ̃λ.
Lemma 5.5. Let λ1 > 0. Then for all λ < 0 there exists a solution of
(4.18)λ.
Proof. We will apply again the monotone iteration scheme. Define ŠB :=
minSB (note that B is compact).
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 25
• 0 ≤ ŠB: Clearly, the strictly positive zero γλ of fλ is a supersolution
(5.10) − β∆Bu+ (SB + ν)u = fλ(u) + νu,
for all ν ∈ R.
On the other hand, for 0 < ǫ = ǫ(λ) small enough,
(5.11) LB(ǫu1) = ǫλ1u1 ≤ fλ(ǫu1).
Then ǫu1 is a subsolution of (5.10) for all ν ∈ R.
By taking ε possibly smaller, we also have
(5.12) 0 < ǫu1 < γλ.
We note that for large enough values of ν ∈ R>0, the nonlinearity
on the right hand side of (5.10), namely fλ(t) + νt, is an increasing
function on [0, γλ].
Thus applying the monotone iteration scheme we obtain a strictly
positive solution of (5.10), and hence a solution of (4.18)λ (see [3],
[4], [54]).
• ŠB < 0: In this case, like in Lemma 5.4 we consider f̃λ(t) := λtp −
q − ŠBt. Then, the problem (4.18)λ is equivalent to
(5.13)
−β∆Bu+ (SB − ŠB)u = f̃λ(u),
u ∈ C∞>0(B),
where the potential is nonnegative and the function f̃λ has a similar
behavior to fλ with a positive zero γ̃λ on the right side of the positive
zero γλ of fλ.
Here, it is clear that γ̃λ is a positive supersolution of
(5.14) − β∆Bu+ (SB − ŠB + ν)u = f̃λ(u) + νu,
for all ν ∈ R. Hence, we complete the proof similarly to the case of
ŠB ≥ 0.
Lemma 5.6. Let λ1 ≤ 0, λ < 0, ŠB := minSB and also let γλ be a positive
zero of fλ and γ̃λ be a positive zero of f̃λ := fλ − ŠBidR≥0 . Then there
exists a solution u of (4.18)λ. Furthermore any solution of (4.18)λ satisfies
γλ ≤ ‖u‖∞ ≤ γ̃λ.
Proof. First of all we observe that if SB ≡ 0 (so λ1 = 0), then u ≡ γλ is the
searched solution of (4.18)λ.
Now, we assume that SB 6≡ 0. Since λ1 ≤ 0, there results ŠB < 0. In this
case, one can notice that 0 < γλ < γ̃λ.
26 FERNANDO DOBARRO & BÜLENT ÜNAL
On the other hand, the problem (4.18)λ is equivalent to
(5.15)
−β∆Bu+ (SB − ŠB)u = f̃λ(u),
u ∈ C∞>0(B).
By the second part of the proof of Lemma 5.4, if u is a solution of (4.18)λ
(or equivalently (5.15)), then ‖u‖∞ ≤ γ̃λ. Besides, since
u1(fλ ◦ u) = λ1
u, u1 > 0 and λ1 ≤ 0 results γλ ≤ ‖u‖∞.
From this point on, the proof of the existence of solutions for (5.15) follows
the lines of the second part of Lemma 5.5. �
6. Conclusions and future directions
Now, we would like to summarize the content of the paper and to propose
our future plans on this topic.
We remark to the reader that several computations and proofs, along with
other complementary results mentioned in this article and references can be
obtained in [29]. We have chosen this procedure to avoid the involved long
computations.
In brief, we introduced and studied curvature properties of a particular
family of warped products of two pseudo-Riemannian manifolds which we
called as a base conformal warped product. Roughly speaking the metric
of such a product is a mixture of a conformal metric on the base and a
warped metric. We concentrated our attention on a special subclass of this
structure, where there is a specific relation between the conformal factor c
and the warping function w, namely c = wµ with µ a real parameter.
As we mentioned in §1 and the first part of §2, these kinds of metrics and
considerations about their curvatures are very frequent in different physi-
cal areas, for instance theory of general relativity, extra-dimension theories
(Kaluza-Klein, Randall-Sundrum), string and super-gravity theories; also in
global analysis for example in the study of the spectrum of Laplace-Beltrami
operators on p-forms, etc.
More precisely, in Theorems 3.1 and 3.2, we obtained the classical relations
among the different involved Ricci tensors (respectively, scalar curvatures)
for metrics of the form c2gB⊕w2gF . Then the study of particular families of
either scalar or tensorial nonlinear partial differential operators on pseudo-
Riemannian manifolds (see Lemmas 3.3 and 3.7) allowed us to find reduced
expressions of the Ricci tensor and scalar curvature for metrics as above with
c = wµ, where µ a real parameter (see Theorems 2.2 and 2.3). The operated
reductions can be considered as generalizations of those used by Yamabe in
[79] in order to obtain the transformation law of the scalar curvature under
ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 27
a conformal change in the metric and those used in [27] with the aim to
obtain a suitable relation among the involved scalar curvatures in a singly
warped product (see also [52] for other particular application and our study
on multiply warped products in [28]).
In §4 and 5, under the hypothesis that (B, gB) be a “compact” and con-
nected Riemannian manifold of dimension m ≥ 3 and (F, gF ) be a pseudo
- Riemannian manifold of dimension k ≥ 0 with constant scalar curvature
SF , we dealt with the problem (Pb-sc). This question leads us to ana-
lyze the existence and uniqueness of solutions for nonlinear elliptic partial
differential equations with several kinds of nonlinearities. The type of non-
linearity changes with the value of the real parameter µ and the sign of SF .
In this article, we concentrated our attention to the cases of constant scalar
curvature SF ≤ 0 and accordingly the central results are Theorems 2.5 and
2.6. Although our results are partial so that there are more cases to study
in forthcoming works, we obtained also other complementary results under
more restricted hypothesis about the sign of the scalar curvature of the base.
Throughout our study, we meet several types of partial differential equa-
tions. Among them, most important ones are those with concave-convex
nonlinearities and the one so called Lichnerowicz-York equation. About the
former, we deal with the existence of solutions and leave the question of
multiplicity of solutions to a forthcoming study.
We observe that the previous problems as well as the study of the Ein-
stein equation on base conformal warped products, (ψ, µ)-bcwp’s and their
generalizations to multi-fiber cases, give rise to a reach family of interesting
problems in differential geometry and physics (see for instance, the several
recent works of R. Argurio, J. P. Gauntlett, M. O. Katanaev, H. Kodama, J.
Maldacena, H. -J. Schmidt, A. Strominger, K. Uzawa, P. S. Wesson among
many others) and in nonlinear analysis (see the different works of A. Am-
brosetti, T. Aubin, I. Choquet-Bruat, J. Escobar, E. Hebey, J. Isenberg, A.
Malchiodi, D. Pollack, R. Schoen, S. -T. Yau among others).
Appendix A.
Let us assume the hypothesis of Theorem 2.3 (i), the dimensions of the
base m ≥ 2 and of the fiber k ≥ 1. In order to describe the classification of
the type of nonlinearities involved in (2.11), we will introduce some notation
(for a complete study of these nonlinearities see [29, Section 5]). The example
in Figure 1 will help the reader to clarify the notation.
Note that the denominator in (2.12) is
(A.1) η := (m− 1)(m − 2)µ2 + 2(m− 2)kµ + (k + 1)k
28 FERNANDO DOBARRO & BÜLENT ÜNAL
and verifies η > 0 for all µ ∈ R. Thus α in (2.12) is positive if and only if
µ > − k
m− 1 and by the hypothesis µ 6= −
m− 1 in Theorem 2.3 (i), results
α 6= 0.
We now introduce the following notation:
(A.2)
p = p(m,k, µ) = 2µα+ 1 and
q = q(m,k, µ) = 2(µ− 1)α + 1 = p− 2α,
where α is defined by (2.12).
Thus, for all m,k, µ given as above, p is positive. Indeed, by (A.1), p > 0
if and only if ̟ > 0, where
̟ := ̟(m,k, µ)
:= 4µ[k + (m− 1)µ] + (m− 1)(m− 2)µ2 + 2(m− 2)kµ + (k + 1)k
= (m− 1)(m+ 2)µ2 + 2mkµ + (k + 1)k.
But discr (̟) ≤ −4km2 ≤ −16 and m > 1, so ̟ > 0.
Unlike p, q changes sign depending on m and k. Furthermore, it is im-
portant to determine the position of p and q with respect to 1 as a function
of m and k. In order to do that, we define
(A.3) D := {(m,k) ∈ N≥2 × N≥1 : discr (̺(m,k, ·)) < 0},
where N≥l := {j ∈ N : j ≥ l} and
̺ := ̺(m,k, µ)
:= 4(µ − 1)[k + (m− 1)µ] + (m− 1)(m− 2)µ2 + 2(m− 2)kµ + (k + 1)k
= (m− 1)(m + 2)µ2 + 2(mk − 2(m− 1))µ + (k − 3)k.
Note that by (A.1), q > 0 if and only if ̺ > 0. Furthermore q = 0 if and
only if ̺ = 0. But here discr (̺(m,k, ·)) changes its sign as a function of m
and k.
We adopt here the notation in [29, Table 4] below, namely CD = (N≥2×
N≥1)\D if D ⊆ N≥2×N≥1 and CI = R\I if I ⊆ R. Thus, if (m,k) ∈ CD, let
µ− and µ+ two roots (eventually one, see [29, Remark 5.3]) of q, µ− ≤ µ+.
Besides, if discr (̺(m,k, ·)) > 0, then µ− < 0; whereas µ+ can take any sign.
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(F. Dobarro) Dipartimento di Matematica e Informatica, Università degli
Studi di Trieste, Via Valerio 12/b, I-34127 Trieste, Italy
E-mail address: dobarro@dmi.units.it
(B. Ünal) Department of Mathematics, Bilkent University, Bilkent, 06800
Ankara, Turkey
E-mail address: bulentunal@mail.com
http://arxiv.org/abs/hep-th/9905221
http://arxiv.org/abs/hep-th/9906064
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http://arxiv.org/abs/gr-qc/0105059
	1. Introduction
	2. Motivations and Main results
	3. The curvature relations - Sketch of the proofs
	4. The problem (Pb-sc) - Existence of solutions
	5. Proof of the Theorem ??
	6. Conclusions and future directions
	Appendix A.  
	References
