hrant hakobyan - mathematics

MODULI ESTIMATES IN SLIT DOMAINS AND APPLICATIONS

Hrant Hakobyan

Kansas State University

09/29/2011Institut Mittag-Leffler

Outline

PreliminariesModulusQS/QC and modulus

Previous resultsSlit domainsco-Hopficity

Results

Proofs

Questions

Modulus in Rn

I Given a curve family Γ ⊂ Rn we say a Borel function ρ : Rn → R[0,∞) isadmissible for Γ, denoted by ρ ∧ Γ, if∫

γρds ≥ 1, ∀γ ∈ Γ.

I Let p ≥ 1. p-modulus of Γ is

modpΓ = infρ∧Γ

∫RnρpdLn.

I Ln is n-regular, i.e. there is a constant C ≥ 1 s.t. for every x ∈ R and R > 0 holds

C−1Rn ≤ LnB(x ,R) ≤ CRn.

I In Rn the n-modulus modnΓ is called conformal modulus.I Examples. Upper bounds.I Conformal invariance in C.I Relation to harmonic and p-harmonic functions.

Modulus in Rn


γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫RnρpdLn.



I In Rn the n-modulus modnΓ is called conformal modulus.

I Examples. Upper bounds.I Conformal invariance in C.I Relation to harmonic and p-harmonic functions.

Modulus in Rn


γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫RnρpdLn.



I In Rn the n-modulus modnΓ is called conformal modulus.I Examples. Upper bounds.

I Conformal invariance in C.I Relation to harmonic and p-harmonic functions.

Modulus in Rn


γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫RnρpdLn.



I In Rn the n-modulus modnΓ is called conformal modulus.I Examples. Upper bounds.I Conformal invariance in C.

I Relation to harmonic and p-harmonic functions.

Modulus in Rn


γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫RnρpdLn.



I In Rn the n-modulus modnΓ is called conformal modulus.I Examples. Upper bounds.I Conformal invariance in C.I Relation to harmonic and p-harmonic functions.

1

mod

2(Γ

)

Γ

EF

Γ(E ,F ; W )

W

Conformal invariance of 2-modulus in C

EF

W

Φ− conformal

Γ(E ,F ; W )

F ′ = Φ(F )

Γ′ = Γ(E ′,F ′; W ′)

W ′ = Φ(W )

E ′ = Φ(E)

For any conformal mapping Φ : W → W ′ let Γ′ = Φ(Γ). Then

mod2Γ′ = mod2Γ.

Extremal Metrics and Mappings: simply connected case.

EF

Γ(E ,F ; W )

Φ

mod

2 Γ(E

,F;W

)

1

There is a unique conformal mapping which maps W onto a rectangle so that E ,F aremapped onto the vertical sides. No analogs in higher dimensions.

Extremal Metrics and Mappings: multiply connected case.

EF

Γ(E ,F ; W )

Φ

mod

2 Γ(E

,F;W

)

1

There is a unique conformal mapping which maps W onto a rectangle with horizontalslits removed and so that E ,F are mapped onto the vertical sides.

Relation to harmonic functions.

u|E = 0u|F = 1

∂u∂n = 0

∆u = 0

I Let U be the solution to this mixed Dirichlet-Neumann problem.

I Then ρ(z) = ‖∇U(z)‖ is the extremal metric for the 2-modulusI mod2Γ(E ,F ; W ) =

∫ ∫W ‖∇U‖2dA.

I Φ = U + iU∗.


u|E = 0u|F = 1

∂u∂n = 0

∆u = 0

I Let U be the solution to this mixed Dirichlet-Neumann problem.I Then ρ(z) = ‖∇U(z)‖ is the extremal metric for the 2-modulus

I mod2Γ(E ,F ; W ) =∫ ∫

W ‖∇U‖2dA.I Φ = U + iU∗.


u|E = 0u|F = 1

∂u∂n = 0

∆u = 0

I Let U be the solution to this mixed Dirichlet-Neumann problem.I Then ρ(z) = ‖∇U(z)‖ is the extremal metric for the 2-modulusI mod2Γ(E ,F ; W ) =

∫ ∫W ‖∇U‖2dA.

I Φ = U + iU∗.

Modulus in Rn


γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫RnρpdLn.



I In Rn the n-modulus modnΓ is called conformal modulus.

Modulus in an Ahlfors regular metric measure space (X ,d , µ).

I Given a metric measure space (X , d , µ) and a curve family Γ ⊂ X we sayρ : X → R≥0 is admissible for Γ, denoted by ρ ∧ Γ, if∫

γρds ≥ 1, ∀γ ∈ Γ.


modpΓ = infρ∧Γ

∫Xρpdµ.

I µ is Q-regular, if there is a constant C ≥ 1 s.t. for every x ∈ X and R ≤ diamXholds

C−1RQ ≤ µB(x ,R) ≤ CRQ .

I If (X , d , µ) is Q regular then modQΓ is called conformal modulus.

QS and QC

I A homeomorphism F : X → Y between metric spaces is (weakly)Quasisymmetric (QS) if ∃H ≥ 1 such that for every triple of distinct pointsx , y , z ∈ X the following holds

|x − y ||x − z|

≤ 1 =⇒|f (x)− f (y)||f (x)− f (z)|

≤ H.

I A homeomorphism F : (X , µ)→ (Y , ν) between Q regular metric spaces is(geometrically) Quasiconformal (QC) if ∃K ≥ 1 s.t. for every family of curvesΓ ⊂ X

K−1modQΓ ≤ modQ f (Γ) ≤ K modQΓ.

Theorem (Tyson’2001)If X ,Y are locally connected, compact Q-regular metric spaces (Q > 1) andF : X → Y is QS then

K−1modQΓ ≤ modQ f (Γ) ≤ K modQΓ.

QUASISYMMETRY

R

f

r

R/H

xf(x)

Self-similar slit domains in R2

Sequence of slit domains corresponding to 12 = { 1

2 ,12 , . . .}.

Self-similar slit domains in R2

horizontal families in S1,S2, and S3.

Theorem (Merenkov’2010)mod2Γi (1/2)→ 0, as i →∞ in R2.

Used conformal mappings and conformal invariance of 2-modulus. Not generalizableto higher dimensions and p 6= 2.

Theorem (H.)modpΓi (s)→ 0, as i →∞ in Rk if s /∈ `k , ∀p ≥ 1.

Works in any dimension. Construct explicit ρ-s. Will only show in the cases = ( 1

2 ,12 , . . .).

CO-HOPFICITY (INTO⇒ ONTO

I A group G is co-Hopf if every monomorphism of G is an isomorphism.

Finite groups. Many infinite groups.I X is topologically co-Hopf if every continuous embedding of X into itself is onto.

Spheres.I (X , dX ) is quasi-isometrically co-Hopf if every quasi-isometric embedding of X into

itself is a quasi-isometry.Euclidean spaces.

I Every topologically co-Hopf space is QS co-Hopf.I Want examples of bounded complete metric spaces which are QS co-Hopf but not

topologically co-Hopf.


I A group G is co-Hopf if every monomorphism of G is an isomorphism.Finite groups. Many infinite groups.

I X is topologically co-Hopf if every continuous embedding of X into itself is onto.Spheres.

I (X , dX ) is quasi-isometrically co-Hopf if every quasi-isometric embedding of X intoitself is a quasi-isometry.Euclidean spaces.





I X is topologically co-Hopf if every continuous embedding of X into itself is onto.

Spheres.I (X , dX ) is quasi-isometrically co-Hopf if every quasi-isometric embedding of X into

itself is a quasi-isometry.Euclidean spaces.






I (X , dX ) is quasi-isometrically co-Hopf if every quasi-isometric embedding of X intoitself is a quasi-isometry.

Euclidean spaces.I Every topologically co-Hopf space is QS co-Hopf.I Want examples of bounded complete metric spaces which are QS co-Hopf but not






I Every topologically co-Hopf space is QS co-Hopf.

I Want examples of bounded complete metric spaces which are QS co-Hopf but nottopologically co-Hopf.

QUASISYMMETRIC CO-HOPFICITY

DefinitionA metric space X is quasisymmetrically co-Hopfian if every QS mapping of X into itselfis onto.

I Examples: Sn

= ∂Hn. Rn.I Want examples of bounded complete metric spaces which are QS co-Hopf but not

topologically co-Hopf. E.g. something without manifold points (boundaries ofgroups/hyperbolic buildings).



I Examples: Sn = ∂Hn.

Rn.I Want examples of bounded complete metric spaces which are QS co-Hopf but not




I Examples: Sn = ∂Hn. Rn.

I Want examples of bounded complete metric spaces which are QS co-Hopf but nottopologically co-Hopf. E.g. something without manifold points (boundaries ofgroups/hyperbolic buildings).



I Examples: Sn = ∂Hn. Rn.I Want examples of bounded complete metric spaces which are QS co-Hopf but not


Standard Sierpinski carpet and Menger curve

Sierpinski carpet :M2,1. Menger curve :M3,1.

DefineMn,k .

Standard Sierpinski carpet and Menger curve

Sierpinski carpet :M2,1. Menger curve :M3,1.DefineMn,k .

Theorem (Merenkov’10)There is a metric space X homeomorphic to the Sierpinski carpet S =M2,1 which isQS co-Hopfian.

Slit carpet

I Define S(1/2) and DS(1/2).I These are both porous.I Let s = {si}∞i=0 be a sequence of positive reals. Define DM(s).


Slit carpet

I Define S(1/2) and DS(1/2).

I These are both porous.I Let s = {si}∞i=0 be a sequence of positive reals. Define DM(s).


Slit carpet

I Define S(1/2) and DS(1/2).I These are both porous.

I Let s = {si}∞i=0 be a sequence of positive reals. Define DM(s).


Slit carpet

I Define S(1/2) and DS(1/2).I These are both porous.I Let s = {si}∞i=0 be a sequence of positive reals. Define DM(s).

QuestionWhat about the Menger curveM3,1 (orMn,k , n ≥ 2, k ∈ [1, n − 1])?

QuestionIs it possible that si → 0 but DMn,k (s) is QS co-Hopfian?

QuestionWhen is DM(s) QS co-Hopfian?

Theorem (H.)There is a metric space X homeomorphic toMn,k which is QS co-Hopfian for everyn = 2, 3, . . . and k = 1, . . . , n − 1.

Theorem (H.)DMn,n−1(s) is QS co-Hopfian if s /∈ `n for every n = 2, 3, . . ..

ProblemDMn,n−1(s) is QS co-Hopfian if and only if s /∈ `n (i.e.

∑∞i=1 sn

i =∞).

Proof I.

Proof I. Main Lemma.

M M(ε)

1

ε

1

0 0

Lemma

Aε2 ≤ M(ε)−M ≤ Bε2.

Proof II. Step 1, generation 1.

Proof II. Step 1, generation 2. Wrong approach.

Proof II. Step 1, generation 2: Short curve.

Proof II. Step 1, generation 2: Even shorter curve.

Proof II. Step 1, generation 1: ρ1,1.

Proof II. Step 1, generation 2 : ρ1,2

Proof II. Step 1, generation 2 : ρ1,2.

Proof II. Step 1: mass estimate of ρ1.

I Let ρ1,2 be 0 on white squares and 1 otherwise.I ρ1,2 is admissible for Γ4.I Continue by induction and obtain a metric ρ1 = limn ρ1,n which is supported only

on the dark grey “buffer" squares.I The mass of ρ1 is 1

2 .


I Let ρ1,2 be 0 on white squares and 1 otherwise.

I ρ1,2 is admissible for Γ4.I Continue by induction and obtain a metric ρ1 = limn ρ1,n which is supported only


2 .


I Let ρ1,2 be 0 on white squares and 1 otherwise.I ρ1,2 is admissible for Γ4.

I Continue by induction and obtain a metric ρ1 = limn ρ1,n which is supported onlyon the dark grey “buffer" squares.

I The mass of ρ1 is 12 .



on the dark grey “buffer" squares.




on the dark grey “buffer" squares.I The mass of ρ1 is

12 .




2 .

Proof II: Admissibility of ρ1.

Proof II: Admissibility of ρ1, avoiding the “buffers”.

zout

zin

zin

Proof II. Step 2.

I Let ρ2,1 be 0 on white squares and 1 otherwise.I Obtain a metric ρ2 which is supported only on the dark grey “buffer" squares.I The mass of ρ2 is 1

22 .

I Similarly construct an admissible ρn with mass 12n .

Proof II. Step 2.

I Let ρ2,1 be 0 on white squares and 1 otherwise.

I Obtain a metric ρ2 which is supported only on the dark grey “buffer" squares.I The mass of ρ2 is 1

22 .


Proof II. Step 2.

I Let ρ2,1 be 0 on white squares and 1 otherwise.I Obtain a metric ρ2 which is supported only on the dark grey “buffer" squares.



Proof II. Step 2.

I Let ρ2,1 be 0 on white squares and 1 otherwise.I Obtain a metric ρ2 which is supported only on the dark grey “buffer" squares.I The mass of ρ2 is

122 .


Proof II. Step 2.

I Let ρ2,1 be 0 on white squares and 1 otherwise.I Obtain a metric ρ2 which is supported only on the dark grey “buffer" squares.I The mass of ρ2 is 1

22 .


Barriers in 3d.

Buffers in 3-d.

Buffers in 3-d, generation 2.

Menger Curve.

I Is modpΓi (s)→ 0, as i →∞ ⇐⇒ s ∈ `n?

I How fast is the convergence?I Is the double of the square carpet QS co-Hopfian?I Is there a quasi-isometrically co-Hopfian Gromov hyperbolic group G with ∂∞G

homeomorphic to Sierpinski carpet or the Menger curve?I Relations to Schramm’s transboundary extremal length.I Are the resistance exponents for S2k+1 different for different k -s.I QC rigidity, estimates for conformal dimension of carpets, etc.

I Is modpΓi (s)→ 0, as i →∞ ⇐⇒ s ∈ `n?I How fast is the convergence?

I Is the double of the square carpet QS co-Hopfian?I Is there a quasi-isometrically co-Hopfian Gromov hyperbolic group G with ∂∞G


I Is modpΓi (s)→ 0, as i →∞ ⇐⇒ s ∈ `n?I How fast is the convergence?I Is the double of the square carpet QS co-Hopfian?

I Is there a quasi-isometrically co-Hopfian Gromov hyperbolic group G with ∂∞Ghomeomorphic to Sierpinski carpet or the Menger curve?

I Relations to Schramm’s transboundary extremal length.I Are the resistance exponents for S2k+1 different for different k -s.I QC rigidity, estimates for conformal dimension of carpets, etc.

I Is modpΓi (s)→ 0, as i →∞ ⇐⇒ s ∈ `n?I How fast is the convergence?I Is the double of the square carpet QS co-Hopfian?I Is there a quasi-isometrically co-Hopfian Gromov hyperbolic group G with ∂∞G

homeomorphic to Sierpinski carpet or the Menger curve?

I Relations to Schramm’s transboundary extremal length.I Are the resistance exponents for S2k+1 different for different k -s.I QC rigidity, estimates for conformal dimension of carpets, etc.


homeomorphic to Sierpinski carpet or the Menger curve?I Relations to Schramm’s transboundary extremal length.

I Are the resistance exponents for S2k+1 different for different k -s.I QC rigidity, estimates for conformal dimension of carpets, etc.


homeomorphic to Sierpinski carpet or the Menger curve?I Relations to Schramm’s transboundary extremal length.I Are the resistance exponents for S2k+1 different for different k -s.

I QC rigidity, estimates for conformal dimension of carpets, etc.

Thank You.

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hrant hakobyan - mathematics

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