Billiards and Teichmüller theory M g = Moduli space moduli space of Riemann surfaces X of genus g { } -- a complex variety, dimension 3g-3 Curtis T McMullen Harvard University Teichmüller metric: every holomorphic map f : H M g is distance-decreasing. Unit ball at X determines X Holomorphic -forms Ω(X) = {holomorphic forms!(z) dz} C g Complex geodesics in genus two ΩM g M g bundle over moduli space Let f : H M be such a complex geodesic. Then f(h ) is either: SL (R) acts on ΩM g Orbits project to complex geodesics: f : H M g an isometrically immersed algebraic curve V, a Hilbert modular surface H D, or the full space M. Compare Ratner, Shah: f(h ) M n = H n /Γ is an immersed finite-volume hyperbolic k-manifold.
Dynamics of SL(R) Optimal Billiards (rigidity) Every orbit closure and every ergodic invariant measure for the action of SL (R) on ΩM is algebraic. (Veech) SL(X,!) = stabilizer of (X,!) in SL(R) PROBLEM: When is SL(X,!) a lattice? = every trajectory is periodic or uniformly distributed. Trapped trajectories Applications
(X, ω) = Billiards and Riemann surfaces ( ρp, dz ) / P P is a Lattice Polygon SL(X,!) is a lattice (X,!) generates a Teichmüller curve (Veech, Masur): If P is a lattice polygon, then billiards in P is optimal.! has just one zero! (renormalization) 0th century lattice billiards Square SL_(Z) Genus Regular 5-8- and 0-gon Tiled by squares Regular polygons ~ SL_(Z) ~ (,n, ) triangle group 8 Problem Are the infinitely many primitiveteichmüller curves V in the moduli space M? Various triangles triangle groups
Jacobians with real multiplication (X,!) generates a Teichmüller curve V Jac(X) admits real multiplication by OD. Corollary V lies on a Hilbert modular surface V H D M The Weierstrass curves WD = {X in M : Jac(X) admits real multiplication by OD with an eigenform! with a double zero.}. WD is a finite union of Teichmüller curves. Corollaries Pd Pd is a lattice polygon " for all integers d>0. " = ( +!d)/ There are infinitely many primitive V in genus. Classification Holomorphic pentagon-to-star map P5 Ex: pentagon and golden table F Thm. WD is irreducible unless D= mod 8 and D>9, in which case it has two components. W5
The regular decagon A genus two billiard table P is dynamically optimal P is tiled by squares or symmetric triangles, P is L-shaped with quadratic irrational dimensions,or P is the regular decagon (up to scissors congruence)!!. The only other primitive Teichmüller curve in genus two is generated by the regular decagon. b Torsion divisors in genus two Rational limits P - -0.5 0.5 - - - Q - - (Möller) [P-Q] is torsion in Jac(X) - y = ( x )(x + cos πα) (x cos πβ)
Ratios of Sines P Q torsion = α, β Q / Exceptional triangular billiards E Jac(X) has real multiplication = sin(πα)/ sin(πβ) Q( D) 5/ /9 /3 E 7 5 possibilities, including /9 /3 sin(π/5) sin(π/5) = + 5 /5 E 8 7/5 /3 Pryms systems in genus, 3 and 3 3 5 3 There exist infinitely many primitive Teichmüller curves V in Mg for genus g =, 3 and. 3 5 7 5 5 3 8 3 7 8