On prescribing scalar curvature on bun- dles and applicationsarXiv:2012.13272v2 [math.DG] 27 Dec...

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1

Prescribing scalar curvature on bundles

and applications

Leonardo Francisco Cavenaghi and Llohann Dallagnol Sperança

Abstract. In the first part of this paper we discuss the problem ofprescribing scalar curvature on the total space of some fiber bundlesvia direct applying some of the Kazdan–Warner results. We then studywhich functions are realizable as scalar curvature functions on the totalspace of several bundles over exotic manifolds. In the second part weprovide reasonable conditions to smooth functions on the total spacesof some fiber bundles to be realized as scalar curvature functions forsome Riemannian submersions metrics. We apply these last results tobundles over manifolds with positive Ricci curvature and tori symmetryand some Calabi–Yau manifolds.

Keywords. Fiber Bundles, Compact Structure Group, Exotic mani-folds, Prescribed Scalar Curvature, Tori symmetry, Calabi–Yau mani-folds.

1. Introduction

As examplified by The Bonnet–Meyers Theorem, The Differentiable SphereTheorem ([Bre10]), The Poincaré Conjecture ([Per02, Per03a, Per03b]), a wellknown application of geometry consists of its own usage to the understandingof manifolds as topological spaces. However, the converse question, given aclass of smooth manifolds, which are the admissible geometries on this class?remains unsolved for almost every manifold. Moreover, there are the veryinteresting examples of exotic spheres Σn, firstly introduced By J. Milnor in[Mil56]. These manifolds are homeomorphic to the standard sphere Sn butnot diffeomorphic to it. More generally, there are other examples of exoticmanifolds such as exotic tori, exotic R4, among others.

Part of this work was developed while the first named author was a PhD student atIME-USP financed by FAPESP 2017/24680-1. The second named author was financiallysupported by CNPq 131875/2016-7, 404266/2016-9.

http://arxiv.org/abs/2012.13272v3

2 Cavenaghi and Sperança

Concerning exotic spheres, Gromoll–Meyer constructed the first exoticsphere with a metric of non-negative sectional curvature (see [GM74]); Wil-helm constructed metrics of positive Ricci curvature and almost non-negativesectional curvature in every exotic sphere of dimension 7 (see [Wil01]); laterGrove–Ziller and Goette–Kerin–Shankar built metrics of non-negative sec-tional curvature on these examples (see [GZ00], [GKS20]). Grove–Verdiani–Wilking–Ziller showed that some exotic spheres do not support metrics ofnon-negative sectional curvature with lots of symmetries (see [GVWZ06]);and Grove–Verdiani–Ziller and [Dea11] built an exotic unitary tangent space(see [GVZ11]) with positive sectional curvature.

Nash ([Nas79]), Poor ([Poo75]), Searle–Wilhelm ([SW15]), Wraith, Joachimand Crowley ([Wra97, Wra07], [JW08] and [CW17a, CW17b]) proved the ex-istence of metrics of positive Ricci curvature on some exotic manifolds. In[CS18, CS19], the authors built metrics of positive Ricci curvature on severalexotic manifolds and bundles with fibers and/or bases of exotic manifolds.On the other hand, it is now known is there exists an exotic sphere with ametric of positive sectional curvature and Hitchin proved that there are ex-otic spheres that do not even admit metrics of positive scalar curvature (see[Hit74]).

Considering the aforementioned facts it is natural to ask

Question 1. To which extend does the smooth structure determines the ge-ometry?.

More particularly,

Question 2. Do exotic manifolds admit similar geometries to their “classi-cal” counterpart?

On the other hand, the problem on which functions can be realized asthe scalar curvature for some Riemannian metric on a closed connected man-ifold had great development in the seminal work of Kazdan and Warner (seee.g [KW75a, KW75b, KW75c]). In this paper we approach natural general-izations of this problem, namely:

(i) Given a fiber bundle F →֒ M → B with compact total space andstructure group and a smooth function f : M → R, can f be realizedas the scalar curvature of some Riemannian submersion metric on M?

(ii) Given a smooth manifold P with an effective G-action by a compact andconnected Lie group G and a smooth G-invariant function f : P → R,can f be realized as the scalar curvature of some G-invariant metric onP?

Our first result is obtained as direct consequence of the work of Kazdan–Warner, which unfortunately does not allow us to solve neither (i) nor (ii).

Theorem A. Let F →֒ Mπ→ B be a fiber bundle where M,F,B and the

structure group G are compact. Assume that:

1. A principal orbit of G on F has finite fundamental group,

Prescribing scalar curvature on bundles 3

2. F carries a G-invariant metric such that RicF reg/G ≥ 1.

Then

1. There is λ ∈ (0, 1], depending only on the geometry of the fiber, such

that any smooth function f : M → R satisfyingminp∈M f

maxp∈M f≤ λ is the

scalar curvature for some Riemannian metric on M , except maybe iff = constant ≥ 0;

2. If F has constant scalar curvature, then any smooth function f :M → Ris the scalar curvature of some Riemannian metric on M , except maybeif f = constant ≥ 0;

Theorem A is related to the Question 2 in the following manner. Eellsand Kuiper in [EK62] computed the number of 7 (respectively 15)-exoticspheres that are realized as total spaces of sphere bundles. Therefore, bysetting G = O(n+1), n = 7, 15, F = Sn, Theorem A enlightens Question 2:

Theorem B (Grou–Rigas). 16 (resp. 4.096) from the 28 (resp. 16.256)diffeomorphisms classes of the 7-dimensional (resp. 15)-exotic spheres, aresuch that any smooth real function f : Σ7 → R (resp. f : Σ15 → R) isthe scalar curvature for some Riemannian metric on Σ7 (resp. Σ15), exceptmaybe if f = constant ≥ 0.

Theorem A is a natural generalization of the following Theorem in [BG](and Theorema A in [Rig76]), from which Theorem B also follows:

Theorem C (Grou–Rigas). Let F →֒ Mπ→ B be a fiber bundle with

M,F,B and structure group G compact. Assume that:

1. F carries a G-invariant metric with positive sectional curvature;2. The action of G (dimG ≥ 2) on F has only one orbit type.

Then there exists λ ∈ (0, 1) such that any smooth function f : M → R

satisfying min fmax f ≤ λ is the scalar curvature for some Riemannian metric onM , except maybe if f = constant ≥ 0.

Theorem D below generalizes Theorem B to several classes of bundlesover exotic spheres and some connected sums. Moreover, in Section 2 wealso provide larges classes of bundles to which Theorem A implies similarconclusions to the ones of Theorem D.

We adopt the following convention: F →֒ P ×G F → B denotes theassociated bundle to the G-principal bundle G →֒ P → B with fiber F .

Theorem D. Let Σ7 and Σ8 be any homotopy spheres of dimensions 7 and8, respectively; Σ10 be any homotopy 10-sphere which bounds a spin mani-fold; Σ4m+1,Σ8m+5 be Kervaire spheres of dimensions 4m+1, 8m+5, respec-tively. Then, there are explicit G-manifolds P such that any smooth functionf (except possibly if f = constant ≥ 0) is the scalar curvature for someRiemannian metric on the total space of the following bundles

Sl →֒ P ×G Sl → Σl,


Sl →֒ P ×G Sl →M l#Σl,

S8r+k →֒ P ×G S8r+k → (M8r+k ×N5−k)#Σ8r+5,

for l = 7, 8, 10, 4m+1, k = 0, 1, where

(i) M7 is any 3-sphere bundle over S4,(ii) M8 is either a 3-sphere bundle over S5 or a 4-sphere bundle over S4,(iii) M10 =M8 × S2 with M8 as in item (ii),(iv) M10 is any 3-sphere bundle over S7, 5-sphere bundle over S5 or 6-sphere

bundle over S4,(v) M4m+1#Σ4m+1 where

(a) S2m →֒ M4m+1 → S2m+1 is the sphere bundle associated to any multi-ple1 of O(2m+1) →֒ O(2m+2) → S2m+1, the frame bundle of S2m+1

(b) CPm →֒ M4m+1 → S2m+1 is the CPm-bundle associated to any multipleof the bundle of unitary frames U(m) →֒ U(m+ 1) → S2m+1

(c) M4m+1 = U(m+2)SU(2)×U(m)(d) N5−k is any manifold with positive Ricci curvature and

(d.i) S4r+k−1 →֒ M8r+k → S4r+1 is the k-th suspension of the unitarytangent S4r−1 →֒ T1S4r+1 → S4r+1,

(d.ii) for k = 1, HPm →֒ M8m+1 → S4m+1 is the HPm-bundle associ-ated to any multiple of Sp(m) →֒ Sp(m+ 1) → S4m+1

(d.iii) for k = 0, M = Sp(m+2)Sp(2)×Sp(m) ,

(d.iv) for k = 1, M =M8m+1 is as in item (v).

In addition, there is λ ∈ (0, 1) such that if f is also smooth as real valuedfunction on (except maybe if f = constant ≥ 0)

(i′) P ×G Σl,(ii′) P ×G M l#Σl,(iii′) P ×G (M8r+k ×N5−k)#Σ8r+5

and it satisfies min fmax f ≤ λ, then f is also the scalar curvature of some Rie-mannian metric on the total space of the following bundles

Σl →֒ P ×G Σl → Σl,

M l#Σl →֒ P ×G Ml#Σl →M l#Σl,

(M8r+k×N5−k)#Σ8r+5 →֒ P×G(M8r+k×N5−k)#Σ8r+5 → (M8r+k×N5−k)#Σ8r+5.

We also could solve the problems (i) and (ii) for some classes of Rie-mannian submersions, Theorems E, F. As hypotheses we assume specificconstraints on the geometry of the fibers that are different in nature to theones in Theorem A. These are obtained via a straightforward application ofVariational Methods for PDE’s.

Let π : (F k, gF ) →֒ (Mn+k, g) → (Bn, h) be a Riemannian submersion.Recall that a smooth function f :M → R is said to be basic if it is constant

1That is, a bundle whose transition function α : Sn−1 → G is a multiple of τ2m : S2m →O(2m + 1), τC

m: S2m → U(m) or τH

m: S4m+2 → Sp(m), for G = O(2m), U(m + 1)

or Sp(m), the transition functions of the orthonormal frame bundle and its reductions,respectively.


along the fibers of, or equivalently, if ∇f is basic, meaning that it is horizontaland projectable.

Define the constants

bk :=k + 1

8k, ck :=

(k + 1)2

8(k − 1)k, θ :=

2(k − 1)

k + 1, k ≥ 2.

Theorem E. Let π : (F k, gF ) →֒ (Mn+k, g) → (Bn, h) be a Riemanniansubmersion where M is orientable closed and connected and F is compactand connected. Assume that:

1. scalF = c ≤ 0,2. (F, gF ) is a minimal submanifold of (M, g),3. If A denotes the Gray–O’Neill tensor of the submersion π, then

maxM

3

∑

i,j

|Aeiej |2 − 2

∑

i,r

|A∗eivr|2

≤ 0,

where {ei} is a g-orthonormal basis for H and {vi} is a g-orthonormalbasis for V .

Then any smooth basic function f ∈ C∞(M ;R) satisfying

λ12 + ǫ

− bk maxM

(f − scalg + c) + ckc (vol(B))2/θ−1

+ bk minM

δA > 0, (1)

for some ǫ > 0, where λ1 denotes the first positive eigenvalue of −∆B, is thescalar curvature for some Riemannian submersion metric on M .

If M is isometric to a product, once its fibers are totally geodesic sub-manifolds and A ≡ 0 one has

Theorem F. Let M = (Bn × F k, gB × gF ) be a closed and connected Rie-mannian manifold such that B is orientable. Suppose that scalF = c ≤ 0.Then any smooth basic function f ∈ C∞(M ;R) satisfying

λ12 + ǫ

− bk maxB

(f − scalB) + ckc (vol(B))2/θ−1

> 0, (2)

for some ǫ > 0, where λ1 denotes the first positive eigenvalue of −∆B, is thescalar curvature for some Riemannian submersion metric on M .

Theorems E and F are natural generalizations of the problem of prescrib-ing constant scalar curvature via warped products, largely studied in [DD87],[EYTK96]. The hypothesis of f being basic is consequence of the techniquessince we exploit General Vertical Warpings of the initial Riemannian Sub-mersion metric (see the Appedix A for definitions and formulae).

Under the hypothesis of positive Ricci curvature on B and assumingthat M is isometric to a product, one derives a different condition:


Corollary G. Let Mn+k = (Bn × F k, gB × gF ) be a closed and connectedRiemannian manifold such that B is orientable. Assume that Ric(gB) ≥ (n−1) and that scalF = c ≤ 0. Then any smooth basic function f : M → R suchthat

n

(8k

(2 + ǫ)(k + 1)+ (n− 1)

)> max

Bf +

ckbkc (vol(B))

2/θ−1, (3)

for some ǫ > 0, is the scalar curvature function for some Riemannian sub-mersion metric on M .

Our approach to prove Theorems E, F and Corollary G follows closelythe work [EYTK96]. In this, the authors deal essentially with the case inwhich the manifold F has constant scalar curvature, scalF = c, consideringseparately the cases c > 0, c < 0, c = 0. In this sense our results are naturalgeneralizations of theirs.

In addition, we justify the relevance of our results by the fact that thepresented hypotheses are easily verifiable, for instance:

Theorem H. For each n ≥ 1, there exist infinitely many diffeomorphismtypes of n-tori bundles

T n →֒ (M, g) → B, (4)

over closed simply-connected smooth (n+ 4)-manifolds B that realizes infin-itely many spin and non-spin diffeomorphism types, such that any smoothfunction f on M that is constant along the fibers T n and satisfies

8n(n+ 4)

(2 + ǫ)(n+ 1)> max

M(f − scalg),

for some ǫ > 0, is the scalar curvature for some Riemannian submersionmetric on M .

In a similar manner we were able to prove the following:

Theorem I. Let X,Y be compact Kähler manifolds and π : X → Y be aholomorphic fiber bundle with base Y and fiber F (with real dimension k) ofCalabi–Yau manifolds. If either b1(F ) = 0 or F is a torus and b1(Y ) = 0,

then there is a Riemannian covering map p̃ : Ỹ → Y with fibers consisting ofa finite number of points such that any smooth basic function f : p∗(X) → Rsatisfying (

8k

k + 1

)(λ1

2 + ǫ

)> max

p∗(X)f,

for some ǫ > 0, is the scalar curvature for some Riemannian submersion

metric on the total space of the pullback bundle F →֒ p∗(X) → Ỹ , whereλ1 denotes the first positive eigenvalue of the minus Laplace operator on themetric p∗(gY ), where gY is the fixed Calabi–Yau metric on Y .


Structure of the article. In Section 2 we prove Theorems A and D. Section3 is divided in two subsections separating the proofs of Theorems E, F andCorollary G. In subsection 3.1 we prove Theorem F to well motivate theproof of Theorem E on the subsequent subsection. Finally, in subsection 3.3.1we prove Theorems H and I. We also decided to include an appendix withsome needed formulae for the results on the paper. These are extracted from[GW09] up to errata.

2. Prescribing scalar curvature on fiber bundles with

compact structure group and applications

In this section we prove Theorems A and D.

To the proof of Theorem A we make a simple and straight applicationof the following result by Kazdan-Warner:

Theorem 2.1 (Kazdan-Warner). Let (M, g) be a compact manifold. De-note by scalg the scalar curvature of g. Let f ∈ C∞(M) be a smooth functionon M . If there exists a constant c > 0 such that

min cf < scalg(p) < max cf, ∀p ∈M, (5)

then, there exists a Riemannian metric g̃ on M such that

scalg̃ = f.

Proof. See [KW75a, Theorem A]. �

Proof of Theorem A. According to the hypothesis 2 and Theorem A in [SW15],F carries a G-invariant Riemannian metric gF with positive Ricci curvature.Given any Riemannian metric gB on B, consider on M the unique Riemann-ian submersion metric g such that its fibers are totally geodesic submanifoldsisometric to (F, gF ) (see [GW09, Proposition 2.7.1, p. 97]). We reinforce thatfact that, to the case of principal bundles, this metric can be madeG-invariantby defining a Kaluza-Klein G-invariant metric on the total space (see [CS18,Proposition 5.1, p. 27]).

Fix p ∈M and let {ei}ki=1 be an orthonormal base forHp and {ej}nj=k+1

be an orthonormal base for Vp. Note that {ei}ki=1 ∪ {e−tej}nj=k+1 is a gt-

orthonormal base to TpM. Using the formulae given by Proposition A.3 weobtain an expression for the scalar curvature of gt:

scalt(p) = scaltH(p) + 2e2t

∑

i,j

|A∗eiej |2g + e

−2tscalF (p).

Moreover,

scaltH(p) = scalB(p)(1− e

2t) + e2tscalHg (p).


Denote by st := minp∈M scalt(p) and by St := maxp∈M scalt(p). Thennote that

λ := limt→−∞

stSt

=minp∈M scalFmaxp∈M scalF

≤ 1.

Therefore, for each λ′ < λ one can find t > 0 such that

λ′ =min f

max f<s−tS−t

≤ 1,

so the result follows from Theorem 2.1. Indeed, note that S−t > 0, therefore,s−t > λ

′S−t. Therefore,

S−t ≥ scalt ≥ s−t >min f

max fS−t.

So choose c = S−tmax f . One gets,

cmax f > scalt > cmin f. (6)

The case when λ = 1 is obvious and when f = constant < 0 it follows fromTheorem C in [KW75a]. �

We now prove Theorem D and generalize it to several bundles. Theseconstructions are deeply relied on the ones in [CS19].

Consider a compact connected principal bundle G →֒ P → M with aprincipal action •. Assume that there is another action on P , which we denoteby ⋆, that commutes with •. This makes P a G×G-manifold. If one assumesthat ⋆ is free, one gets a ⋆-diagram of bundles:

G

•

G⋆

P

π

��

π′// M ′

M

(7)

In (7), M is the quotient of P by the •-action and M ′ is the quotient of Pby the ⋆-action.

Once • and ⋆ commute, • descends to an action on M ′ and ⋆ descendsto an action on M . We denote the orbit spaces of these actions by M ′/•and M/⋆, respectively. Corollary 5.2 in [CS18] implies that one can choose aRiemannian metric g′ on M ′ such that the orbit spaces M/⋆ and M ′/• areisometric, as metric spaces. Furthermore, it can be shown that the orbits ofthe ⋆-action on M have finite fundamental group if, and only if, the orbits ofthe •-action does [CS18, Theorem 6.4]. This implies that if the G-manifoldM satisfies the hypotheses of Theorem A, then M ′ also does.

The idea on considering diagrams like (7) is that M and M ′ can betaken as homeomorphic manifolds that are not diffeomorphic to each other.Moreover, one can consider the following associated bundles


1. The associated bundle M →֒ P ×G M →M to π : P →M,2. The associated bundle M →֒ P ×G M →M ′ to π′ : P →M ′,

On Theorem D the manifolds M are always standard spheres with metricsof constant sectional curvature. Therefore, one obtains the following:

Theorem 2.2. Let M be a standard sphere with a metric of constant sec-tional curvature. Then any smooth function f : P ×G M → R (except maybeif f = constant ≥ 0) is the scalar curvature of some Riemannian metric onthe total space of M →֒ P×GM →M if, and only if, it is the scalar curvatureof some Riemannian metric on the total space of M →֒ P ×G M →M ′.

On the other hand, any smooth function f : P ×G M → R can de-fines a function (possible not smooth) on f : P ×G M ′ → R after fixing ahomeomorphism. So we can conclude the following:

Theorem 2.3. Any smooth function f : P ×G M → R that is also smoothas a real valued function on P ×G M ′ (except maybe if f = constant ≥ 0)and satisfies

minP×GM ′ f

maxP×GM ′ f≤

minM ′ scalg′

maxM ′ scalg′,

is the scalar curvature of some Riemannian metrics on P×GM and P×GM ′.

Theorem 2.2 and 2.3 finish the proof of Theorem D since the mentionedexamples are constructed by the means of cross diagrams such as (7) in [CS18,Theorem A].

3. Prescribing scalar curvature on some Riemannian

submersion and applications

3.1. Proof of Theorem F

Once the proof of Theorem F is written, the proof of Theorem E followssimilarly after small modifications. Let Bn be a closed smooth manifold andF be a compact manifold with constant scalar curvature c ≤ 0 and f :B×F → R be a smooth basic function. Theorem F is proved using VariationalMethods to ensure the existence of positive solutions for the following PDE:

∆Bu+k + 1

4k(f − scalB)u−

k + 1

4ku

k−3k+1 c = 0, (8)

Indeed, equation (8) is obtained from Lemma 1.

Lemma 1. Let (B, gB) and (F, gF ) be Riemannian manifolds and φ : B → Rbe a smooth function. Denote by g̃ the warped metric on B×e2φ F . Then thescalar curvature of g̃ has the following expression

s̃cal = scalB + e−2φscalF − k(k − 1)|∇φ|

2 − 2k|∇φ|2 − 2k∆Bφ. (9)


Note then that since scalF = c ≤ 0, then equation (9) implies that s̃calis basic.

Given a basic function f ∈ C∞(B × F ;R), consider the PDE

scalB + e−2φscalF − f = k(k − 1)|∇φ|

2 + 2k{|∇φ|2 +∆Bφ

}. (10)

The solution φ is such that f is the scalar curvature of the warped metricg̃ = gB + e

2φgF .

To obtain equation (8) we note that if φ = logϕ then ∇φ =1

ϕ∇ϕ.

Hence,

∆Bφ = −1

ϕ2|∇ϕ|2 +

1

ϕ∆Bϕ.

Once, e−2φ = ϕ−2 then

k(k − 1)|∇φ|2 + 2ke−2φ{|∇φ|2 +∆Bφ

}

=k(k − 1)

ϕ2|∇ϕ|2+

2k

{1

ϕ2|∇ϕ|2 −

1

ϕ2|∇ϕ|2 +

1

ϕ∆Bϕ

}.

Therefore,

scalB + ϕ−2scalF − f =

k(k − 1)

ϕ2|∇ϕ|2 +

2k

ϕ∆Bϕ. (11)

By changing variable ϕ = u2

k+1 we obtain

∇ϕ =2

k + 1u

1−kk+1 ∇u, ∆Bϕ =

(2(1− k)

(k + 1)2u−

2kk+1 |∇u|2 +

2

k + 1u

1−kk+1 ∆Bu

).

Consequently, substituting the above term on equation (11) we obtain

scalB + u− 4

k+1 scalF − f = 4k(k − 1)

(k + 1)2u−

4k+1 u

2(1−k)k+1 |∇u|2+

2ku−2k+1

(2(1− k)

(k + 1)2u−

2kk+1 |∇u|2 +

2

k + 1u

1−kk+1 ∆Bu

)

scalB + u−4k+1 scalF − f = 4

k(k − 1)

(k + 1)2u−2|∇u|2

+ 4k(1− k)

(k + 1)2u−2|∇u|2 +

4k

k + 1u−1∆Bu

=4k

k + 1u−1∆Bu.

To show that PDE (8) has a positive solution, assume that f, scalBare continuous functions. Denote by H1(B) the Sobolev Space W 1,2(B) and


define the following functional J : H1(B) → R by

J(u) =1

2

∫

B

|∇u|2−

(k + 1

8k

)∫

B

(f − scalB)u2+

(k + 1)2

8(k − 1)k

∫

B

cu2(k−1)/(k+1),

∀u ∈ H1(B). (12)

Setting

bk :=k + 1

8k, ck :=

(k + 1)2

8(k − 1)k, θ := 2

k − 1

k + 1,

the functional (12) can be written as

J(u) =1

2

∫

B

|∇u|2 − bk

∫

B

(f − scalB)u2 + ck

∫

B

cuθ. (13)

Remark. Observe that 0 < θ ≤ 2.

We obtain the desired solution u to equation (8) as a minimum for J onthe set M := {u ∈ H1(B) : u ≥ ǫ0,

∫Buθ ≥ 1}. Precisely, Theorem F follows

from the following Lemma.

Lemma 2. Let ǫ0 > 0 be arbitrarily small and define M := {u ∈ H1(B) :u ≥ ǫ0,

∫Buθ ≥ 1}. Assume that f, scalB are continuous functions and that

c ≤ 0. Let λ1 be the first positive eigenvalue of −∆B. Assume that there isǫ > 0 such that the following inequality holds

α :=λ1

2 + ǫ− bk max

B(f − scalB) + ckc (vol(B))

2/θ−1 > 0. (14)

Then

1. There is a constant c0 > 0 such that J∣∣M> c0,

2. J∣∣M

is coercive,3. M is weakly closed,4. J

∣∣M

is weakly lower semi-continuous.

Remark. The necessity of ǫ on the hypothesis of Lemma 2 is justified toguarantee that J

∣∣M

is coercive.

We now prove Lemma 2.

Proof of Lemma 2. 1. It is easy to see that M 6= ∅. Therefore, let u ∈ M.Recall the classical Poincaré inequality

∫

B

|∇u|2 ≥ λ1

∫

B

u2. (15)

According to the continuity of f, scalB we have

J(u) ≥1

2

∫

B

|∇u|2 − bk maxB

(f − scalB)

∫

B

u2 + ckc

∫

B

uθ.


Once B is compact and θ ≤ 2 the Hölder inequality implies thatthere is a continuous immersion L2(B) →֒ Lθ(B). Moreover,

(∫

B

uθ)2/θ

≤ (vol(B))2/θ−1∫

B

u2.

Since∫uθ ≥ 1 and θ ≤ 2 one has

∫

B

uθ ≤ (vol(B))2/θ−1∫

B

u2. (16)

Since we assumed that c ≤ 0 one has

ckc

∫

B

uθ ≥ ckc (vol(B))2/θ−1

∫

B

u2. (17)

Therefore,

J(u) ≥1

2

∫

B

|∇u|2 − bk maxB

(f − scalB)

∫

B

u2 + ckc (vol(B))2/θ−1

∫

B

u2.

(18)According to equation (15)

J(u) ≥

(λ12

− bk maxB


)∫

B

u2. (19)

By hypothesis,

λ12

− bk maxB


> 0.

According to equation (16) and the definition of M, there is c0 > 0 suchthat J(u) ≥ c0, ∀u ∈ M.

2. To prove coerciveness we observe that

J(u) ≥1

2

∫

B

|∇u|2 −(bk max

B(f − scalB)− ckc (vol(B))

2/θ−1)∫

B

u2. (20)

From equation (14) we get

J(u) >1

2

∫

B

|∇u|2 −λ1

2 + ǫ

∫

B

u2. (21)

According to the Poincaré inquality(15)

J(u) > λ1

(1

2−

1

2 + ǫ

)∫

B

|∇u|2 (22)

J(u) >λ1ǫ

2(2 + ǫ)

∫

B

|∇u|2, (23)

from where it follows that J is coercive.3. This follows from the compact embedding of H1(B) into L2(B), that

admits a continuous immersion into Lθ(B). More precisely, if {un} ⊂ M


weakly converges to u ∈ H1(B), then {un} strongly converges to u inLθ(B). Therefore,

1 ≤ limn→∞

∫

B

uθn =

∫

B

uθ.

Moreover, there exists a subsequence {unj} almost everywhere pointwiseconverging to u. Hence, u = limj→∞ unj ≥ ǫ0, what finishes the proof.

4. To see that J is weakly lower semi-continuous, note that according toKondrachov’s Theorem, one has that H1(B) compactly embeds intoL2(B). Let {un} ⊂ M weakly converging to u ∈ H1(B). Then u ∈ Maccording to the previous item and un → u ∈ L2(B). Once B is compactand and scalB, scalF , f are continuous functions, one has∫

B

|∇u|2 ≤ lim infn→∞

∫

B

|∇un|2.

Moreover,

lim infn→∞

J(un) = lim infn→∞

(∫

M

|∇un|2 − bk

∫

B

(f − scalB)u2n + ck

∫

B

cuθn

)

≥

∫

M

|∇u|2 − bk

∫

B

(f − scalB)u2 + ck

∫

B

cuθ = J(u).

�

Proof of Theorem F. According to Lemma 2, it follows that J∣∣M

has a min-

imal point on M. Note, by substituting u = ǫ0 on the expression (10), that fmust coincide with the scalar curvature of a canonical deformation of the ini-tial metric (see the Appendix A). Hence, if this is not the case, the obtainedcritical point u satisfies u 6≡ ǫ0. Therefore, given any function v ∈ H1(B),one has that

0 =d

dt

∣∣∣t=0

J(u+ tv) =

∫

B

(∆Bu+

k + 1

4k(f − scalB)u −

k + 1

4kcuk−3/k+1

)v.

(24)Hence, it follows that u is a weak solution to PDE (8). Moreover, Assumingthat f, scalB are smooth functions, according to classical theory to the regu-larity of elliptic PDE’s (see [Aub98, Theorem 3.58, p. 87]), the solution u issmooth indeed. �

We finish this subsection by proving Lemma 1.

Proof of Lemma 1. Consider a gB-orthonormal basis {wj} and a gF -orthonormal{vi}. Then the set

{wj} ∪{e−φvi

}

defines a g̃-orthonormal basis to T (B × F ) .According to the equations on Proposition A.2 one has

K̃(e−φvi, e−φvj) = e

−2φKF (vi, vj)− |∇φ|2,

K̃(wj , e−φvi) = −dφ(wj)

2 −Hess φ(wj , wj).


Assume that {wj} is such that dφ(w1) = |∇φ|, dφ(wj) = 0, ∀j ≥ 2. Then thescalar curvature of g̃ is given by

s̃cal = scalB + e−2φscalF − k(k − 1)|∇φ|

2 − 2k{|∇φ|2 +∆Bφ

}.

�

3.2. Proof of Theorem E

We now prove Theorem E. In what follows we consider General VerticalWarpings by smooth and basic functions (see the Appendix A for furtherdetails).

Lemma 3. Let F →֒ (M, g)π→ be a Riemnnian submersion. Let g̃ be a

general vertical warping metric on M via the function u4

k+1 where u :M → Ris a basic function, i.e, it is constant along F . Then the scalar curvature ofg̃ is given by

s̃cal = scalg −4k

k + 1u−1∆Bu+ (4 + 2(k − 1))

2

k + 1u−1du(H)

+(u−

4k+1 − 1

)scalF +

(1− u

4k+1

)3

∑

i,j

|Aeiej |2 − 2

∑

i,r

|A∗eivr|2

.

(25)

Proof. According to the formulae on Proposition A.1, let Ti = e−φvi, vi ∈ V

e X ∈ H. Then,

K̃(e−φv1, e−φv2) = (e

−2φ−1)KF (v1, v2)+Kg(v1, v2)−|∇φ|2+dφ(σ(v1, v1)+σ(v2, v2)),

K̃(X, e−φv) = Kg(X, v)−(1−e2φ)|A∗Xv|

2−Hess φ(X,X)−dφ(X)2+2dφ(X)g(SXv, v).

Take a g-orthonormal {ei} to H. Then,∑

i,j

K̃(ei, ej) = (1− e2φ)scalB + e

2φscalH,

∑

r,s

K̃(e−φvr, e−φvs) = (e

−2φ−1)scalF +scalV−k(k−1)|∇φ|2+2(k−1)dφ(H)

2∑

i,r

K̃(e−φvr, ei) = 2∑

i,r

K(ei, vr)− 2(1− e2φ)

∑

i,r

|A∗eivr|2

− 2k(∆Bφ+ |∇φ|

2)+ 4

∑

r,i

dφ(ei)g(Seivr, vr).

Assume that dφ(e1) = |∇φ|, i.e, e1 =∇φ|∇φ| , dϕ(ei) = 0, i ≥ 2. One has,

4∑

r,i

dφ(ei)g(∇eivr, vr) = 4tr S∇ϕ = 4dϕ(H).


So we conclude that

s̃cal = scalg−2(1−e2φ)

∑

i,r

|A∗eivr|2+(1−e2φ)(scalB−scal

H)+(e−2φ−1)scalF

− k(k − 1)|∇φ|2 − 2k(∆Bφ+ |∇φ|2) + (4 + 2(k − 1)) dφ(H). (26)

Once equation (26) only differs to equation (10) bt the terms

2(1− e2φ)∑

i,r

|A∗eivr|2 + (4 + 2(k − 1)) dφ(H),

by introducing the change of variables φ = logϕ and ϕ = u2

k+1 we concludethat

s̃cal = scalg −4k

k + 1u−1∆Bu+ (4 + 2(k − 1))

2

k + 1u−1du(H)

+(u−

4k+1 − 1

)scalF +

(1− u

4k+1

)scalB − scalH − 2

∑

i,r

|A∗eivr|2

.

(27)

�

Lemma 4. Let (F, gF ) →֒ (M, g) → (B, h) be a Riemannian submersion withM closed connected oriented and (F, gF ) minimal. If ∆M denotes the Laplaceoperator on the metric g, then the restriction of ∆M to basic functions definesa strongly elliptic operator.

Proof. Let u :M → R be a basic function. Once M is compact, it is possibleto choose a collection of open sets {Un} ⊂ M trivializing the submersion πin the following sense (see [Her60])

Un = Bn × F, Bn ⊂ B.

Once u is a basic function,

u|Un(p) = u(b, f) = u(b, f′), ∀f, f ′ ∈ F, ∀b ∈ Bn.

If {ψn} denotes a unity partition on {Bn}, then u =∑

n ψnu and there is awell defined injection

ζ : H1(M) → H1(B)

u 7→ v,

where v =∑

n vn, vn(b) = ψnu(b, f), ∀b ∈ Bn.Since (F, gF ) is minimal, by identifying ζu = u one has that ∆Bu =

∆Mu since ∆Bu = ∆Mu − du(H) and H ≡ 0 (see [GW09, Section 2.1.4,p.53]). Hence, once ∆B is strongly elliptic the result follows. �

Proof of Theorem E. Define

δA := 3∑

i,j

|Aeiej|2 − 2

∑

i,j

|A∗eivr|2, (28)


where {ei} is a g-orthonormal basis to H and {vr} is a g-orthonormal basisto V . According to Lemma 4, given a basic function f : M → R, we shallstudy the following elliptc problem(k + 1

4k

)u(f−scalg) = −∆Bu+

(k + 1

4k

){(u

k−3k+1 − u

)c+

(u− u

k+5k+1

)δA

},

(29)where 0 ≥ c = scalF .

Recalling that θ = 2(k−1)k+1 and thatk+14k = 2bk, define γ :=

2k+6k+1 . Let

ǫ0 > 0 be arbitraly small and define

Mb := {u ∈ H1(M) : u ≥ ǫ0, u is basic and

∫

B

uθ ≥ 1}. (30)

Consider the following functional J defined on Mb :

J(u) :=1

2

∫

B

|∇u|2

+

(k + 1

4k

)∫

B

{−u2

2(f − scalg) +

(k + 1

2(k − 1)u

2(k−1)k+1 −

1

2u2

)c

}

+

(k + 1

4k

)∫

B

(u2

2−

k + 1

2k + 6u

2k+6k+1

)δA. (31)

We conveniently rewrite J as

J(u) =1

2

∫

B

|∇u|2

+ 2bk

∫

B

{(scalg + δA− c− f)

u2

2+ cθ−1uθ − δAγ−1uγ

}(32)

It is worth recalling that we are assuming that

1. maxB δA ≤ 0,2. c ≤ 0.

Therefore,

J(u) ≥1

2

∫

B

|∇u|2 + bk(minM

scalg −maxM

f − c+minM

δA) ∫

B

u2

+ 2bkθ−1c (vol(B))2/θ−1

∫

B

u2 − 2bk maxM

δAγ−1∫

B

uγ , (33)

where the penultimate term follows from equation (16).

According to the Hölder inequality applied to the continuous immersionLγ(B) →֒ L2(B) one has

(∫

B

u2) γ

2

≤ vol(B)γ2 −1

∫

B

uγ . (34)


Exploiting the equation (33), the Poincaré inequality and the equation (34)imply that

J(u) ≥{λ12

− bk maxM

(f − scalg + c) + ckc (vol(B))2/θ−1

+ bk minM

δA

}∫

B

u2

− 2bk{γ−1vol(B)1−

γ2 max

MδA

}(∫

B

u2) γ

2

. (35)

The proof of Theorem E is finished using equation (35) and a straightforwardadaptation of Lemma 2 once realized that

−2bkγ−1vol(B)1−

γ2 max

MδA ≥ 0.

�

Proof of Corollary G. The proof of Corollary G follows easily once under thehypothesis Ric(B) ≥ (n− 1), the eigenvalue λ1 of −∆B satisfies

λ1 ≥ n (36)

with equality if, and only if, B is isometric to the unit sphere (see [OBA62]).Therefore, once Ric(B) ≥ (n− 1) implies that scalgB ≥ n(n− 1), it fol-

lows from equation (14) that a sufficient condition to the existence of solutionis given by

n

2 + ǫ> bk max

Bf − bkn(n− 1) + ckc (vol(B))

2/θ−1 , ǫ > 0. (37)

In particular, if c = 0 it reduces to

8kn

(2 + ǫ)(k + 1)+ n(n− 1) > max

Bf, (38)

what finishes the proof. �

3.3. Examples

3.3.1. Tori bundles and generalizations. We proceed by sketching someapplications of Theorem E. The following constructions are obtained fromSection 5.3 on [AB15].

Let (B, gB) and (F, gF ) be compact connected Riemannian manifolds.Fix b ∈ B and assume that there is Lie group homomorphism ρ : π1(B, b) →

Iso(gB). Let π̃ : B̃ → B be the projection of the universal covering of B. It

is possible to define an action of π1(B, b) on M̃ := B̃ × F in the followingmanner

[α] · (̃b, f) := (̃b · [α], ρ(α−1)f), (39)

where b̃·[α] denotes the Deck transformation associated to α applied to b̃ ∈ B̃.Denote by M the orbit space according to the action (39) and let Π :

M̃ → M be the quotient map projection. Then there is a well defined fiberbundle

F →֒Mπ→ B


with projection π defined as

π(Π(̃b, f)) := π̃(̃b).

The structure group of π is precisely ρ(π1(B, b)). The total space M carriesa Riemannian submersion metric g such that its fibers are totally geodesic.Moreover, the horizontal distribution is integrable, meaning that A ≡ 0.

The previous construction furnishes lots of bundles such that the onlyrestriction to a smooth function on its total space to be the scalar curvaturefunction to some Riemannian submersion metric is an upper bound, depend-ing only on the dimension of the base and the fiber, to the quantity f − scalg.More precisely, the following Theorem of Corro–Galaz-Garćıa ([CGG16, The-orem A]) provides infinitely many examples:

Theorem 3.1 (Corro–Galaz-Garćıa). For each integer n ≥ 1, the follow-ing hold:

(i) There exist infinitely many diffeomorphism types of closed simply-connectedsmooth (n+4)-manifolds B with a T n-invariant Riemannian metric withpositive Ricci curvature.

(ii) The manifolds B realize infinitely many spin and non-spin diffeomor-phism types.

(iii) Each manifold B supports a smooth, effective action of a torus T n+2

extending the isometric T n-action in item (i).

We can then simply take F = T n and B as in Theorem 3.1 to obtainthe following results as applications of Theorem E and Corollary G:

Theorem 3.2. For each integer n ≥ 1, there exist infinitely many diffeomor-phism types of closed simply-connected smooth (n+4)-manifolds B, realizinginfinitely many spin and non-spin diffeomorphism types, such that any smoothfunction f on the total space of the following bundles that is constant alongthe fibers

T n →֒ (M, g) → B (40)

and satisfies8n(n+ 4)

(2 + ǫ)(n+ 1)> max

M(f − scalg),

for some ǫ > 0, is the scalar curvature for some Riemannian submersionmetric on M .

3.3.2. Some Calabi–Yau bundles. We proceed by furnishing some in-formation about how to prescribe scalar curvature functions on Calabi–Yaubundles. We make use of the following result by Tosatti–Zhang ([TZ14, The-orem 4.1, p. 912]):

Theorem 3.3 (Tosatti–Zhang). Let X,Y be compact Kähler manifoldsand π : X → Y be a holomorphic fiber bundle with base Y and fiber F ofCalabi–Yau manifolds. If either b1(F ) = 0 or F is a torus and b1(Y ) = 0,

then there is a finite unramified covering p : Ỹ → Y such that the pullbackbundle to Ỹ is holomorphically trivial.


We make some remarks: in this context, the term unramified covering

map can be reduced to that p̃ : Ỹ → Y is a smooth covering map. Thehypothesis of p̃ being finite implies that this smooth covering has sheetsconsisting of a finite number of points. For the sake of completeness we givea precise definition to it: there exists a (finite) covering {Vi} of Y by affine

varieties such that p̃∣∣∣p̃−1(Vi)

: p̃−1(Vi) → V are finite maps, in the sense that

C[p̃−1(Vi)] is a finite C[Vi]-algebra.

The idea is the following: let π̃ : F →֒ p∗(X) → Ỹ be the pullback

bundle of π to Ỹ . Denote by p̃∗(gY ) the pullback of the Calabi–Yau metric

gY of Y to Ỹ via p̃. If gF is a Calabi–Yau metric on F , then consider onp∗(X) the unique Riemannian submersion metric on π̃ with totally geodesicfibers (as in the proof of Theorem A). In this case this is just the productmetric. Now we apply Theorem F to conclude that:

Theorem 3.4. Let X,Y be compact Kähler manifolds and π : X → Y be aholomorphic fiber bundle with base Y and fiber F (with real dimension k) ofCalabi–Yau manifolds. If either b1(F ) = 0 or F is a torus and b1(Y ) = 0,

then there is a Riemannian covering map p̃ : Ỹ → Y with fibers consisting ofa finite number of points such that any smooth basic function f : p∗(X) → Rsatisfying (

8k

k + 1

)(λ1

2 + ǫ

)> max

p∗(X)f,

for some ǫ > 0, is the scalar curvature for some Riemannian submersion

metric on F →֒ p∗(X) → Ỹ , where λ1 denotes the first positive eigenvalueof the minus Laplace operator on the metric p∗(gY ), where gY is the fixedCalabi–Yau metric on Y .

Acknowledgments

The authors are thankful to Prof. Marcus Marrocos for useful comments onthe analytical part of this paper and to Prof. Marcos Alexandrino for pointingout the procedure presented in subsection 3.3.1.

Appendix A. General Vertical Warpings

Let F →֒ (M, g)π→ B be a Riemannian submersion. It is possible to obtain

new Riemannian submersions from π by introducing some metric deforma-tions changing the g on vertical directions. More precisely, let φ :M → R bea smooth function. We define a new metric g̃ on π in the following way

g̃(E,F ) := g(EH, FH

)+ e2φg

(EV , FV

), ∀E,F ∈ TpM, ∀p ∈M.

Since this metric preserves the horizontal distribution, π : (M, g̃) → B re-

mains a Riemannian submersion. Denote by ∇̃, R̃ the Levi-Civita connectionand the Riemann curvature tensor of g̃.


Definition 1. A Riemannian submersion π : (M, g̃) → B such that φ isconstant along the fibers (or equivalently, ∇φ is basic) is called general verticalwarping.

Proposition A.1. Let F →֒ (M, g)π→ B be a Riemannian submersion and

g̃ be a general vertical warping of g with respect to the function e2φ, φ ∈C∞(M ;R). Fix p ∈M . Let X,Y ∈ Hp, V, Vi ∈ Vp, i ∈ {1, 2}. If g(V1, V2) =

0, the following formulae hold true for the sectional curvature K̃ of g̃:

K̃(X,Y ) = (1− e2φ)KB(X,Y ) + e2φK(X,Y ),

K̃(V1, V2) = (e2φ − e4φ)KF (V1, V2) + e

4φKg(V1, V2)

− e4φ|V1|2|V2|

2|∇φ|2 + e4φdφ(σ(V1 , V1))|V2|2 + e4φdφ(σ(V2, V2))|V1|

2,

K̃(X,V ) = Kg(X,V )e2φ − e2φ

(1− e2φ

)|A∗XV |

2−{Hess φ(X,X) + dφ(X)2

}e2φ|V |2 + 2e2φdφ(X)g(SXV, V ).

Proof. See [GW09, Section 2.1.3, p. 52] �

A.1. Warped products

Let (B, gB) e (F, gF ) be Riemannian manifolds. Assume that π : M = B ×F → B is a trivial Riemannian submersion with the product metric on M .Let φ : B → R be a smooth function. Then the metric g̃ := gB × e2φgFis an example of general vertical warping known as warped product. TheRiemannian manifold (M, g̃) is called warped product of B and F , beingusually denoted by B ×e2φ F .

On warped products the Gray–O’Neill tensor A vanishes identically.Moreover, the second fundamental form of the fibers satisfy

σ̃ (T1, T2) = −e2φg (T1, T2)∇φ = −g̃ (T1, T2) . (41)

The following formulae for the sectional curvature of a warped productholds true

Proposition A.2. Consider a warped product π :(B × F, g̃ = gB × e2φgF

)→

M and let K̃ be the sectional curvature of g̃. Fix (p, f) ∈ M × F and takeX,Y ∈ Hp, V, Vi ∈ Vp, i ∈ {1, 2}. Then

K̃(X,Y ) = KB(X,Y ); (42)

K̃(V1, V2) = e2φ

{KF (V1, V2)− e

2φ|∇φ|2(|V1|

2|V2|2 − 〈V1, V2〉

)2}; (43)

K̃(X,V ) = −e2φ|V |2(〈∇φ,X〉2 +Hess φ(X,X)

), (44)

(45)


A.2. Canonical deformation

Another very simple case of general vertical warping happens when one takesφ(p) = t ∈ R, ∀p ∈M . The metric g̃ is usually known as canonical variation

of g. Let g̃ = g∣∣∣B+ e2tg

∣∣∣V.

Proposition A.3. Let π : F →֒ (M, g) → B be a Riemannian submersion

with totally geodesic fibers. Let K̃, K, KB, KF denote the non-reduced sec-tional curvatures of g̃, gB, gF , respectively, where g̃ is the canonical variationof g; gB is the submersion metric on B, and gF the metric on F . Then, ifX,Y, Z ∈ H, and V,W ∈ V ,

1. K̃(X,Y ) = KB(π∗X, π∗Y )(1− e2t) + e2tK(X,Y ),

2. K̃(X,V ) = e4t|A∗XV |2,

3. K̃(V,W ) = e2tK(V,W ),

4. R̃(X,Y, Z,W ) = e2tg((∇XA)Y Z,W ).

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Leonardo Francisco CavenaghiDepartamento de Matemática – Universidade Federal da Paráıba - Campus 1, 1ºfloor - Lot. Cidade Universitaria 58051-900, João Pessoa, PB, Brazile-mail: [email protected]

Llohann Dallagnol SperançaInstituto de Ciência e Tecnologia – Unifesp, Avenida Cesare Mansueto Giulio Lat-tes, 1201, 12247-014, São José dos Campos, SP, Brazile-mail: [email protected]

1. IntroductionStructure of the article

2. Prescribing scalar curvature on fiber bundles with compact structure group and applications3. Prescribing scalar curvature on some Riemannian submersion and applications3.1. Proof of Theorem F3.2. Proof of Theorem E3.3. Examples3.3.1. Tori bundles and generalizations3.3.2. Some Calabi–Yau bundles

Acknowledgments

Appendix A. General Vertical WarpingsA.1. Warped productsA.2. Canonical deformation

References

On prescribing scalar curvature on bun- dles and applicationsarXiv:2012.13272v2 [math.DG] 27 Dec...

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Transcript of On prescribing scalar curvature on bun- dles and applicationsarXiv:2012.13272v2 [math.DG] 27 Dec...