Introduction
Torus Method
Applications
Isospectral Alexandrov spaces
Martin Weilandt
funded by FAPESP
2011-04-08 / IME-USP
Summary
Introduction
Torus Method
Applications
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
The Isospectrality Problem
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
The Isospectrality Problem
∆ on manifolds
Definition
Let M be a Riemannian manifold.
The Laplace Operator is given by
∆ : C ∞ (M) ∋ f 7→ δdf = − div grad f ∈ C ∞ (M).
Theorem
Let M be a compact Riemannian manifold. Then:
The eigenspaces of ∆ are finite-dimensional.
The eigenvalues (and dim. of eigenspaces) of ∆
determine:
vol(M),
dim(M)
R
scal
and
other curvature integrals
M
Summary
Introduction
Torus Method
Applications
The Isospectrality Problem
The Isospectrality Problem
Question (L. Green, M. Kac 1960s)
Are compact Riem. manifolds with the same ∆-eigenvalues
(and multiplicities) isometric?
Summary
Introduction
Torus Method
Applications
Summary
The Isospectrality Problem
The Isospectrality Problem
Question (L. Green, M. Kac 1960s)
Are compact Riem. manifolds with the same ∆-eigenvalues
(and multiplicities) isometric?
Counterexamples (selection)
Milnor (1964): Isospectral flat tori (dim=16)
Sunada (1985): Condition for isospectrality of properly
discontinuous quotients M/Γ1 , M/Γ2
Gordon (1993): Isospectral, not locally isometric manifolds
via “torus method”
Schüth (1999): Isospectral metrics on simply connected
manifolds
Introduction
Torus Method
Applications
Alexandrov spaces
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
Alexandrov spaces
Definition of Alexandrov spaces
Definition
Let K ∈ R. An Alexandrov space of curvature ≥ K is a
complete locally compact intrinsic metric space X such that
every point in X has a neighbourhood in which every geodesic
triangle has a comparison triangle ∆āb̄c̄ in MK such that
|ad| = |ād̄| implies |bd| ≥ |b̄d̄|.
c
d
c̄
d̄
b̄
b
a
ā
X
MK
Introduction
Torus Method
Applications
Summary
Alexandrov spaces
Definition of Alexandrov spaces
Definition
Let K ∈ R. An Alexandrov space of curvature ≥ K is a
complete locally compact intrinsic metric space X such that
every point in X has a neighbourhood in which every geodesic
triangle has a comparison triangle ∆āb̄c̄ in MK such that
|ad| = |ād̄| implies |bd| ≥ |b̄d̄|.
c
d
c̄
d̄
b̄
b
a
ā
X
MK
Example (Topogonov’s Theorem)
Complete Riemannian manifold with sect. curv. bounded below
Introduction
Torus Method
Applications
Alexandrov spaces
Examples and Properties of Alexandrov spaces
Example (Burago, Gromov, Perel’man 1992)
Given:
(M, g) complete Riem. manifold with sect. curv. ≥ K ∈ R
G compact group of isometries on (M, g)
Then: (M/G, dg ) Alexandrov space of curvature ≥ K .
Summary
Introduction
Torus Method
Applications
Alexandrov spaces
Examples and Properties of Alexandrov spaces
Example (Burago, Gromov, Perel’man 1992)
Given:
(M, g) complete Riem. manifold with sect. curv. ≥ K ∈ R
G compact group of isometries on (M, g)
Then: (M/G, dg ) Alexandrov space of curvature ≥ K .
Special case: If all stabilizers Gx , x ∈ M, are finite, then
M/G is a Riemannian orbifold.
A good orbifold is a quotient M/Γ with Γ ⊂ Isom(M)
discrete.
Remark
Within the class of compact metric spaces we have:
manifolds ⊂ good orbifolds ⊂ orbifolds ⊂ Alexandrov spaces
Summary
Introduction
Torus Method
Applications
The Laplacian on Alexandrov spaces
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
The Laplacian on Alexandrov spaces
The set of regular points as a manifold
Let X be an Alexandrov space of dimension n ≥ 2 and consider
the dense subset
X reg := {x ∈ X ;
1
Bǫ (x) → B1 (0) ⊂ Rn for ε → 0}.
ε
Assume that X reg is open in X (automatically satisfied in our
special case M/G with G a torus). Then
Theorem (Kuwae, Machigashira, Shioya 2001)
There is a unique continuous Riemannian metric g on X reg
such that the distance function dg induced by g coincides
with the original metric on X reg .
X reg carries a unique C ∞ -manifold structure such that g is
a limit of C ∞ -Riemannian metrics on X reg
These observations enable us to define the Laplacian on X ...
Introduction
Torus Method
Applications
The Laplacian on Alexandrov spaces
Definition (Kuwae, Machigashira, Shioya 2001)
Let X be an Alexandrov space. Then set
H 1 (X , R) := {u : X → R; u measurable, u|X reg ∈ H 1 (X reg , R)
∆ : D(∆) → L2 (X , R) maximal self-adjoint operator such that
D(∆) ⊂ H 1 (X , R)
R
R
1
X u∆v = X h∇u, ∇v i ∀u ∈ H (X , R), v ∈ D(∆)
Summary
Introduction
Torus Method
Applications
The Laplacian on Alexandrov spaces
Definition (Kuwae, Machigashira, Shioya 2001)
Let X be an Alexandrov space. Then set
H 1 (X , R) := {u : X → R; u measurable, u|X reg ∈ H 1 (X reg , R)
∆ : D(∆) → L2 (X , R) maximal self-adjoint operator such that
D(∆) ⊂ H 1 (X , R)
R
R
1
X u∆v = X h∇u, ∇v i ∀u ∈ H (X , R), v ∈ D(∆)
Theorem
For a compact Alexandrov space one has as for manifolds:
1
Spectrum of ∆: eigenvalues 0 = λ0 ≤ λ1 ≤ . . . ր ∞,
L2 (O) has ON base (φi ) with ∆φi = λi φi (Kuwae et al.).
2
(1) implies Min-Max-Principle:
R
λk = infU∈Lk supf ∈U\{0}
2
XR k∇f k
2
X |f |
with L the set of k -dim. subspaces of H 1 (X ).
Summary
Introduction
Torus Method
Applications
The Laplacian on Alexandrov spaces
Special Case: Orbifolds
Heat kernel expansion shows: The spectrum on orbifolds
determines volume, dimension and certain curvature
integrals. (Not known for general Alexandrov spaces.)
Summary
Introduction
Torus Method
Applications
Summary
The Laplacian on Alexandrov spaces
Special Case: Orbifolds
Heat kernel expansion shows: The spectrum on orbifolds
determines volume, dimension and certain curvature
integrals. (Not known for general Alexandrov spaces.)
This motivates constructions of isospectral orbifolds:
Theorem (Sunada 1985, Bérard 1992)
Let M be a compact Riem. manifold, Γ1 , Γ2 ⊂ Isom(M) finite
subgroups with a bijection φ : Γ1 → Γ2 such that
φ(γ) = gγ γgγ−1 . Then M/Γ1 and M/Γ2 are isospectral orbifolds.
Theorem (Rossetti, Schüth, Weilandt 2007)
Let G be a compact Lie group with bi-invariant metric, let
Γ1 , Γ2 ⊂ G satisfy the conditions above and set M := G/Γ1 .
Then Γ1 \M, Γ2 \M are isospectral orbifolds and have different
maximal isotropy orders.
Introduction
Torus Method
Applications
The Laplacian on Alexandrov spaces
Other Possibilities
Formula for dimensions of eigenspaces on compact flat
orbifolds Rn /Γ (Miatello, Rossetti 2001)
Given “equivariantly isospectral” G-manifolds M1 , M2 , one
can construct isospectral orbifolds M1 /Γ1 , M2 /Γ2 (with
Γi ⊂ G) (Sutton 2006 based on Pesce).
Note: All these constructions yield isospectral good orbifolds,
i.e., orbifolds isometric to quotients M/Γ with M a Riem.
manifold and Γ ⊂ Isom(M) discrete.
Summary
Introduction
Torus Method
Applications
Summary
The Laplacian on Alexandrov spaces
Open Question
Central Problem
Can a singular space (sing. orbifold/sing. Alexandrov space) be
isospectral to a manifold?
Still open. However, there are some obstructions, e.g.:
Theorem (Gordon, Rossetti 2003)
Let Γi act properly discontinuously on M and assume that
M/Γ1 , M/Γ2 are isospectral. Then M/Γ1 is a manifold if and
only if M/Γ2 is.
Together with a similar obstruction this implies: No known
construction of isospectral orbifolds can provide a
(positive) answer to the question above.
Therefore: Study bad orbifolds - or more generally
Alexandrov spaces
Introduction
Torus Method
Applications
The Manifold Case
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
The Manifold Case
The Torus Method on Manifolds
Theorem (Schüth 2001)
Let T be a (non-trivial) torus acting effectively and
isometrically on two connected Riem. manifolds M1 , M2 .
b i ⊂ Mi denote the (dense) subset of points in which T
Let M
acts freely.
Assume: For every subtorus W ⊂ T of codimension 1
there is a T -equivariant diffeo FW : M1 → M2 satisfying:
∗
b 1 /W → M
b 2 /W is an isometry.
FW
dvol2 = dvol1 , F W : M
Then: M1 and M2 isospectral.
Proof.
Idea: Use Min-Max-Principle, representation theory and
b i ⊂ Mi (based on ideas of Gordon 1993).
denseness M
Introduction
Torus Method
Applications
Summary
The Manifold Case
The Torus Method on Manifolds in Practice
Let T act isometrically on a fixed Riem. manifold (M, g0 )
For λ a T -invariant and T -horizontal t-valued 1-form on M
d
exp(tλx (X ))x]
define [with λ(X )∗x := dt
|t=0
gλ (X , Y ) := g0 (X + λ(X )∗ , Y + λ(Y )∗ )
L := ker(exp(t → T )) and L∗ := {µ ∈ t∗ ; µ(Z ) ∈ Z ∀Z ∈ L}
Introduction
Torus Method
Applications
Summary
The Manifold Case
The Torus Method on Manifolds in Practice
Let T act isometrically on a fixed Riem. manifold (M, g0 )
For λ a T -invariant and T -horizontal t-valued 1-form on M
d
exp(tλx (X ))x]
define [with λ(X )∗x := dt
|t=0
gλ (X , Y ) := g0 (X + λ(X )∗ , Y + λ(Y )∗ )
L := ker(exp(t → T )) and L∗ := {µ ∈ t∗ ; µ(Z ) ∈ Z ∀Z ∈ L}
Theorem (λ-torus method, Schüth 2001)
Let λ1 , λ2 be 1-forms as above such that: For every µ ∈ L∗
there is a T -equivariant Fµ ∈ Isom(M, g0 ) with
µ ◦ λ1 = Fµ∗ (µ ◦ λ2 ). Then: (M, gλ1 ) and (M, gλ2 ) are isospectral.
Proof.
For W ⊂ T from theorem above choose µ ∈ L∗ with
ker µ = Te W and use FW := Fµ .
Introduction
Torus Method
Applications
Generalization to Alexandrov Spaces
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
Generalization to Alexandrov Spaces
The Torus Method on (Special) Alexandrov Spaces
Fix a manifold M with an action of a torus G and two
metrics g1 , g2 on M. Under which condition are (M/G, dg1 )
and (M/G, dg2 ) isospectral Alexandrov spaces?
Introduction
Torus Method
Applications
Summary
Generalization to Alexandrov Spaces
The Torus Method on (Special) Alexandrov Spaces
Fix a manifold M with an action of a torus G and two
metrics g1 , g2 on M. Under which condition are (M/G, dg1 )
and (M/G, dg2 ) isospectral Alexandrov spaces?
Theorem (M.W.)
Let T be a non-triv. torus acting on M, isom. w.r.t. g1 and g2 .
Assume that the T - and G-actions commute and that the
T -action on M/G is effective and set
b := {x ∈ M; Gx = {IdM }, T[x] = {IdM/G }} ⊂ M.
M
Assume that for every W ⊂ T of codim. 1 there is a G- and
∗ dvol = dvol and
T -equiv. diffeo EW on M which satisfies EW
g2
g1
b
b
, g1W ) → ((M/G)/W
, g2W ).
induces an isometry FW : ((M/G)/W
Then the Alexandrov spaces (M/G, dg1 ), (M/G, dg2 ) are
isospectral.
Introduction
Torus Method
Applications
Summary
Generalization to Alexandrov Spaces
Proof.
(following Schüth 2001)
Consider the complex Sobolev spaces
H i := H 1 (M/G, dgi ), i = 1, 2.
L
Decompose H i = µ∈L∗ Hµi with L = ker expT and
Hµi = {f ∈ H i ; [Z ]f = e2πiµ(Z ) f ∀Z ∈ t}.
L
L
i
i :=
i
i
i
With SW
µ∈L∗ \{0} Hµ we obtain H = H0 ⊕
W SW .
Te W =ker µ
∗ : S 2 → S 1 is an L2 -preserving
Assumptions imply that FW
W
W
isometry.
Patch together to L2 -preserving isometry H 2 → H 1 .
Min-Max-Principle gives isospectrality of (M/G, dg1 ) and
(M/G, dg2 ).
Introduction
Torus Method
Applications
Summary
Generalization to Alexandrov Spaces
In analogy to the manifold setting the theorem above yields the
λ-torus method on Alexandrov spaces (of the form M/G):
Fix (M, g0 ), commuting tori G, T ⊂ Isom(M).
A t-valued 1-form λ on M is called admissible if it is
invariant and horizontal w.r.t. G and T .
gλ (X , Y ) := g0 (X + λ(X )∗ , Y + λ(Y )∗ )
Theorem
Let λ1 , λ2 be admissible 1-forms on M such that for every
µ ∈ L∗ there is a G- and T -equivariant Eµ ∈ Isom(M, g0 )
satisfying
µ ◦ λ1 = Eµ∗ (µ ◦ λ2 )
Then (M/G, dλ1 ) and (M/G, dλ2 ) are isospectral Alexandrov
spaces.
Introduction
Torus Method
Applications
Non-Isometry
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Non-Isometry
When are (M/G, dλ1 ), (M/G, dλ2 ) not isometric?
Idea: Use criterion from Schüth 2001. Gives criterion on
b
b
λ1 , λ2 for non-isometry of (M/G,
gλ1 ), (M/G,
gλ2 ).
Summary
Introduction
Torus Method
Applications
Isospectral non-isometric Alexandrov spaces
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
Isospectral non-isometric Alexandrov spaces
Applications of the λ-torus method
Schüth 2001: M.W.:
T := S 1 × S 1 ⊂ C × C acts effectively and isometrically on
M := S 2n+1 via z(u, v ) = (u, zv ) with u ∈ Cn−1 , v ∈ C2 .
Let G := S 1 act on S 2n+1 via σ(u, v ) := (σu, v ). T -action
above induces an effective isometric action of T on
S 2n+1 /S 1 .
For j : t → su(n − 1) linear define t ≃ R2 -valued 1-form
λ = (λ1 , λ2 ) on S 2n+1 :
λk(u,v ) (U, V ) := kuk2 hjZk u, Ui − hU, iuihjZk u, iui
with k = 1, 2 and Z1 = (i, 0), Z2 = (0, i) basis of
t = T(1,1) (S 1 × S 1 ) ⊂ C × C.
Observe:
λ is T -invariant and -horizontal (Schüth 2001).
λ is also S 1 -invariant und -horizontal.
Introduction
Torus Method
Applications
Summary
Isospectral non-isometric Alexandrov spaces
Applications of the λ-torus method
Schüth 2001: M.W.:
T := S 1 × S 1 ⊂ C × C acts effectively and isometrically on
M := S 2n+1 via z(u, v ) = (u, zv ) with u ∈ Cn−1 , v ∈ C2 .
Let G := S 1 act on S 2n+1 via σ(u, v ) := (σu, v ). T -action
above induces an effective isometric action of T on
S 2n+1 /S 1 .
For j : t → su(n − 1) linear define t ≃ R2 -valued 1-form
λ = (λ1 , λ2 ) on S 2n+1 :
λk(u,v ) (U, V ) := kuk2 hjZk u, Ui − hU, iuihjZk u, iui
with k = 1, 2 and Z1 = (i, 0), Z2 = (0, i) basis of
t = T(1,1) (S 1 × S 1 ) ⊂ C × C.
Observe:
λ is T -invariant and -horizontal (Schüth 2001).
λ is also S 1 -invariant und -horizontal.
Introduction
Torus Method
Applications
Isospectral non-isometric Alexandrov spaces
Theorem (M.W.)
Let j1 , j2 : t → su(n − 1) be isospectral linear maps, i.e.:
For every Z ∈ t there is AZ ∈ SU(n − 1) such that
j2 (Z ) = AZ j1 (Z )A−1
Z .
Define 1-forms λ1 , λ2 on M as on the preceding slide.
Then the Alexandrov spaces (M/S 1 , gλ1 ), (M/S 1 , gλ2 ) are
isospectral.
Proof.
Use λ-torus-method: Let µ ∈ L∗ ⊂ t∗ . Set
Z := µ(Z1 )Z1 + µ(Z2 )Z2 and choose AZ as in the assumption.
Eµ := (AZ , Id) ∈ SU(n − 1) × SU(2) satisfies
µ ◦ λ1 = Eµ∗ (µ ◦ λ2 ) on S 2n+1 and is T -equivariant (Schüth
2001). Eµ is is also S 1 -equivariant.
Summary
Introduction
Torus Method
Applications
Isospectral non-isometric Alexandrov spaces
Theorem (M.W.)
Let j1 , j2 : t → su(n − 1) be isospectral linear maps, i.e.:
For every Z ∈ t there is AZ ∈ SU(n − 1) such that
j2 (Z ) = AZ j1 (Z )A−1
Z .
Define 1-forms λ1 , λ2 on M as on the preceding slide.
Then the Alexandrov spaces (M/S 1 , gλ1 ), (M/S 1 , gλ2 ) are
isospectral.
Proof.
Use λ-torus-method: Let µ ∈ L∗ ⊂ t∗ . Set
Z := µ(Z1 )Z1 + µ(Z2 )Z2 and choose AZ as in the assumption.
Eµ := (AZ , Id) ∈ SU(n − 1) × SU(2) satisfies
µ ◦ λ1 = Eµ∗ (µ ◦ λ2 ) on S 2n+1 and is T -equivariant (Schüth
2001). Eµ is is also S 1 -equivariant.
Summary
Introduction
Torus Method
Applications
Isospectral non-isometric Alexandrov spaces
Theorem (Schüth 2001)
For n ≥ 4 there are continuous families j(t) : t → su(n − 1) of
isospectral 1-forms which via λ(t) induce isospectral pairwise
non-isometric metrics on S 2n+1 .
The same reasoning applied to S 2n+1 /S 1 shows:
Theorem (M.W.)
For the families j(t) from the theorem above the Alexandrov
spaces (S 2n+1 /S 1 , dλ(t) ) are pairwise non-isometric.
Moreover, results from foliation theory (due to Lytchak,
Alexandrino) show:
Theorem
None of our Alexandrov spaces (S 2n+1 /S 1 , dλ(t) ) can be
isometric to a Riemannian orbifold.
Summary
Introduction
Torus Method
Applications
Isospectral non-isometric Alexandrov spaces
Theorem (Schüth 2001)
For n ≥ 4 there are continuous families j(t) : t → su(n − 1) of
isospectral 1-forms which via λ(t) induce isospectral pairwise
non-isometric metrics on S 2n+1 .
The same reasoning applied to S 2n+1 /S 1 shows:
Theorem (M.W.)
For the families j(t) from the theorem above the Alexandrov
spaces (S 2n+1 /S 1 , dλ(t) ) are pairwise non-isometric.
Moreover, results from foliation theory (due to Lytchak,
Alexandrino) show:
Theorem
None of our Alexandrov spaces (S 2n+1 /S 1 , dλ(t) ) can be
isometric to a Riemannian orbifold.
Summary
Introduction
Torus Method
Applications
Isospectral Metrics on Orbifolds
Outline
1
Introduction
The Isospectrality Problem
Alexandrov spaces
The Laplacian on Alexandrov spaces
2
The Torus Method
The Manifold Case
Generalization to Alexandrov Spaces
Non-Isometry
3
Applications
Isospectral non-isometric Alexandrov spaces
Isospectral Metrics on Orbifolds
Summary
Introduction
Torus Method
Applications
Summary
Isospectral Metrics on Orbifolds
Isospectral Metrics on Weighted Projective Spaces
Let p, q ∈ N pairwise prime, n ≥ 4.
Again consider M = S 2n+1 , G = S 1 , but with the action
σ(u, v ) := (σ p u, σ q v )
All stabilizers are finite (G(0,v ) ≃ Zq , G(u,0) ≃ Zp ), hence
M/G is an orbifold (a weighted projective space W(p, q))
The same construction as before applies
metrics on any fixed W(p, q).
isospectral
The special case CPn (i.e. p = q = 1) has been treated by
Rückriemen (2006).
For the spectrum of weighted projective spaces with their
standard metrics see work by Abreu, Freitas, Godinho,
Dryden (2008) and Guillemin, Uribe, Wang (2008).
Introduction
Torus Method
Applications
Summary
Summary
Theorem
For every n ≥ 4 there are families of isospectral non-isometric
bad Riemannian orbifolds (weighted projective spaces) or even
more general Alexandrov spaces of dimension 2n.
Appendix
References I
K. Kuwae, Y. Machigashira, T. Shioya.
Sobolev spaces, Laplacian, and heat kernel on Alexandrov
spaces
Mathematische Zeitschrift 238 (2001)
D. Schüth.
Isospectral metrics on five-dimensional spheres.
J. Differential Geometry 58 (2001).
M. Weilandt
Isospectral Alexandrov spaces.
preprint on arxiv, 2011.