Rigidity of Circle Packings Ken Stephenson University of Tennessee - - PowerPoint PPT Presentation

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rigidity of circle packings

Rigidity of Circle Packings Ken Stephenson University of Tennessee - - PowerPoint PPT Presentation

Jan 30, 2024 440 likes •1.22k views

Rigidity of Circle Packings Ken Stephenson University of Tennessee Oded Schramm Memorial, 8/2009 Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 1 / 31 Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 2

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Rigidity of Circle Packings

Ken Stephenson

University of Tennessee

Oded Schramm Memorial, 8/2009

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 1 / 31

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SLIDE 2

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 2 / 31

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SLIDE 3

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 3 / 31

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Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 4 / 31

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SLIDE 5

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 5 / 31

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SLIDE 6

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 6 / 31

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SLIDE 7

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 7 / 31

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SLIDE 8

Circle Packing – Background

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31

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SLIDE 9

Circle Packing – Background

Definition: A circle packing is a configuration P of circles satisfying a specified pattern of tangencies.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31

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SLIDE 10

Circle Packing – Background

Definition: A circle packing is a configuration P of circles satisfying a specified pattern of tangencies.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31

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SLIDE 11

Circle Packing – Background

Definition: A circle packing is a configuration P of circles satisfying a specified pattern of tangencies.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 9 / 31

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SLIDE 12

Existence and Uniqueness

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 13

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 14

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 15

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 16

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 17

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms. Upshot: Circle packings endow combinatorial situations with geometry.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 18

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms. Upshot: Circle packings endow combinatorial situations with geometry.

Local rigidity

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 19

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms. Upshot: Circle packings endow combinatorial situations with geometry.

Local rigidity
Global flexibility

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 20

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms. Upshot: Circle packings endow combinatorial situations with geometry.

Local rigidity
Global flexibility
and this is a particularly familiar geometry — it’s conformal!

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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SLIDE 21

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of a topological sphere, there exists a univalent circle packing PK of the Riemann sphere having the combinatorics of K. Moreover, PK is unique up to Möbius transformations and inversion. Theorem: Given a triangulation K of any oriented topological surface S, there exists a conformal structure on S and a univalent circle packing PK in its intrinsic metric, so that PK “fills” S. Moreover, the conformal structure is unique and PK is unique up to its conformal automorphisms. Upshot: Circle packings endow combinatorial situations with geometry.

Local rigidity
Global flexibility
and this is a particularly familiar geometry — it’s conformal!

Oded’s frequent collaborator, Zheng-Xu He, will say more about this in the next talk.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31

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Thurston’s Conjecture, 1985

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31

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Thurston’s Conjecture, 1985

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31

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SLIDE 24

Thurston’s Conjecture, 1985

Conjecture: Under refinement, the discrete conformal maps f : PK − → P converge uniformly on compacta to the classical conformal map F : D − → Ω.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31

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SLIDE 25

Thurston’s Conjecture, 1985

Conjecture: Under refinement, the discrete conformal maps f : PK − → P converge uniformly on compacta to the classical conformal map F : D − → Ω. Rodin and Sullivan proved the conjecture, which has been vastly extended

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31

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SLIDE 26

Thurston’s Conjecture, 1985

Conjecture: Under refinement, the discrete conformal maps f : PK − → P converge uniformly on compacta to the classical conformal map F : D − → Ω. Rodin and Sullivan proved the conjecture, which has been vastly extended — under refinement, objects in the discrete world of circle packing invariably converge to their classical counterparts.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31

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SLIDE 27

Rigidity

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31

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Rigidity

Claim: If P and P′ are two circle packings of the sphere sharing the combinatorics of K, then they are Möbius images of one another.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31

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SLIDE 29

Rigidity

Claim: If P and P′ are two circle packings of the sphere sharing the combinatorics of K, then they are Möbius images of one another. The crucial tool?

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31

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SLIDE 30

Rigidity

Claim: If P and P′ are two circle packings of the sphere sharing the combinatorics of K, then they are Möbius images of one another. The crucial tool? Two circles can intersect in at most two points.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31

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SLIDE 31

The setup, I

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 13 / 31

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The setup, I

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 14 / 31

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The setup, I

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 15 / 31

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SLIDE 34

The setup, I

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 16 / 31

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SLIDE 35

The setup, I

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 17 / 31

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SLIDE 36

The setup, II

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31

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SLIDE 37

The setup, II

Put ∞ in the chosen interstice and project both packings to the plane to get these juxtaposed configurations:

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31

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SLIDE 38

The setup, II

Put ∞ in the chosen interstice and project both packings to the plane to get these juxtaposed configurations:

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31

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SLIDE 39

The setup, II

Put ∞ in the chosen interstice and project both packings to the plane to get these juxtaposed configurations: Scale P away from a to put the packings in general position:

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 19 / 31

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SLIDE 40

The “elements” of P and P′

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31

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SLIDE 41

The “elements” of P and P′

Elements:

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31

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The “elements” of P and P′

Elements: circle elements

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31

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SLIDE 43

The “elements” of P and P′

Elements: circle elements

interstice elements

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 21 / 31

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SLIDE 44

The “elements” of P and P′

Elements: circle elements

interstice elements

= E

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 21 / 31

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SLIDE 45

The “elements” of P and P′

Elements: circle elements

interstice elements

= E

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 22 / 31

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SLIDE 46

The “elements” of P and P′

Elements: circle elements

interstice elements

= E Likewise for P’ E ← → E′

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 22 / 31

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SLIDE 47

The “elements” of P and P′

Elements: circle elements

interstice elements

= E Likewise for P’ E ← → E′

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 23 / 31

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SLIDE 48

Comparison via “Fixed point index”

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31

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SLIDE 49

Comparison via “Fixed point index”

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31

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SLIDE 50

Comparison via “Fixed point index”

Definition: Given simple closed curves γ and σ and an orientation preserving, fixed-point-free homeomorphism f : γ

fpf

− → σ, the fixed point index η(f; γ) is the winding number of g(z) = f(z) − z about γ.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31

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SLIDE 51

Compatibility

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31

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SLIDE 52

Compatibility

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31

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SLIDE 53

Compatibility

If γ and σ are both circles then for every f : γ

fpf

− → σ, η(f; γ) ≥ 0.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31

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SLIDE 54

Compatibility

If γ and σ are both circles then for every f : γ

fpf

− → σ, η(f; γ) ≥ 0.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31

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SLIDE 55

Compatibility

If γ and σ are both circles then for every f : γ

fpf

− → σ, η(f; γ) ≥ 0. If γ = a, b, c and σ = a′, b′, c′, then there exists f : γ

fpf

− → σ with η(f; γ) ≥ 0.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31

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SLIDE 56

The Proof

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 57

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 58

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 59

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 60

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

The element maps induce a homeomorphism F : Γ

fpf

− → Σ between the outer boundaries of our two configurations.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 61

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

The element maps induce a homeomorphism F : Γ

fpf

− → Σ between the outer boundaries of our two configurations.

Taking account of cancellations on interior segments,

η(F; Γ) =

ej∈E

η(fj; ej).

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 62

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

The element maps induce a homeomorphism F : Γ

fpf

− → Σ between the outer boundaries of our two configurations.

Taking account of cancellations on interior segments,

η(F; Γ) =

ej∈E

η(fj; ej).

In particular,

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 63

The Proof

∀ interstice element ej ∈ E choose fj : ej

fpf

− → e′

j so that η(fj; ej) ≥ 0.

∀ circle element ek, define fk : ek

fpf

− → e′

k to agree

with the maps of neighboring interstices.

The element maps induce a homeomorphism F : Γ

fpf

− → Σ between the outer boundaries of our two configurations.

Taking account of cancellations on interior segments,

η(F; Γ) =

ej∈E

η(fj; ej).

In particular,

η(F; Γ) ≥ 0

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31

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SLIDE 64

but ...

F : Γ

fpf

− → Σ and η(F; Γ) ≥ 0

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31

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SLIDE 65

but ...

F : Γ

fpf

− → Σ and η(F; Γ) ≥ 0

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31

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SLIDE 66

but ...

F : Γ

fpf

− → Σ and η(F; Γ) ≥ 0 By observation, η(F; Γ) = −1

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31

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SLIDE 67

The other bookend

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31

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SLIDE 68

The other bookend

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31

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SLIDE 69

The other bookend

Theorem: [Schramm/He] The KAT Theorem on circle packings of the sphere implies the Riemann Mapping Theorem for plane domains.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31

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SLIDE 70

Existence

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31

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SLIDE 71

Existence

Theorem: [Schramm/He] Given any Jordan region Ω, there exists a univalent circle packing with heptagonal combinatorics which fills Ω.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31

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SLIDE 72

Existence

Theorem: [Schramm/He] Given any Jordan region Ω, there exists a univalent circle packing with heptagonal combinatorics which fills Ω. Moreover, the packing is unique subject to standard normalization.

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31

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SLIDE 73

Existence

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 30 / 31

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SLIDE 74

Existence

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 30 / 31

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SLIDE 75

Thanks

“Packing two-dimensional bodies ...”, “Existence and uniqueness of packings with specified combinatorics”, “Rigidity of infinite (circle) packings”, “How to cage an egg”, “Conformal uniformization and packings”, “Circle patterns with the combinatorics of the square grid”, With Zheng-Xu He: “Fixed points, Koebe uniformization and circle packings”, “Rigidity of circle domains whose boundary has σ-finite linear measure” “Hyperbolic and Parabolic Packings”, “The inverse Riemann Mapping Theorem for relative circle domains”, “On the convergence of circle packings to the Riemann map”, “The C∞-convergence of hexagonal disk packings to the Riemann map”,

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 31 / 31

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SLIDE 76

Thanks

“Packing two-dimensional bodies ...”, “Existence and uniqueness of packings with specified combinatorics”, “Rigidity of infinite (circle) packings”, “How to cage an egg”, “Conformal uniformization and packings”, “Circle patterns with the combinatorics of the square grid”, With Zheng-Xu He: “Fixed points, Koebe uniformization and circle packings”, “Rigidity of circle domains whose boundary has σ-finite linear measure” “Hyperbolic and Parabolic Packings”, “The inverse Riemann Mapping Theorem for relative circle domains”, “On the convergence of circle packings to the Riemann map”, “The C∞-convergence of hexagonal disk packings to the Riemann map”,

Thanks, Oded

Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 31 / 31