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Billiards, quadrilaterals and moduli spaces
Curtis McMullenHarvard University
coworkers Eskin, Mukamel and Wright
RigidityRenormalization
Dynamics on moduli spaces
We are still in the Age of Discovery
Polygonal billiards
Dynamics on moduli spaces
•There exists a totally geodesic complex surface F in Mg,n.
• Billiards in suitable darts have optimal dynamics.
RigidityRenormalization
A Little History
Zdxp1� x
2= sin�1
x
Zdx
3p1� x
3=
3 3
qx�1
1+ 3p�1+ 1 3
qx�1
1�(�1)2/3+ 1(x� 1)F1
⇣23 ;
13 ,
13 ;
53 ;�
x�11�(�1)2/3
,� x�11+ 3p�1
⌘
2 3p1� x
3
F1 = Appel hypergeometric function
Zdx
Q(x)
Zdx
Q(x)1/d
Q(x) a polynomial
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Zdx
Q(x)1/dA Little History
Zdxp1� x
2= sin�1
x
(X,ω) = (The curve yd = Q(x), the form dx/y)
periods of (X,ω) = {∫ ω }C
↝ Riemann surfaces, homology, Hodge theory, automorphic forms, ...
Moduli space
-- a complex variety, dimension 3g-3
{ }
The exceptional triangular billiard tables
2/9
1/4
5/12
4/9
7/15
1/3
1/5
1/3
1/3
7EE
8EE
6EE
Mg = moduli space of Riemann surfaces X of genus g
f : H2 �Mg
Teichmüller metric: every holomorphic map
is distance-decreasing.
Static
How to describe X in Mg ?
g>1: X = ? Uniformization Theorem
Every X in Mg can be built from a polygon in C
g=1: X = C/Λ
P
X = P / gluing by translations
How to describe X in Mg ?
g>1: X = ? Uniformization Theorem
Every (X,ω) in ΩMg can be built from a polygon in C
g=1: X = C/Λ
P
(X,ω) = (P,dz) / gluing by translations
Mg
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Moduli space ΩMg
SL2(R) acts on ΩMg
Dynamic:
Polygon for A ⋅ (X,ω) = A ⋅ (Polygon for (X,ω))
Complex geodesics f : H ⟶ Mg
τ1
τ2
f(τ1) f(τ2)
Teichmüller curves
The complex geodesic generated by (X,ω) factors:
V = H / SL(X,ω)
Hf
Mg
SL(X,ω) lattice ⇔ f : V → Mg is an algebraic,
isometrically immersed Teichmüller curve.
stabilizer of (X,ω)
in SL2(R)
Are there any examples of
Teichmüller curves?
Billiards in polygons
Neither periodic nor evenly distributed
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Optimal Billiards
Theorem. In a regular n-gon, every billiard path is eitherperiodic or uniformly distributed.
(Veech)
Billiards and Riemann surfaces
P
X has genus 2ω has just one zero!
(X,ω) = P/~
Theorem (Veech, Masur):
If P is a lattice polygon, then billiards in P is optimal.
P is a Lattice Polygon
⇔ SL(X,ω) is a lattice
⇔ (X,ω) generates a Teichmüller curve
(Renormalization)V ⊂ Mg
Every geodesic on V is either recurrent orexits a cusp
Optimal Billiards
CorollaryBilliards in a regular polygon
has optimal dynamics.
Theorem (Veech, 1989): Any regular polygon generatesa Teichmüller curve in moduli space.
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Beyond Teichmüller curves?
Mg ⊂ PN is a projective variety
PROBLEM
Are there any totally geodesic* subvarieties V ⊂ Mg with dim(V)>1?
Almost all subvarieties V ⊂ Mg arecontracted.
(*Every complex geodesic tangent to V is contained in V.)
The flex locus F
Main Theorem
There is a primitive, totally geodesic complexsurface F ⊂ M1,3.
•T is not isomorphic to any traditionalTg,n.
• T is not a symmetric space (Antonakoudis)
The universal cover T → F: a new type of Teichmüller space?
A TREATISE
ON THE
HIGHER PLAM CURVES :
INTENDED AS A SEQUEL
TO
A TREATISE ON CONIC SECTIONS.
BY
GEORGE SALMON, D.D., D.C.L., LL.D., F.R.S.,REGIUS PROFESSOR OF DIVINITY IN THE UNIVERSITY OP DUBLIN.
THIRD EDITION.
HODGES, FOSTER, AND FIGGIS, GKAFTON STREET,BOOKSELLERS TO THE UNIVERSITY.
MJDCCCLXXIX,
1879
-4 -2 0 2 4
-4
-2
0
2
4
-4 -2 0 2 4
-4
-2
0
2
4
-4 -2 0 2 4
-4
-2
0
2
4
The Polar Conic
A
Polar(S,A)
S
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-4 -2 0 2 4
-4
-2
0
2
4
-4 -2 0 2 4
-4
-2
0
2
4
HA
The Hessian
A
HA = {S : Pol(A,S) is 2 lines}
= Z(det D2f)
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6 The Solar Configuration
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
Sun
The Solar Configuration
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
SunDawn
Dusk
The Solar Configuration
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-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
SunDawn
Codaw
n
F = {(A,P=Codawn) : A is in M1}
The Solar Configuration
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6 The Space QF(A,P)
P = Codaw
n
QF(A,P) ⇔ {q : (q) = Ray - P)}
Ray
Sun
A
The gothic locus ΩG ⊂ ΩM4
(X, ω) in ΩG ⟺ (A,q) = (X/J, ω2/J) in QF
Definition: ΩG = √QF
Theorem: ΩG is an SL2(R) invariant subvariety of
ΩM4 of dimension 4
Cor:G is geodesically ruled
From G back to F
(X, ω) in ΩG → (A,q) = (X/J, ω2/J) in QF
Corollary: F is totally geodesic
Gruled
F
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Proof of SL2(R) invariance of ΩG:
• Variety ΩG is linear and defined/R in period coordinates on ΩM4 !
B
2
3X
A
P1
3
2g=4
g=1
g=0
Jac(X) ~ A x B x C
4 = 1 + 1 + 2
ω
solar
√
Cathedral polygons
a a2a
1 2
1b
b
1
Complement. We have a new infinite series of Teichmüller curves GD ⊂G ⊂ M4.
Known Teichmüller curves
M3
M2
M6
M5
M4
Veech 1989
. . .. .
.
. . . Calta, M 2002(Jacobian)
. . .
. . .
M 2005(Prym)
. . . MMW 2016
?
Bouw-Möller 2006
. . .
E6, E7, E8
Does the flex surface fit into
a broader pattern?
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Classical perspective: Triangles
(1,1,n)
(1,2,n)
(3,4,5)
(2,3,4)
(3,5,7)
(1,4,7)
These generate Teichmüller curves
1989 Veech, Ward, Vorobets, Kenyon-Smillie, Hooper 2013
New perspective: Quadrilaterals
P→ (X,ω) = (P,dz)/~
in ΩM4
X : y6 = x (x-1) (x-t) ω : C dx/y5
Q(1,1,1,9)
New perspectives on ΩG:
• The gothic locus is the closure of the SL2(R) orbits of the (1,1,1,9) quadrilateral forms.
• A Zariski open subset of ΩG coincides with an explicit space of dihedral forms.
• The variety of dihedral forms is linear in period coordinates.
(Z/6 symmetry)
(D12 symmetry)
1
y √3
New optimal billiards
• Suitable (1,1,1,9) quadrilaterals have optimal dynamics and generate Teichmüller curves in M4.
y2 + (3c+1)y + c = 0
Shown: y = 1 +√2
some c in Q
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Does the (1,1,1,9) quadrilateral
fit into a broader pattern?
(1,1,1,7)
(1,2,2,11)(1,1,2,12)
(1,1,1,9)
(1,2,2,15)
(1,1,2,8)
A suite of quadrilaterals
(1,1,1,7)
(1,2,2,11)(1,1,2,12)
(1,1,1,9)
(1,2,2,15)
(1,1,2,8)
Six SL2(R) invariant 4-folds ΩG ⊂ ΩMg
(1,1,1,7)
(1,2,2,11)(1,1,2,12)
(1,1,1,9)
(1,2,2,15)
(1,1,2,8)
Two series of optimal billiard tables / V ⊂ M4
(cf. Bouw-Möller)
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(1,1,1,7)
(1,2,2,11)(1,1,2,12)
(1,1,1,9)
(1,2,2,15)
(1,1,2,8)
Three totally geodesic surfaces
M2,1M1,4
M1,3
Pentagons, reprise
• The closure of the SL2(R) orbits of the (1,1,2,2,12) pentagonal forms gives a new, 6-dimensional
invariant subvariety of ΩM4.