Kenneth Stephenson- A probabilistic proof of Thurston's conjecture on circle packings
Rigidity of Circle Packings - David B. Wilsondbwilson.com/schramm/workshop/slides-stephenson.pdf ·...
Transcript of 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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 2 / 31
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 3 / 31
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 4 / 31
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 5 / 31
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 6 / 31
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 7 / 31
Circle Packing – Background
Definition: A circle packing is a configuration P of circles satisfying aspecified pattern of tangencies.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31
Circle Packing – Background
Definition: A circle packing is a configuration P of circles satisfying aspecified pattern of tangencies.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31
Circle Packing – Background
Definition: A circle packing is a configuration P of circles satisfying aspecified pattern of tangencies.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 8 / 31
Circle Packing – Background
Definition: A circle packing is a configuration P of circles satisfying aspecified pattern of tangencies.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 9 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 10 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 11 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 12 / 31
The setup, I
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 13 / 31
The setup, I
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 14 / 31
The setup, I
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 15 / 31
The setup, I
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 16 / 31
The setup, I
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 17 / 31
The setup, II
Put∞ in the chosen interstice and project both packings to the plane to getthese juxtaposed configurations:
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31
The setup, II
Put∞ in the chosen interstice and project both packings to the plane to getthese juxtaposed configurations:
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31
The setup, II
Put∞ in the chosen interstice and project both packings to the plane to getthese juxtaposed configurations:
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 18 / 31
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:
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 19 / 31
The “elements” of P and P ′
Elements:
circle elements
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31
The “elements” of P and P ′
Elements:
circle elements
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31
The “elements” of P and P ′
Elements:
circle elements
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 20 / 31
The “elements” of P and P ′
Elements:
circle elements⋃interstice elements
= E
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 21 / 31
The “elements” of P and P ′
Elements:
circle elements⋃interstice elements
= E
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 21 / 31
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
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
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
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 24 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 25 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
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
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 26 / 31
but ...
F : Γfpf−→ Σ and η(F ; Γ) ≥ 0
By observation, η(F ; Γ) = −1
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31
but ...
F : Γfpf−→ Σ and η(F ; Γ) ≥ 0
By observation, η(F ; Γ) = −1
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31
but ...
F : Γfpf−→ Σ and η(F ; Γ) ≥ 0
By observation, η(F ; Γ) = −1
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 27 / 31
The other bookend
Theorem: [Schramm/He] The KAT Theorem on circle packings of the sphereimplies the Riemann Mapping Theorem for plane domains.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31
The other bookend
Theorem: [Schramm/He] The KAT Theorem on circle packings of the sphereimplies the Riemann Mapping Theorem for plane domains.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31
The other bookend
Theorem: [Schramm/He] The KAT Theorem on circle packings of the sphereimplies the Riemann Mapping Theorem for plane domains.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 28 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31
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.
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 29 / 31
Existence
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 30 / 31
Existence
Ken Stephenson (UTK) Circle Packing Oded Schramm Memorial, 8/2009 30 / 31
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
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