weiping zhang chern institute of mathematics … scalar curvatures on foliations the vanishing...

IntroductionScalar curvatures on foliations

The Vanishing theorem

Positive scalar curvature on foliations

Weiping Zhang

Chern Institute of Mathematics

Workshop on Geometric Analysis

Dalian, September 1st, 2016

Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations


The Vanishing theoremIntroduction

Introduction

I Mn smooth manifold, gTM Riemannian metric on TM

I For any p ∈M ,

vol(BMp (r)

)vol(BRn

0 (r)) = 1− kg

TM

6(n+ 2)r2 +O

(r3).

I kgTM

the scalar curvature of gTM .

I Basic question : when exists gTM with kgTM

> 0 ?

(Every manifold Mn with n ≥ 3 carries gTM with

kgTM

< 0)




Introduction

I Lichnerowicz vanishing theorem. If a compact spinmanifold M carries a Riemannian metric of positive scalarcurvature, then the Hirzebruch A-genus vanishes. That is,A(M) = 0.

A(M) =⟨A(TM), [M ]

⟩=

∫M

det1/2(

RTM/4πi

sinh(RTM/4πi)

).

(a la Chern-Weil)

I spin condition essential, as A(CP 2) = −18 .

(A complex manifold is spin if c1(M) is even)




Introduction

I Proof : Using the Dirac operator on spin manifolds :

I D± : S±(TM) −→ S∓(TM).

I D2 = −∆ + kTM

4 > 0 (Lichnerowicz formula)

I 0 = ind(D+) = A(M). (Atiyah-Singer index theorem)

I Aim : Present a generalization to the case of foliation




Scalar curvatures on foliationsConnes fibration

Foliation

I (M,F ) is a foliation if F ⊆ TM is an integrable subbundle

I i.e., X, Y ∈ Γ(F ) implies [X,Y ] ∈ Γ(F )

I For any x ∈M , (unique) leaf through x, denoted by Fx (asubmanifold of M)

I For any x ∈M , F |Fx = TFx

I Locally looks like fibration near each x ∈M

I “space of leaves” (M/∪x∈MFx) might be non-Hausdorff





Leafwise scalar curvature

I gF a Euclidean metric on F

I For any x ∈M , gF induces a Riemannian metricgTFx = gF |Fx on TFx

I kgTFx

be the scalar curvature along Fx associated to gTFx

I kgF ∈ C∞(M) with

kgF

(x) = kgTFx

(x)

well-defined, called leafwise scalar curvature of gF

I When F = TM , kgTM

is the usual scalar curvature on M .





An open question

I Open question : If

kgF> 0

over M , whether there exists gTM such that

kgTM

> 0 ?

I Easy case : if F = T VM of a fibration π : M → B, set

gTMε =π∗gTB

ε2⊕ gTVM ,

thenkg

TMε = kg

F+O(ε2).





Foliation : the Bott connection

I Let p⊥ : TM → F⊥ be the orthogonal projection withrespect to a splitting : TM = F ⊕ F⊥, gTM = gF ⊕ gF⊥

.

I Let ∇B be the Bott connection on F⊥ ' TM/F :for any X ∈ Γ(F ), U ∈ Γ(F⊥),

∇BXU = p⊥[X,U ].

I For any X ∈ Γ(F ), define ω(X) ∈ Γ(End(F⊥)) by that forany U, V ∈ Γ(F⊥),

〈ω(X)U, V 〉 = X〈U, V 〉 − 〈∇BXU, V 〉 − 〈U,∇BXV 〉.

I Riemannian foliation : For any X ∈ Γ(F ), ω(X) ≡ 0.





Adiabatic limit of kgTMε

I In general, set gTMε = gF ⊕ gF⊥

ε2.

I Let f1, · · · , fq be an orthonormal basis of (F, gF ) ;

h1, · · · , hq1 an orthonormal basis of (F⊥, gF⊥

).

I Set

|ω (fi)|2 =

q1∑s=1

|ω (fi)hs|2 .





Adiabatic limit of kgTMε

II (Based on earlier computations with Kefeng Liu andYong Wang)

kgTMε = kg

F+

3

4

q∑i=1

|ω(fi)|2 −1

4

q∑i=1

(q1∑s=1

〈ω(fi)hs, hs〉

)2

−q∑i=1

q1∑s=1

fi (〈ω(fi)hs, hs〉) + 2

q∑i=1

q1∑s=1

⟨ω(fi)hs, p

⊥[fi, hs]⟩

+

q∑i=1

q1∑s=1

⟨ω(p∇TMfi fi

)hs, hs

⟩+O

(ε2).





Easy case : Riemannian foliation

II If ω = 0, i.e., in the case of Riemannian foliation,

I as ε→ 0,kg

TMε = kg

F+O

(ε2)

I General case : Much more complicated relations ...





Adiabatic limit of kgTMε : the codimension one case

I Assume q1 = rk(TM/F ) = 1, then

I

kgTMε = kg

F − 1

2

q∑i=1

|ω(fi)|2 −q∑i=1

fi (〈ω(fi)h1, h1〉)

+

q∑i=1

⟨ω(p∇TMfi fi

)h1, h1

⟩+O

(ε2).





Connes vanishing theorem

I Theorem of Connes (1986) : Let (M,F ) be a foliationsuch that M is compact and oriented, while F is spin. If

there is a metric gF on F such that kgF> 0 over M , then

A(M) = 0.

I F = TM , Lichnerowicz theorem.

I In general, M need not spin, A(M) is not a priori aninteger

I Highly non-trivial even in the q1 = 1 case.





Connes vanishing theorem

I If (M,F ) is a fibration, easy consequence of Atiyah-Singerfamilies index theorem

A(M) =

∫BA(TB)

∫Mb

A(T VM) =

∫BA(TB)ch(ind(Db))

I Connes’ proof : noncommutative (families) index theory onfoliations + cyclic cohomology

I Relies essentially on the spin structure on F

I Natural question : What happens if one assumesTM spin instead of F spin ?

I If TM spin, one has Dirac operators on M .





Main result

I Theorem (Zhang, arXiv : 1508.04503) : Let (M,F ) bea foliation such that M is compact and spin. If there is a

metric gF on F such that kgF> 0 over M , then A(M) = 0.

I F = TM , Lichnerowicz theorem.

I Corollary. If M4k (k > 1) is also simply connected, then

M4k carries a metric with positive scalar curvature.

I Main difficulty : ω 6= 0.

I Resolution : need Connes fibration.





The Connes fibration

I Let π : M →M be the fibration

I For any x ∈M ,

π−1(x) = {Euclidean metrics on TxM/Fx}

I Each fiber π−1(x) is a space of non-positive curvature

I non-positive curvature property plays an essential role inConnes’ proof of his vanishing theorem






I F ⊆ TM lifts to an integrable subbundle F of TM.

I T V M vertical tangent bundle

I Natural splitting : TM = F ⊕ T V M ⊕ F⊥

I T V M carries a natural metric of (fiberwise) non-positivecurvature

I From F⊥ ' π∗(TM/F ), by definition, F⊥ carries acanonically induced metric

I (Any p ∈ M, determines a metric onTπ(p)M/Fπ(p) ' π∗F⊥p , which determines a metric on F⊥p )






I gF lifts to a metric on F

I Orthogonal splitting

TM = F ⊕ T V M ⊕ F⊥, gTM = gF ⊕ gTV M ⊕ gF⊥

I Connes : Hol(M, F ) acts almost isometrically on

T V M ⊕ F⊥

I Infinitesimally, for any X ∈ Γ(F ), Y, Z ∈ Γ(T V M),U, V ∈ Γ(F⊥), one has

X〈Y,Z〉 = 〈[X,Y ], Z〉+ 〈Y, [X,Z]〉,

X〈U, V 〉 = 〈[X,U ], V 〉+ 〈U, [X,V ]〉,

〈[X,Y ], U〉 = 0.






I Following Connes, take an embedded section s : M ↪→ M

I Equivalently : take a metric on F⊥ ' TM/F

I F ⊕ F⊥ = π∗(TM) spin

I For any β > 0, ε > 0, rescale the metric to

TM = F ⊕ T V M ⊕ F⊥, gTM = β2gF ⊕ gTV M ⊕ gF⊥

ε2.




A Lichnerowicz vanishing theorem on foliationsSummary

The vanishing theorem

I Theorem (Zhang, arXiv : 1508.04503) : Let (M,F ) bea foliation such that M is compact and spin. If there is a

metric gF on F such that kgF> 0 over M , then A(M) = 0.

I Proof. Instead of working near s(M) ⊂ M as usual, we

work on the whole M.

I s ◦ π : M → s(M) looks like a vector bundle over s(M).

I Apply analytic Riemann-Roch property on M.





Vanishing theorem : outline of proof

I For any p ∈ M, let dMp denote the fiberwise distant onMp = π−1(π(p))

I Denote ρ(p) = dMp(p, s ◦ π(p)).

I For any R > 0, set

MR = {p ∈ M : ρ(p) ≤ R}

I MR is a smooth manifold with boundary






I On M, consider the Dirac type operator(called sub-Dirac operator, go back to Liu-Zhang 1999)

(1) Dβ,ε : Γ(Sβ,ε(F ⊕ F⊥)⊗Λ∗(T V M))

−→ Γ(Sβ,ε(F ⊕ F⊥)⊗Λ∗(T V M))

I Set

Dβ,ε,R = Dβ,ε +c(ρ dT

V Mρ)

β R.

I No R in the usual Riemann-Roch.






I Key estimate 1 :

(Dβ,ε)2 = −∆β,ε +

kgF

4β2+ o

(1

β2

).

I This step uses essentially the almost isometric property ofthe Connes fibration.

I Key estimate 2 :[Dβ,ε,

c(ρ dTV Mρ)

β R

]= O

(1

β2R

).

I This step uses essentially the nonpositive curvatureproperty of each fiber Mp






I By key estimates 1 and 2 :(Dβ,ε,R

)2=

(Dβ,ε +

c(ρ dTV Mρ)

β R

)2

= (Dβ,ε)2 +

ρ2

β2R2+O

(1

β2R

)= −∆β,ε +

kgF

4β2+

ρ2

β2R2+O

(1

β2R

)+ o

(1

β2

)I Thus, when R >> 0 and β > 0, ε > 0 are very small,(

Dβ,ε,R

)2≥ kg

F

8β2

in the interior of MR.






I Proposition. There are R >> 0, β, ε > 0 sufficientlysmall, such thata). Dβ,ε,R is invertible over the interior of MR ;

b). (Dβ,ε,R)|∂MR

is invertible on the boundary ∂MR.

I Proof of b). This is because ρ = R on ∂MR. QED

I By anaytic Riemann-Roch,

0 = ind(Dβ,ε,R,+) = A(s(M)) = A(M).

I Easy modification gives a purely geometric proof ofthe Connes vanishing theorem.





Summary

I Theorem. M spin + kgF> 0 => A(M) = 0.

I if M4k (k > 1) also simply connected => kgTM

> 0.

I Open question : kgF> 0 => kg

TM> 0

I Positive anwser if dimM ≥ 5 and M simply connected.

I Theorem. On Tn, no kgF> 0.

(F = TM due to Schoen-Yau and Gromov-Lawson)

I General case still open.





Two more open questions (1)

I Theorem. M spin + kgF> 0 =>⟨

A(F )e(TM/F )p(TM/F ), [M ]⟩

= 0,

where e(TM/F ) is the Euler class of TM/F .

I Corollary. 〈A(F )e(TM/F )3, [M ]〉 = 0.

I Open question : e(TM/F )3 = 0 in general ?

I Bott : There is no integrable codimension 2 subbundle ofT (CP 2n+1) for n ≥ 2. (Reason : e(TM/F )4 = 0)

I Open question. How about n = 1, i.e. CP 3 ?I Partial answer : No if assume further that kg

F> 0.





Two more open questions (2)

I Witten : M4 oriented, closed, b+2 > 1, spinc-structure c,

then kgTM

> 0 implies SW(M4, c) = 0,

where “SW” stands for the Seiberg-Witten invariant.

I Question : (M4, F ) foliation, · · ·

whether kgF> 0 implies SW(M4, c) = 0 ?





Thanks !


weiping zhang chern institute of mathematics … scalar curvatures on foliations the vanishing...

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