rigidity of circle packings - david b. wilsondbwilson.com/schramm/workshop/slides-stephenson.pdf ·...

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

Circle Packing – Background

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

Existence and Uniqueness

Theorem: [KAT, Koebe-Andreev-Thurston] Given any triangulation K of atopological sphere, there exists a univalent circle packing PK of the Riemannsphere having the combinatorics of K . Moreover, PK is unique up to Möbiustransformations 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 itsintrinsic metric, so that PK “fills” S. Moreover, the conformal structure isunique 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 thenext talk.

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

Moreover, PK is unique up to Möbiustransformations 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 itsintrinsic metric, so that PK “fills” S.

Moreover, the conformal structure isunique 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!

Upshot: Circle packings endow combinatorial situations with geometry.• Local rigidity

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

Upshot: Circle packings endow combinatorial situations with geometry.• Local rigidity• Global flexibility

• and this is a particularly familiar geometry — it’s conformal!

Thurston’s Conjecture, 1985

Conjecture: Under refinement, the discrete conformal maps f : PK −→ Pconverge 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 invariablyconverge to their classical counterparts.

Rodin and Sullivan proved the conjecture, which has been vastly extended

—under refinement, objects in the discrete world of circle packing invariablyconverge to their classical counterparts.

Rigidity

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

The crucial tool? Two circles can intersect in at most two points.

Rigidity

The crucial tool?

Two circles can intersect in at most two points.

Rigidity

The setup, I

The setup, II

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

The setup, II

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

The “elements” of P and P ′

Elements:

circle elements

Elements:

circle elements

Elements:

circle elements

Elements:

circle elements⋃interstice elements

Elements:

Likewise for P’E ←→ E ′

Elements:

Comparison via “Fixed point index”

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

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

Compatibility

If γ and σ are both circles thenfor 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.

Compatibility

fpf−→ σ,

η(f ; γ) ≥ 0.

fpf−→ σ with

η(f ; γ) ≥ 0.

Compatibility

fpf−→ σ,

η(f ; γ) ≥ 0.

fpf−→ σ with

η(f ; γ) ≥ 0.

Compatibility

fpf−→ σ,

η(f ; γ) ≥ 0.

fpf−→ σ with

η(f ; γ) ≥ 0.

Compatibility

fpf−→ σ,

η(f ; γ) ≥ 0.

fpf−→ σ with

η(f ; γ) ≥ 0.

The Proof

• ∀ interstice element ej ∈ E choose fj : ejfpf−→ e′j so that η(fj ; ej) ≥ 0.

• ∀ circle element ek , define fk : ekfpf−→ e′k to agree

with the maps of neighboring interstices.

• The element maps induce a homeomorphism F : Γfpf−→ 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

The Proof

η(fj ; ej).

The Proof

η(fj ; ej).

The Proof

η(fj ; ej).

The Proof

η(fj ; ej).

The Proof

η(fj ; ej).

The Proof

η(fj ; ej).

• In particular,

η(F ; Γ) ≥ 0

The Proof

η(fj ; ej).

but ...

F : Γfpf−→ Σ and η(F ; Γ) ≥ 0

By observation, η(F ; Γ) = −1

but ...

The other bookend

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

The other bookend

Existence

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

Existence

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

Moreover, thepacking is unique subject to standard normalization.

Existence

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

Existence

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

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

rigidity of circle packings - david b. wilsondbwilson.com/schramm/workshop/slides-stephenson.pdf ·...

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