spectral flow and the riesz stability of the atiyah-singer...

Spectral flow and the Riesz stability of theAtiyah-Singer Dirac operator under boundedperturbations of local boundary conditions

Lashi BandaraJoint with Andreas Rosen (GU)

16 March 2017

Oxford University

Oxford, UK

Lashi Bandara Riesz continuity of the Dirac operator of local boundary conditions 1/31

Dedicated to the memory of Alan G. R. McIntosh


Spectral flow

H a separable Hilbert space, BF sa the set of bounded, self-adjointFredholm operators on this space.

Function f : [0, 1]→ BF sa norm continuous. Spectral flow SF(f) isthe “net number of eigenvalues crossing 0”.

Connections to the study of particle physics. First defined by Atiyahand Singer in 1969 in [AS69] on the index theory of skew-adjointFredholm operators using topological language.


In 1996, Phillips in [P96] gave an equivalent definition:

SF(f) =

n∑j=1

rank(χ[0,εj)(f(tj)))− rank(χ[0,εj)(f(tj−1))),

where 0 ≤ t0 < t1 < · · · < tn = 1 such that εj > 0 with:

(i) ±εj 6∈ σ(f(t)) for all t ∈ [0, 1] and

(ii) [−εj , εj ] ∩ σess(f(t)) = ∅ for t ∈ [tj−1, tj ].

Right hand expression is well-defined, path additive, homotopyinvariant and integer valued. These properties essentially characterisethe spectral flow. The times tj and εj exist given f .


Unbounded operators

Let F sa denote the set of unbounded self-adjoint Fredholm operatorson H .

Need to equip F sa with a topology. There are two natural candidates:

ρG(T, S) =

∥∥∥∥T + i

T − i− S + i

S − i

∥∥∥∥ , (Graph metric)

ρR(T, S) =

∥∥∥∥ T√1 + T 2

− S√1 + S2

∥∥∥∥ . (Riesz metric)


For graph continuous maps f : [0, 1]→ (F sa, ρG), the spectral flowtakes the form

SF(f) = winding number(κ ◦ f),

where κ(x) = x+ix−i , the Cayley transform.

This topology is weak (easier to show continuity) but homotopy typeof (F sa, ρG) is complicated.

Hard to establish continuity in the Riesz topology but carries overproperties of the bounded case for Riesz continuous mapsf : [0, 1]→ (F sa, ρR).

Spectral flow is given by:

SF(f) = SF(R ◦ f),

where R(x) = x1+x2

.


Boundary conditions

Let D be: first-order, elliptic, symmetric differential operator on V, aRiemannian vector bundle, over a complete manifold (M, g) withsmooth compact boundary ∂M = Σ.

A local boundary condition is a subspace B = H12 (E), where E is a

smooth subbundle of V|Σ. The gap between two such subspaces is

δ∞(B, B) = supx∈M

|πE(x)− πE(x)|,

where πE and πE are orthogonal projectors.

The associated operator DB = D with domain

D(DB) = {u ∈ D(Dmax) : R u ∈ B} ,

where R : D(Dmax)→ H−1(V|∂M) is the restriction map.


Functional calculus

Suppose B are self-adjoint; then we can form functions ψ(DB) of DBfor holomorphic functions ψ on the hyperbolic region:

Soω,σ =

{x+ ıy : y2 < tan2(ωx2) + σ2

}with |ψ(ζ)| ≤ min {|ζ|−α, |ζ|α} for some α > 0.

This functional calculus given by the Cauchy integral formula:

ψ(DB)u =1

2πı

˛γψ(ζ)(ζI−DB)−1u dζ.

It can be extended for functions f ∈ Hol∞(Soω,σ) functions, and

choosing

f(ζ) =ζ√

1 + ζ2

allows us to access the Riesz topology.


Main Theorem

Theorem

Let (M, g) be a smooth, complete Spin manifold with smooth,compact boundary Σ = ∂M and suppose that there exists:

(i) κ > 0 such that inj(M, g) > κ, and

(ii) CR <∞ such that |Ricg| ≤ CR and |∇Ricg| ≤ CR.

Let B and B be two local self-adjoint /D-elliptic boundary conditionswhich satisfy:

(iii) ‖B‖Lip + ‖B‖Lip ≤ CB, and

(iv) /D-ellipticity constants of orders 1 and 2 in a sufficiently smallcompact neighbourhood of Σ.


Theorem (Continued...)

Then, for ω ∈ (0, π/2) and σ > 0, whenever we havef ∈ Hol∞(So

ω,σ), we have the perturbation estimate

‖f(/DB)− f(/DB)‖L2→L2 . ‖f‖∞ δ∞(B, B),

where the implicit constant depends on dimM and the constantsappearing in (i)-(iv).


Local boundary conditions in even dimensions

For M even dimensional, we have /∆M = /∆+M⊕⊥ /∆

−M and

/D =

(0 /D

−

/D+

0

).

Let B ∈ End( /∆+

Σ) smooth and invertible, and define /∆B given bythe fibres

/∆B,x Σ ={

(ψ,~n(x) · B(x)ψ) : ψ ∈ /∆+x Σ}

An important class of local boundary conditions are then given by:

BB = H12 ( /∆B Σ).

These boundary conditions are /D-elliptic and self-adjoint by askingthe defining endomorphism to satisfy: B(x)∗ = B(x).


Pullback of operator

Without the loss of generality, we can assume that δ∞(B, B) ≤ 1/2.Since the projectors πE and πE on /∆ Σ to E and E respectively areorthogonal, ‖2πE − I‖∞ = 1 and so we obtain:

(i) ‖πE − πE‖∞ ≤1

2‖2πE−I‖∞ and

(ii) ‖∇πE‖∞ + ‖∇πE‖∞ ≤ CB

Then, there exists a U ∈ Lip(L( /∆M)) with ‖U− I‖∞ ≤ 12 and

‖∇U‖ ≤ CB such that

/DB = U−1 /DBU.


Geometric properties of the operators and their difference

We need to establish the following:

∗ /DB −U−1 /DBU = A1∇ + div A2 + A3 distributionally with

‖∇A2‖∞ ≤ CB and

‖A1‖∞ + ‖A2‖∞ + ‖A3‖∞ . ‖I−U‖∞,

∗ the embedding D(/DB) ↪→ H1( /∆M) and D(/DB) ↪→ H1( /∆M)with equivalence of norms on D(/DB) and D(/DB) respectively,

∗ that D(/DB) = D(/DBU) with equivalence of norms, and

∗ the embedding D(/D2B) ↪→ H2( /∆M).


A good cover

These properties are reduced to proving the following:

Proposition

There exist rH > 0 and a constant 1 ≤ C <∞ depending on κ andCR such that at each x ∈M, ψx : B(x, rH)→ Rn corresponds to acoordinate system and inside that coordinate system with coordinatebasis {∂j} satisfying:

(i) C−1|u|ψ∗xδ(y) ≤ |u|g(y) ≤ C|u|ψ∗xδ(y),

(ii) |∂kgij(y)| ≤ C, and

(iii) |∂k∂lgij(y)| ≤ C.for all y ∈ B(x, rH).

For a manifold without boundary, the lower bound on injectivityradius and bounds on Ricci curvature and its first derivatives yieldC2,α-harmonic coordinates which gives us what we want.


Doubling the manifold

Let M =MtM be the double of the manifold M which isobtained by taking two copies of M and pasting along the boundaryto obtain a manifold without boundary.

Since the boundary is smooth, this manifold is again smooth.

By reflection, we obtain an extension gext of the metric g to thewhole of M. This metric, a priori, is only guaranteed to becontinuous, but smooth on M \ Σ.

However, working a little harder we can obtain a smooth “close”metric with the right properties.


Reduction to quadratic estimates

Let D = /DB and D = U−1 /DBU. Reduce

‖f(D)− f(D)‖ . ‖f‖∞‖I−U‖∞

for ω ∈ (0, π/2) and σ ∈ (0,∞), and f ∈ Hol∞(Soω,σ) to

ˆ 1

0‖(Qt −Qt)u‖2

dt

t. ‖I−U‖2∞‖u‖2,

where Qt = tDPt and Qt = tDPt with Pt = (I + t2D2)−1. Then tothe quadratic estimates:

ˆ 1

0‖QtA1∇(ıI + D)−1Ptf‖2

dt

t. ‖I−U‖2∞‖f‖2 and

ˆ 1

0‖tPt divA2Ptf‖2

dt

t. ‖I−U‖2∞‖f‖2.


ByQt −Qt = −Pt[t(D−D)]Pt − Qt[t(D−D)]Qt,

the integral´ 1

0 ‖(Qt −Qt)u‖2 dtt bound by

ˆ 1

0‖PttA1∇Ptf‖2

dt

t+

ˆ 1

0‖PttdivA2Ptf‖2

dt

t

+

ˆ 1

0‖PttA3Ptf‖2

dt

t

+

ˆ 1

0‖QttA1∇Qtf‖2

dt

t+

ˆ 1

0‖Qtt divA2Qtf‖2

dt

t

+

ˆ 1

0‖QttA3Qtf‖2

dt

t.


Trouble term via operator theoryBy duality,⟨

QttdivA2Qtf, u⟩

= −⟨A2Qtf, t∇Qtu

⟩+ t⟨A2Qtf, Qtu

⟩L2( /∆ Σ)

.

The boundary term:

‖t⟨A2Qtf, Qtu

⟩L2( /∆ Σ)

‖ . ‖A2‖∞t‖Qtf‖L2( /∆ Σ)‖Qtu‖L2( /∆ Σ),

and√t‖Qtu‖L2( /∆ Σ) . ‖u‖.

We prove that:

t‖Qtf‖2L2( /∆ Σ) . ‖ψ1−s(tD)f‖2 + ‖ψs(tD)f‖2 + ‖Qtf‖2

for some s ∈ (0, 1) where

ψr(ζ) =ζ|ζ|r

1 + ζ2.


Fractional Sobolev spaces

Let ∇N = ∇2 and ∇D = ∇0, where

∇2 : C∞ ∩ L2( /∆M)→ C∞ ∩ L2(T∗M⊗ /∆M) and

∇0 : C∞cc ( /∆M)→ C∞cc (T∗M⊗ /∆M).

Define:

Hs( /∆M) = [L2( /∆M),H1( /∆M)]θ=s = D((I +√

∆N )s),

Hs0( /∆M) = C∞cc ( /∆M)

‖·‖Hs

,

Hs00( /∆M) = [L2( /∆M),H1

0( /∆M)]θ=s = D((I +√

∆D)s).

Lemma

The equality Hs( /∆M) = Hs0( /∆M) = Hs

00( /∆M) holds whenever0 ≤ s < 1/2.


Let ~N be an extension of the normal vectorfield ~n on a compactneighbourhood around Σ. Then

‖Qtf‖2L2( /∆ Σ) . t|⟨∇ ~NQtf,Qtf

⟩|+ t‖Qtf‖2.

For s ≤ 1, we have

|⟨∇ ~NQtf,Qtf

⟩| . ‖∇ ~NQtf‖H−s‖Qtf‖Hs .

Also, ∇ ~N : H1( /∆M)→ L2( /∆M) and on defining

(∇ ~Nu)(v) = −⟨u,∇ ~Nv

⟩for v ∈ C∞cc ( /∆M), we obtain that ∇ ~N : L2( /∆M)→ H1

0( /∆M)∗

(equal to H−1( /∆M)).


By interpolation:

∇ ~N : [H1( /∆M),L2( /∆M)]θ=s → [L2( /∆M),H−1( /∆M)]θ=s.

By lemma and calculation,

[L2( /∆M),H−1( /∆M)]θ=s = ([L2( /∆M),H10( /∆M)]θ=s)

∗

= Hs00( /∆M)

∗= Hs

0( /∆M)∗

= H−s( /∆M).

This yeilds ∇ ~N : H1−s( /∆M)→ H−s( /∆M) and on noting thatD(|D|s) ↪→ Hs( /∆M),

t‖Qtf‖2L2( /∆ Σ) . ‖ψ1−s(tD)f‖2 + ‖ψs(tD)f‖2 + ‖Qtf‖2


Harmonic analysis I

Off-diagonal estimates for the operator D: there exists C > 0 suchthat, for each M > 0, there exists a constant C∆,M > 0 such that

‖χE(I + ıtD)−1(χFu)‖L2( /∆M) ≤C∆,M‖A‖2∞⟨ρ(E,F )

t

⟩−M×

exp

(−C ρ(E,F )

t

)‖χFu‖L2( /∆M)

Local boundary condition assumption: ηD(D) ⊂ D(D) forη ∈ C∞c (M) and [D, ηI] is a pointwise-multiplication operator with

|[D, ηI]u(x)| . |∇η(x)||u(x)|.


Need the following from [BMcR]:

∗ the dyadic cubes{Qkα ⊂M : α ∈ Ik, k ∈ N

}, with centres

zkα ∈ Qkα and constants δ ∈ (0, 1), a0 > 0, η > 0 and C1, C2 <∞(Theorem 5.1 in [BMcR]),

∗ the scale tS = δJ where C1δJ ≤ ρ/5, with ρ = ρT∗M, the the

uniform radii trivialisation of T∗M,

∗ collection of dyadic cubes Qj , Q = ∪j≥JQj , and Qt for t ≤ tS,

∗ the unique ancestor Q ∈ QJ for a dyadic cube Q and the set ofGBG coordinates C and dyadic GBG coordinates CJ,

∗ the cube integral B(xQ, ρ)×Q 3 (x,Q) 7→ (

´Q · )(x) defined on

L1loc( /∆M) and cube average uQ , and

∗ for t > 0, the dyadic averaging operator

Et : L1loc( /∆M)→ L1

loc( /∆M)


Split the quadratic estimate:

ˆ 1

0‖QtA1∇(ıI + D)−1Ptu‖2

dt

t

.ˆ 1

0‖(Qt − γtEt)A1∇(ıI + D)−1Ptu‖2

dt

t

+

ˆ 1

0‖γtEtA1∇(ıI + D)−1(I− Pt)u‖2

dt

t

+

ˆ 1

0‖γtEtA1∇(ıI + D)−1u‖2 dt

t.

γt is the principal part of the operator:

γt(x)w = (Qtωc),

where ωc(x) = w inside the fixed trivialisation containing dyadic cubecontaining x.


∗ The first term is handled by a Poincare inequality coupled with theembedding D(D) ↪→ H1( /∆M) and D(D2) ↪→ H2( /∆M) and via avariant of Proposition 5.4 in [BMcR].

∗ The last term is handled by a straightforward application ofProposition 5.12 in [BMcR].

∗ The remaining term is handled by noting the estimate:∣∣∣∣ˆQ

Du dµ

∣∣∣∣ . µ(Q)12 ‖u‖ and

∣∣∣∣ˆQ∇u dµ

∣∣∣∣ . µ(Q)12 ‖u‖,

for u ∈ D(D) with spt u ⊂ Q ∩ M coupled with the followinglemma.


Lemma

There exist constants k0, η, C3 > 0 such that for all cubes Q ∈ Qk

with k > k0 and Q ∩ Σ 6= ∅, we have

µ {x ∈ Q : ρ(x,Σ) ≤ s `(Q)} ≤ C3sηµ(Q).

In particular, for every Q ∈ Qk with k > k0,

µ {x ∈ Q : ρ(x,M\ (Q \ Σ)) ≤ s `(Q)} ≤ C3sηµ(Q).


Harmonic analysis II

Let Θt = tPt divA2Pt and let γt denote the principal part of Θt.Split

ˆ 1

0‖Θtu‖2

dt

t

≤ˆ 1

0‖Θt(I− Pt)u‖2

dt

t+

ˆ 1

0‖(Θt − γtEt)Ptu‖2

dt

t

+

ˆ 1

0‖γtEt(Pt − I)u‖2 dt

t+

ˆ 1

0‖γtEtu‖2

dt

t

Green terms are handled similar to the previous quadratic estimate,via methods from [BMcR]. Red term required us to reduce theestimate to a Carleson measure.


Carleson’s Theorem: the estimate of this term is is reduced to showing

dν(x, t) = |γt(x)|2 dµ(x)dt

t

is a Carleson measure. Prove for each cube Q ∈ Q, and for Carlesonregions RQ = Q × (0, `(Q)),

¨RQ

|γt(x)|2 dµ(x)dt

t. ‖A‖2∞µ(Q). (Carl)

Trouble: the coefficients A2 might not preserve boundary conditions.


Lemma

Suppose that for every cube Q ∈ Q with `(Q) ≤ ρ(Q,Σ) theCarleson estimate (Carl). Then, (Carl) holds for every cube Q ∈ Q.

The proof of this lemma: cover every off-diagonal t-slice by cubes ofa comparable radius, and then using the inequality

Q′|γt|2 dµ . ‖I−U‖2∞.

Carleson measure estimate (Carl) for Q with ρ(Q,Σ) ≥ `(Q)obtained since we are supported away from the boundary, and hence,we map into D(divmin) which does not see boundary conditions.


References I

[AS69] M. F. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators,

Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 37, 5–26. MR 0285033

[BMcR] L. Bandara, A. McIntosh, and A. Rosen, Riesz continuity of the Atiyah-SingerDirac operator under perturbations of the metric, ArXiv e-prints (2016).

[P96] John Phillips, Self-adjoint Fredholm operators and spectral flow, Canad. Math.Bull. 39 (1996), no. 4, 460–467. MR 1426691


spectral flow and the riesz stability of the atiyah-singer...

Documents