poincaré duality angles on riemannian manifolds with...
TRANSCRIPT
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Poincare Duality Angles on RiemannianManifolds with Boundary
Clayton Shonkwiler
Department of MathematicsHaverford College
September 15, 2009
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Realizing cohomology groups as spaces of differential forms
Let Mn be a compact Riemannian manifold with non-emptyboundary ∂M.
CO-EXACT NEUMANN FORMS HAVE CO-CLOSEDTRACES
CLAYTON SHONKWILER
Hp(M
;R)Hp (M
, !M
;R)
Figure 1. The concrete realizations of the absolute and rel-ative cohomology groups
Department of Mathematics, University of PennsylvaniaE-mail address: [email protected]: http://www.math.upenn.edu/~shonkwil
1
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Realizing cohomology groups as spaces of differential forms
Let Mn be a compact Riemannian manifold with non-emptyboundary ∂M.
CO-EXACT NEUMANN FORMS HAVE CO-CLOSEDTRACES
CLAYTON SHONKWILER
Hp(M
;R)Hp (M
, !M
;R)
Figure 1. The concrete realizations of the absolute and rel-ative cohomology groups
Department of Mathematics, University of PennsylvaniaE-mail address: [email protected]: http://www.math.upenn.edu/~shonkwil
1
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de Rham’s Theorem
de Rham’s TheoremSuppose Mn is a compact, oriented, smooth manifold. Then
Hp(M; R) ∼= Cp(M)/Ep(M),
where Cp(M) is the space of closed p-forms on M and Ep(M) isthe space of exact p-forms.
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Riemannian metric
If M is Riemannian, the metric induces an L2 inner product onΩp(M):
〈ω, η〉 :=
∫Mω ∧ ? η.
When M is closed, the orthogonal complement of Ep(M) insideCp(M) is
Hp(M) := ω ∈ Ωp(M) : dω = 0, δω = 0
Kodaira called this the space of harmonic p-fields on M.
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Riemannian metric
If M is Riemannian, the metric induces an L2 inner product onΩp(M):
〈ω, η〉 :=
∫Mω ∧ ? η.
When M is closed, the orthogonal complement of Ep(M) insideCp(M) is
Hp(M) := ω ∈ Ωp(M) : dω = 0, δω = 0
Kodaira called this the space of harmonic p-fields on M.
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Riemannian metric
If M is Riemannian, the metric induces an L2 inner product onΩp(M):
〈ω, η〉 :=
∫Mω ∧ ? η.
When M is closed, the orthogonal complement of Ep(M) insideCp(M) is
Hp(M) := ω ∈ Ωp(M) : dω = 0, δω = 0
Kodaira called this the space of harmonic p-fields on M.
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Hodge’s Theorem
Hodge’s Theorem
If Mn is a closed, oriented, smooth Riemannian manifold,
Hp(M; R) ∼= Hp(M).
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Hodge–Morrey–Friedrichs Decomposition
Define i : ∂M → M to be the natural inclusion.
The L2-orthogonal complement of the exact forms inside the spaceof closed forms is now:
HpN(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗? ω = 0.
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Hodge–Morrey–Friedrichs Decomposition
Define i : ∂M → M to be the natural inclusion.
The L2-orthogonal complement of the exact forms inside the spaceof closed forms is now:
HpN(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗? ω = 0.
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i∗? ω = 0
Suppose dxn corresponds to the normal direction.
If i∗? ω = 0, then
?ω =(∑
fIdxI
)∧ dxn,
meaning that ω has no dxn in it.
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i∗? ω = 0
Suppose dxn corresponds to the normal direction.
If i∗? ω = 0, then
?ω =(∑
fIdxI
)∧ dxn,
meaning that ω has no dxn in it.
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i∗? ω = 0
Suppose dxn corresponds to the normal direction.
If i∗? ω = 0, then
?ω =(∑
fIdxI
)∧ dxn,
meaning that ω has no dxn in it.
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Hodge–Morrey–Friedrichs Decomposition
Define i : ∂M → M to be the natural inclusion.
The L2-orthogonal complement of the exact forms inside the spaceof closed forms is now:
HpN(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗? ω = 0.
ThenHp(M; R) ∼= Hp
N(M).
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Hodge–Morrey–Friedrichs Decomposition
Define i : ∂M → M to be the natural inclusion.
The L2-orthogonal complement of the exact forms inside the spaceof closed forms is now:
HpN(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗? ω = 0.
ThenHp(M; R) ∼= Hp
N(M).
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Hodge–Morrey–Friedrichs Decomposition (continued)
The relative cohomology appears as
Hp(M, ∂M; R) ∼= HpD(M).
HpD(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗ω = 0.
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Hodge–Morrey–Friedrichs Decomposition (continued)
The relative cohomology appears as
Hp(M, ∂M; R) ∼= HpD(M).
HpD(M) := ω ∈ Ωp(M) : dω = 0, δω = 0, i∗ω = 0.
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Non-orthogonality
The concrete realizations of Hp(M; R) and Hp(M, ∂M; R) meetonly at the origin:
HpN(M) ∩Hp
D(M) = 0
...but they are not orthogonal!
CO-EXACT NEUMANN FORMS HAVE CO-CLOSEDTRACES
CLAYTON SHONKWILER
H pN (M
)
boundarybo
unda
ry
Hp
D(M
)
interio
r
interio
r
Figure 1. The concrete realizations of the absolute and rel-ative cohomology groups
Department of Mathematics, University of PennsylvaniaE-mail address: [email protected]: http://www.math.upenn.edu/~shonkwil
1
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Non-orthogonality
The concrete realizations of Hp(M; R) and Hp(M, ∂M; R) meetonly at the origin:
HpN(M) ∩Hp
D(M) = 0
...but they are not orthogonal!
CO-EXACT NEUMANN FORMS HAVE CO-CLOSEDTRACES
CLAYTON SHONKWILER
H pN (M
)
boundarybo
unda
ry
Hp
D(M
)
interio
r
interio
r
Figure 1. The concrete realizations of the absolute and rel-ative cohomology groups
Department of Mathematics, University of PennsylvaniaE-mail address: [email protected]: http://www.math.upenn.edu/~shonkwil
1
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Non-orthogonality
The concrete realizations of Hp(M; R) and Hp(M, ∂M; R) meetonly at the origin:
HpN(M) ∩Hp
D(M) = 0
...but they are not orthogonal!
CO-EXACT NEUMANN FORMS HAVE CO-CLOSEDTRACES
CLAYTON SHONKWILER
H pN (M
)
boundarybo
unda
ry
Hp
D(M
)
interio
rin
terio
r
Figure 1. The concrete realizations of the absolute and rel-ative cohomology groups
Department of Mathematics, University of PennsylvaniaE-mail address: [email protected]: http://www.math.upenn.edu/~shonkwil
1
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Interior and boundary subspaces
Interior subspace of HpN(M):
ker i∗ where i∗ : Hp(M; R) → Hp(∂M; R)
E∂HpN(M) := ω ∈ Hp
N(M) : i∗ω = dϕ,ϕ ∈ Ωp−1(∂M).
Interior subspace of HpD(M):
? E∂Hn−pN (M) = cE∂Hp
D(M)
= η ∈ HpD(M) : i∗? η = dψ,ψ ∈ Ωn−p−1(∂M).
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Interior and boundary subspaces
Interior subspace of HpN(M):
ker i∗ where i∗ : Hp(M; R) → Hp(∂M; R)
E∂HpN(M) := ω ∈ Hp
N(M) : i∗ω = dϕ,ϕ ∈ Ωp−1(∂M).
Interior subspace of HpD(M):
? E∂Hn−pN (M) = cE∂Hp
D(M)
= η ∈ HpD(M) : i∗? η = dψ,ψ ∈ Ωn−p−1(∂M).
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Interior and boundary subspaces
Interior subspace of HpN(M):
ker i∗ where i∗ : Hp(M; R) → Hp(∂M; R)
E∂HpN(M) := ω ∈ Hp
N(M) : i∗ω = dϕ,ϕ ∈ Ωp−1(∂M).
Interior subspace of HpD(M):
? E∂Hn−pN (M) = cE∂Hp
D(M)
= η ∈ HpD(M) : i∗? η = dψ,ψ ∈ Ωn−p−1(∂M).
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Poincare duality angles
To prove that there is an element of cEHpN (M) having arbitrary preassigned periods on
cp1, . . . c
pg, it su!ces to show that (F1, . . . , Fg) !" (C1, . . . , Cg) is an isomorphism.
Suppose some set of F -values gives all zero C-values, meaning that i!! is zero in the cohomology
of "M . In other words, the form i!! is exact, meaning that ! # E!HpN (M), the interior subspace
of HpN (M). Since E!Hp
N (M) is orthogonal to cEHpN (M), this implies that ! = 0, so !! = ± # ! = 0
and hence the periods Fi of !! must have been zero.
Therefore, the map (F1, . . . , Fg) !" (C1, . . . , Cg) is an isomorphism, completing Step 1.
Step 2: Let $ # HpN (M) and let C1, . . . , Cg be the periods of $ on the above p-cycles
cp1, . . . , c
pg. Let % # cEHp
N (M) be the unique form guaranteed by Step 1 having the same periods
on this homology basis.
Then & = $ $ % has zero periods on the p-cycles cp1, . . . , c
pg; since & is a closed form on M , it
certainly has zero period on each p-cycle of "M which bounds in M . Hence, & has zero periods
on all p-cycles of "M , meaning that i!& is exact, so & # E!HpN (M).
Therefore, $ = % + & # cEHpN (M) + E!Hp
N (M), so HpN (M) is indeed the sum of these two
subspaces, as claimed in (2.1.8). This completes the proof of the theorem.
Theorem 2.1.2 allows the details of Figure 1.1 to be filled in, as shown in Figure 2.1.
H pN (M
)
cEH pN (M
) EHp
D(M
)
Hp
D(M
)
E !H
p N(M
)cE
! HpD
(M)
Figure 2.1: HpN (M) and Hp
D(M)
18
Definition (DeTurck–Gluck)
The Poincare duality angles of the Riemannian manifold M are theprincipal angles between the interior subspaces.
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What do the Poincare duality angles tell you?
GuessIf M is “almost” closed, the Poincare duality angles of M shouldbe small.
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For example...
Consider CP2 with its usual Fubini-Study metric. Let p ∈ CP2.Then define
Mr := CP2 − Br (p).
CP1
p
π/2
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For example...
Consider CP2 with its usual Fubini-Study metric. Let p ∈ CP2.Then define
Mr := CP2 − Br (p).
CP1
p
π/2
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Topology of Mr
∂Mr is a 3-sphere.
Mr is the D2-bundle over CP1 (' S2(1/2)) with Eulercharacteristic 1.
Mr has absolute cohomology in dimensions 0 and 2.
Mr has relative cohomology in dimensions 2 and 4.
Therefore, Mr has a single Poincare duality angle θr betweenH2
N(Mr ) and H2D(Mr ).
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Topology of Mr
∂Mr is a 3-sphere.
Mr is the D2-bundle over CP1 (' S2(1/2)) with Eulercharacteristic 1.
Mr has absolute cohomology in dimensions 0 and 2.
Mr has relative cohomology in dimensions 2 and 4.
Therefore, Mr has a single Poincare duality angle θr betweenH2
N(Mr ) and H2D(Mr ).
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Topology of Mr
∂Mr is a 3-sphere.
Mr is the D2-bundle over CP1 (' S2(1/2)) with Eulercharacteristic 1.
Mr has absolute cohomology in dimensions 0 and 2.
Mr has relative cohomology in dimensions 2 and 4.
Therefore, Mr has a single Poincare duality angle θr betweenH2
N(Mr ) and H2D(Mr ).
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Topology of Mr
∂Mr is a 3-sphere.
Mr is the D2-bundle over CP1 (' S2(1/2)) with Eulercharacteristic 1.
Mr has absolute cohomology in dimensions 0 and 2.
Mr has relative cohomology in dimensions 2 and 4.
Therefore, Mr has a single Poincare duality angle θr betweenH2
N(Mr ) and H2D(Mr ).
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Topology of Mr
∂Mr is a 3-sphere.
Mr is the D2-bundle over CP1 (' S2(1/2)) with Eulercharacteristic 1.
Mr has absolute cohomology in dimensions 0 and 2.
Mr has relative cohomology in dimensions 2 and 4.
Therefore, Mr has a single Poincare duality angle θr betweenH2
N(Mr ) and H2D(Mr ).
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Find harmonic 2-fields
So the goal is to find closed and co-closed 2-forms on Mr whichsatisfy Neumann and Dirichlet boundary conditions.
Such 2-forms must be isometry-invariant.
Isom0(Mr ) = SU(2).
Find closed and co-closed SU(2)-invariant forms on Mr satisfyingNeumann and Dirichlet boundary conditions
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Find harmonic 2-fields
So the goal is to find closed and co-closed 2-forms on Mr whichsatisfy Neumann and Dirichlet boundary conditions.
Such 2-forms must be isometry-invariant.
Isom0(Mr ) = SU(2).
Find closed and co-closed SU(2)-invariant forms on Mr satisfyingNeumann and Dirichlet boundary conditions
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Find harmonic 2-fields
So the goal is to find closed and co-closed 2-forms on Mr whichsatisfy Neumann and Dirichlet boundary conditions.
Such 2-forms must be isometry-invariant.
Isom0(Mr ) = SU(2).
Find closed and co-closed SU(2)-invariant forms on Mr satisfyingNeumann and Dirichlet boundary conditions
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Find harmonic 2-fields
So the goal is to find closed and co-closed 2-forms on Mr whichsatisfy Neumann and Dirichlet boundary conditions.
Such 2-forms must be isometry-invariant.
Isom0(Mr ) = SU(2).
Find closed and co-closed SU(2)-invariant forms on Mr satisfyingNeumann and Dirichlet boundary conditions
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Hypersurfaces
CP1
p
π/2
The hypersurfaces at constant distance t from CP1 are Berger3-spheres:
S3(cos t)sin t
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Hypersurfaces
CP1
p
π/2
The hypersurfaces at constant distance t from CP1 are Berger3-spheres:
S3(cos t)sin t
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The Poincare duality angle for Mr
cos θr =1− sin4 r
1 + sin4 r.
As r → 0, the Poincare duality angle θr → 0.
As r → π/2, θr → π/2.
Generalizes to CPn − Br (p).
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The Poincare duality angle for Mr
cos θr =1− sin4 r
1 + sin4 r.
As r → 0, the Poincare duality angle θr → 0.
As r → π/2, θr → π/2.
Generalizes to CPn − Br (p).
![Page 41: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/41.jpg)
The Poincare duality angle for Mr
cos θr =1− sin4 r
1 + sin4 r.
As r → 0, the Poincare duality angle θr → 0.
As r → π/2, θr → π/2.
Generalizes to CPn − Br (p).
![Page 42: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/42.jpg)
The Poincare duality angle for Mr
cos θr =1− sin4 r
1 + sin4 r.
As r → 0, the Poincare duality angle θr → 0.
As r → π/2, θr → π/2.
Generalizes to CPn − Br (p).
![Page 43: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/43.jpg)
Poincare duality angles of Grassmannians
ConsiderNr := G2Rn − νr (G1Rn−1).
Theorem
• As r → 0, all the Poincare duality angles of Nr go to zero.
• As r approaches its maximum value of π/2, all the Poincareduality angles of Nr go to π/2.
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Poincare duality angles of Grassmannians
ConsiderNr := G2Rn − νr (G1Rn−1).
Theorem
• As r → 0, all the Poincare duality angles of Nr go to zero.
• As r approaches its maximum value of π/2, all the Poincareduality angles of Nr go to π/2.
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A conjecture
Conjecture
If Mn is a closed Riemannian manifold and Nk is a closedsubmanifold of codimension ≥ 2, the Poincare duality angles of
M − νr (N)
go to zero as r → 0.
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A question
What can you learn about the topology of M from knowledge of ∂M?
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Electrical Impedance Tomography
Induce potentials on the boundary of a region and determine theconductivity inside the region by measuring the current fluxthrough the boundary.
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Electrical Impedance Tomography
Induce potentials on the boundary of a region and determine theconductivity inside the region by measuring the current fluxthrough the boundary.
![Page 49: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/49.jpg)
The Voltage-to-Current map
Suppose f is a potential on the boundary of a region M ⊂ R3.
Then f extends to a potential u on M, where
∆u = 0, u|∂M = f .
If γ is the conductivity on M, the current flux through ∂M is givenby
(γ∇u) · ν = −γ ∂u
∂ν
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The Voltage-to-Current map
Suppose f is a potential on the boundary of a region M ⊂ R3.
Then f extends to a potential u on M, where
∆u = 0, u|∂M = f .
If γ is the conductivity on M, the current flux through ∂M is givenby
(γ∇u) · ν = −γ ∂u
∂ν
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The Voltage-to-Current map
Suppose f is a potential on the boundary of a region M ⊂ R3.
Then f extends to a potential u on M, where
∆u = 0, u|∂M = f .
If γ is the conductivity on M, the current flux through ∂M is givenby
(γ∇u) · ν = −γ ∂u
∂ν
![Page 52: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/52.jpg)
The Dirichlet-to-Neumann map
The map Λcl : C∞(∂M) → C∞(∂M) defined by
f 7→ ∂u
∂ν
is the classical Dirichlet-to-Neumann map.
Theorem (Lee-Uhlmann)
If Mn is a compact, analytic Riemannian manifold with boundary,then M is determined up to isometry by Λcl.
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The Dirichlet-to-Neumann map
The map Λcl : C∞(∂M) → C∞(∂M) defined by
f 7→ ∂u
∂ν
is the classical Dirichlet-to-Neumann map.
Theorem (Lee-Uhlmann)
If Mn is a compact, analytic Riemannian manifold with boundary,then M is determined up to isometry by Λcl.
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Generalization to differential forms
Joshi–Lionheart and Belishev–Sharafutdinov generalized theclassical Dirichlet-to-Neumann map to differential forms:
Λ : Ωp(∂M) → Ωn−p−1(∂M)
If ϕ ∈ Ωp(∂M), then let ω solve the boundary value problem
∆ω = 0, i∗ω = ϕ, i∗δω = 0.
DefineΛϕ := i∗? dω.
If f ∈ Ω0(∂M),
Λf = i∗? du =∂u
∂νdvol∂M = (Λclf ) dvol∂M
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Generalization to differential forms
Joshi–Lionheart and Belishev–Sharafutdinov generalized theclassical Dirichlet-to-Neumann map to differential forms:
Λ : Ωp(∂M) → Ωn−p−1(∂M)
If ϕ ∈ Ωp(∂M), then let ω solve the boundary value problem
∆ω = 0, i∗ω = ϕ, i∗δω = 0.
DefineΛϕ := i∗? dω.
If f ∈ Ω0(∂M),
Λf = i∗? du =∂u
∂νdvol∂M = (Λclf ) dvol∂M
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Generalization to differential forms
Joshi–Lionheart and Belishev–Sharafutdinov generalized theclassical Dirichlet-to-Neumann map to differential forms:
Λ : Ωp(∂M) → Ωn−p−1(∂M)
If ϕ ∈ Ωp(∂M), then let ω solve the boundary value problem
∆ω = 0, i∗ω = ϕ, i∗δω = 0.
DefineΛϕ := i∗? dω.
If f ∈ Ω0(∂M),
Λf = i∗? du =∂u
∂νdvol∂M = (Λclf ) dvol∂M
![Page 57: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/57.jpg)
Generalization to differential forms
Joshi–Lionheart and Belishev–Sharafutdinov generalized theclassical Dirichlet-to-Neumann map to differential forms:
Λ : Ωp(∂M) → Ωn−p−1(∂M)
If ϕ ∈ Ωp(∂M), then let ω solve the boundary value problem
∆ω = 0, i∗ω = ϕ, i∗δω = 0.
DefineΛϕ := i∗? dω.
If f ∈ Ω0(∂M),
Λf = i∗? du =∂u
∂νdvol∂M = (Λclf ) dvol∂M
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Boundary data → topology
Theorem (Belishev–Sharafutdinov)
The data (∂M,Λ) completely determines the cohomology groupsof M.
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Connection to Poincare duality angles
Define the Hilbert transform T := dΛ−1.
TheoremIf θ1, . . . , θk are the Poincare duality angles of M in dimension p,then the quantities
(−1)np+n+p cos2 θi
are the non-zero eigenvalues of an appropriate restriction of T 2.
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Connection to Poincare duality angles
Define the Hilbert transform T := dΛ−1.
TheoremIf θ1, . . . , θk are the Poincare duality angles of M in dimension p,then the quantities
(−1)np+n+p cos2 θi
are the non-zero eigenvalues of an appropriate restriction of T 2.
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Idea of the Proof
1
cos θ
θcos
2 θ
The Hilbert transform T recaptures the orthogonal projectionHp
N(M) → HpD(M).
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Idea of the Proof
1
cos θ
θcos
2 θ
The Hilbert transform T recaptures the orthogonal projectionHp
N(M) → HpD(M).
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Cup products
Belishev and Sharafutdinov posed the following question:
Can the multiplicative structure of cohomologies berecovered from our data (∂M,Λ)? Till now, the authorscannot answer the question.
TheoremThe mixed cup product
∪ : Hp(M; R)× Hq(M, ∂M; R) → Hp+q(M, ∂M; R)
is completely determined by the data (∂M,Λ) when the relativeclass is restricted to come from the boundary subspace.
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Cup products
Belishev and Sharafutdinov posed the following question:
Can the multiplicative structure of cohomologies berecovered from our data (∂M,Λ)? Till now, the authorscannot answer the question.
TheoremThe mixed cup product
∪ : Hp(M; R)× Hq(M, ∂M; R) → Hp+q(M, ∂M; R)
is completely determined by the data (∂M,Λ) when the relativeclass is restricted to come from the boundary subspace.
![Page 65: Poincaré Duality Angles on Riemannian Manifolds with Boundaryclayton/research/talks/Duke.pdfPoincar´e Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63fb52224f0d331a231e21/html5/thumbnails/65.jpg)
Thanks!