Teichmüller theory, collapse of flat manifolds and applications to the Yamabe problem

Teichmüller theory, collapse of flat manifolds and applications to the Yamabe problem

Paolo Piccione

Departamento de Matemática Instituto de Matemática e Estatística

Universidade de São Paulo

July 25, 2017

Outiline of this talk

The Yamabe problem

Compact manifolds. Aubin inequality.

Noncompact manifolds. Example:Sⁿ\S^k.

Two arguments for the existence of multiple solutions.

Compact flat manifolds Bieberbach theorems.

Teichmüller space of flat metrics.

Algebraic description.

Existence of flat deformations.

Boundary of the flat Teichmüller space Collapse of flat manifolds;

Flat orbifolds.

Examples of 3-D collapse.

Acknowledgements

My co-authors

Renato Bettiol (UPenn) Andrzej Derdzinski (OSU) Bianca Santoro (CCNY)

My sponsors

Finding the best metric on a manifold

M a compact manifold;

a metricg onMis a smooth choice of measuring length and angles of tangent vectors toM:gp:TpM×TpM →R; metrics give a way of computing length of curves and distances;

metrics determinecurvatureinM.Best metrics are those that have the simplest curvature formulas.Constant.

Whenn=dim(M)>2, there are several notions of curvature:

sectional curvature constant: only if M =Rⁿ,Sⁿ,Hⁿ

Ricci curvature

constant: Einstein field equations in vacuum

scalar curvature constant:

Yamabe problem

The Yamabe problem

Two metricsg₁andg₂areconformalif they give the same anglesbetween vectors.

Conformal class ofg: [g] :=

metrics conformal tog Yamabe problem

Given a compact manifoldMⁿ(n≥3) and a metricg onM, does there existh∈[g]withscal_hconstant?

Solutionshto the Yamabe problem are critical points of the Hilbert–Einstein functionalA: [g]→R:

A(h) =vol(h)²⁻ⁿⁿ Z

M

scalhdM_h

Solution (Yamabe, Trudinger, Aubin, Schoen)

Yamabe problem on round spheres

Consider the round sphere(Sⁿ,g_round) Two special properties

[g_round]containsinfinitely manyconstant scalar curvature metrics (infact, anoncompactset!)

(Aubin inequality)foranycompact manifoldMⁿandany metricgonM:

Y(M,g)≤Y(Sⁿ,g_round)

The Yamabe problem on noncompact manifolds

Asymptotic condition:completeness.

Yamabe problem on noncompact manifolds

Given a noncompact(Mⁿ,g),n≥3, does there exist a completemetrich∈[g]withscalhconstant?

Counterexample (Jin Zhiren, 1980) Me compact, M=Me \ {p₁, . . . ,p_k}.

Chooseg onMe withscalg <0 (Aubin). Then:

there is noh∈[g]withscalh≥0;

Ifh∈[g]andscal_h<0, thenhis noncomplete (a priori estimates on an elliptic PDE).

A nice example: S

\ S

ConsiderM =Sⁿ\S^k (0≥k <n) metric: g_round

Sⁿ\S^k ∼=Rⁿ\R^k (stereographic projection) g_round∼=g_flat

Rⁿ\R^k = R^n−k \ {0}

×R^k g_flat=r²dθ²+dr²+dy²

∼=dθ²+ 1

r² dr²+dy²)

=S^n−k−1 × H^k+1 ⇐= complete and constant scalar curvature

Infinitely many solutions!

Theorem

When2k <n−2, there are infinitely many solutions of the Yamabe problem inSⁿ\S^k.

Proof.

Two different arguments:

(A) Topology (covering spaces) (B) Bifurcation theory (whenk =0,1).

This theorem producesperiodic solutions, i.e., coming from compact quotients.

The topological argument

1 H^k+1has compact quotients that give aninfinite tower of finite-sheeted Riemannian coverings:

(H^k+1,g_hyp)→. . .→(Σ₂,g₂)→(Σ₁,g₁)→(Σ₀,g₀)

2 Multiply by(S^n−k−1,g_round), product metrics:

. . .→(S^n−k−1×Σ₁,g_round⊕g₁)→(S^n−k−1×Σ₀,g_round⊕g₀)

3 pull-back Yamabe metric in[g_round⊕g₀]: Hilbert–Einstein energyAdiverges! (uses assumption2k <n−2)

4 ByAubin inequality, minimum ofAmust be attained at some other metric in the conformal class of the product.

5 Iterate.

Coverings of hyperbolic surfaces

Profinite completion and residually finite groups

Infinite tower of finite-sheeted coverings:

. . .→M_k →Mk−1→. . .→M₁→M₀ iffG=π₁(M₀)hasinfiniteprofinite completionG.b Def. Gb =lim

← G/Γ, ΓEG, G: Γ

<+∞.

Canonical homomorphismι:G→Gb Ker(i) = \

ΓEG [G:Γ]<+∞

Γ

Def. Gisresidually finiteif: \

ΓEG [G:Γ]<+∞

Γ ={1}

The bifurcation argument (k = 1)

1 Choose a compact quotient ofH²: a compact surfaceΣ^g of genusg≥2.

2 Σ^g has many nonisometric hyperbolic metrics.

Deformations: Teichmüller spaceT(Σ^g)∼=R^6g−6.

3 Forh∈ T(Σ^g),g_round⊕his a solution of the Yamabe problem inSⁿ⁻²×Σ^g

4 Each of these solutions has aMorse index, computed in terms of the spectrum of∆h.

5 Jump of Morse index when∆_hhas manysmalleigenvalues

=⇒ bifurcation must occur along paths inT(Σ^g).

Natural candidates to replace H

Simply-connected space(X,g)such that:

(a) scalgconstant;

(b) (X,g)admits an infinite tower of finite sheeted compact Riemannian coveringsX/Γ(Γhas infinite profinite completion)

(c) Arichspace of metrics inX/Γ(Teichmüller space) that are locally isometric tog, withsmallLaplacian eigenvalues.

Symmetric spaces

Theorem (Borel)

Symmetric spacesof noncompact type X admit irreducible compact quotients X/Γ.

X/Γloc. symmetric =⇒ constant scalar curvature Selberg–Malcev lemma

Finitely generated linear groups areresidually finite.

Corollary. Γ =π₁(X/Γ)has infiniteprofinite completion.

Simplest example:(X,g) = (Rⁿ,g_flat)

Γ⊂Iso(Rⁿ)∼=RⁿnO(n)is aBieberbach group.

Rⁿ/Γis a compact flat manifold/orbifold.

Compact flat manifolds and orbifolds

Γ⊂Iso(Rⁿ)is aBieberbachgroup:

(a) discrete;

(b) co-compact;

(c) torsion-free.

Theorem

(M,g)compact flat manifold⇐⇒ M =Rⁿ/Γ ΓBieberbach.

Algebraic structure:

0−→L−→Γ−→H −→1 L⊂Rⁿis a co-compact lattice

H ⊂O(n)is a finite group (holonomy)

Orbifolds:compact flat orbifolds have a similar structure:

Γ⊂Iso(Rⁿ)is acrystallographic, possibly with torsion.

Small eigenvalues

Theorem (Cheng)

(F^d,g_F)closed manifold with nonnegative Ricci curvature and unit volume. Then:

λj(F,g_F)≤2j² d(d+4) diam(F,g_F)².

For bifurcation purposes, need flat metrics with volume 1 and arbitrarily large diameter. Equivalently, fixed diameter and arbitrarily small volume. Collapse flat metrics!

Gromov–Hausdorff convergence and collapse

Hausdorff distance:X,Y ⊂Z: d_H^Z(X,Y) =inf

ε:X ⊂B(Y, ε)andY ⊂B(X, ε)

Gromov–Hausdorff distance: d_GH(X,Y) = inf

X,Y,→Zd_H^Z(X,Y) Gromov

M=

compact metric spaces isometries.

(M,d_GH)is acompletemetric space.

GH-limits can change topology, dimension...

Diameter is acontinuous functionin(M,d_GH).

Collapseof compact Riemannian manifolds:

GH−lim(M_i,g_i) = (X,d), with limvol(M_i,g_i) =0.

Bieberbach’s theorems (geometric version)

Theorem

(I) A compact flat n-orbifold(O,g)is isometrically covered by a flat n-torus.

(II) Compact flat orbifolds of the same dimension and with isomorphic fundamental groups are affinely diffeomorphic.

(III) For all n, there is only a finite number of affine equivalence classe of compact flat n-orbifolds.

n=2:

manifolds: torusT²and Klein bottleK²

orbifolds:17affine classes (wallpaper groups).

n=3: #mnfld=10, #orbfld=219

Flat 2-orbifolds

Underlying top. space: D²,S²,RP²,M²,K²,S¹×[0,1]

Cyclic or dihedral local groups (Leonardo da Vinci)

Conepoints andcornerreflections.

Notation: S(n₁, . . . ,n_k;m₁, . . . ,m_`)

A few pictures

S²(3,3,3; ) D²(;4,4,4,4)

D²(4;2)

Symmetries of the Vitruvian Man

Wallpaper group symmetries

Alhambra, Granada Spain

Copacabana Rio de Janeiro

Brazil

Teichmüller space of flat metrics

M compact manifold (orbifold) admitting a flat metric;

Flat(M)the set of all flat metrics onM;

M_flat =Flat(M)/Diff(M)moduli space of flat metrics onM;

T_flat(M) =Flat(M)/Diff₀(M)space ofdeformationsof flat metrics (Teichmüller space).

Theorem

T_flat(M)is a real-analytic manifold (homogeneous space) diffeomorphic to someR^d.

M_flat(M) =T_flat(M)/MCG(M), where themapping class groupMCG(M)is countable and discrete.

Algebraic description

H ⊂O(n)holonomy representation Rⁿ=

`

L

i=1

W_i, whereW_i isotypical component W_i direct sum ofm_icopies of an irreducible Ki =R,C,HtypeofW_i

Theorem T_flat(M) =

`

Q

i=1

GL(mi,Ki) O(mi,Ki)

GL(m_i,Ki) O(m_i,Ki)

∼=R^di, d_i =







1

2m_i(m_i+1), ifKi =R, m²_i, ifKi =C, m_i(2m_i−1), ifKi =H.

Existence of nontrivial flat deformations

Theorem (Hiss–Szczepa ´nski)

Holonomy repn of a compact flat manifold isneverirreducible.

Corollary

Compact flat manifolds admit nonhomothetic flat deformations.

Proof. dim T_flat(M)

≥2.

Obs.Flat orbifolds can berigid!!(⇐⇒irreducible holonomy) Theorem

(M₀,g₀)closed Riemannian manifold withscalg0 >0 Γ⊂Iso(R^d)Bieberbach, d ≥2.

Then there exist infinitely many branches ofΓ-periodicsolutions to the Yamabe problem on M₀×R^d,g₀⊕g_flat

.

Examples of Teichmüller space

n-torus

T_flat(Tⁿ)∼=GL(n)/O(n)∼=R^12n(n+1) MCG(Tⁿ) =GL(n,Z).

Kummer surface

O=T⁴/Z2(antipodal map on each coordinate).

16 conical singularities

Holonomy rep: 4 copies of nontrivialZ2-rep T_flat(O)∼=GL(4,R)/O(4)∼=R¹⁰

Klein bottle,Möbius band,cylinder

H=Z2(reflection).T_flat(O)∼=R²

Joyce orbifolds

6-dim flat orbifolds (desingularized to Calabi–Yau mnflds)

O₁=T⁶/Z4, holonomy generated by diag(−1,i,i)ofC³∼=R⁶. T_flat(O₁)∼=GL(2,R)/O(2)×GL(2,C)/U(2)∼=R⁷

Boundary of the flat Teichmüller space – 1

Theorem

TheGromov–Hausdorff limitof a sequence(Mⁿ,g_i)of compact flat manifolds is a compact flat orbifold.

Proof.

Result true for flat tori (Mahler’scompactness thm) can assume all holonomy groups equal: H_i =H

(M,g_i)is the quotient of a flat torus(Tⁿ,ge_i)by an isometric free action ofH.

ByFukaya–Yamaguchi:

lim(M,g_i) =lim(Tⁿ/H,g_i) =lim(Tⁿ,g_i)/H.

Boundary of the Teichmüller space – 2

Flat orbifolds admitflat desingularization (through higher dimensional manifolds):

Theorem

Every compact flat orbifold is the limit of a sequence of compact flat manifolds.

Proof.

Oⁿ=Rⁿ/Γ, Γ⊂Iso(Rⁿ)crystallographic

H⊂O(n)holonomy,O=Tⁿ/H(possiblynonfreeaction).

Auslander–Kuranishi:∃BieberbachN-groupΓ⁰ withHol(Γ⁰) =H.

CompactN-manifoldM₀=R^N/Γ⁰=T^N/H(freeaction).

Diagonal action ofHonTⁿ×T^N: this isfree!

SetM = (Tⁿ×T^N)/H, compact flat manifold.

Boundary in the 3-D case

Theorem

The G-H limit of a sequence of compact flat3manifold belongs to one of these classes:

point

closed interval, circle

2-torus, Klein bottle, Möbius band, cylinder

flat disk with two singularities: D²(4;2), D²(3;3)or D²(2,2; )

flat sphere with singularities: S²(3,3,3; )or S²(2,2,2,2; ) (pillowcase)

projective plane with2singularitiesRP²(2,2; ).

A few more pictures

Collapse of flat 3-manifolds Need higher dimensional collapse

Thanks for your attention!!

See you at ICM2018!