Flat geometry on Riemann surfaces and Teichm uller dynamics in - PDF document

Flat geometry on Riemann surfaces and Teichm¨ uller dynamics in moduli spaces Anton Zorich Obergurgl, December 15, 2008 From Billiards to Flat Surfaces 2 Billiard in a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Closed trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Challenge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Motivation for studying billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Gas of two molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Unfolding billiard trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Rational polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Billiard in a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Unfolding rational billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Flat pretzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Surfaces which are more flat than the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Very flat surfaces 16 Very flat surfaces: construction from a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Properties of very flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Conical singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Families of flat surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Family of flat tori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Holomorphic 1-forms versus very flat surfaces 23 From flat to complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 From complex to flat structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Concise geometro-analytic dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Flat surfaces and quadratic differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Volume element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Group action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Masur—Veech Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Hope for a magic wand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Relation to the Teichm¨ uller metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Generic geodesics on very flat surfaces 33 Asymptotic cycle for a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1

Asymptotic cycle for general surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Asymptotic flag: empirical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Saddle connections and closed geodesics 42 Counting of closed geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Exact quadratic asymptotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Phenomenon of multiple saddle connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Homologous saddle connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Billiards in rectangular polygons 49 L-shaped billiard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Rectangular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Billiards versus quadratic differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Number of generalized diagonals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Naive intuition does not help... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Billiard players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2

From Billiards to Flat Surfaces 2 / 55 Billiard in a polygon Moon Duchin plays billiard on an L-shaped table 3 / 55 Closed trajectories It is easy to find a closed billiard trajectory in an acute triangle. Exercise. Prove that the broken line joining the bases of hights of an acute triangle is a billiard trajectory (it is called Fagnano trajectory ). Show that it realizes an inscribed triangle of minimal perimeter. 4 / 55 3

Challenge It is difficult to believe, but a similar problem for an obtuse triangle is open. Open problem. Is there at least one closed trajectory for (almost) any obtuse triangle? It seems like the answer is affirmative (see an extensive computer search performed by R. Schwartz . Hooper: www.math.brown.edu/ ∼ res/Billiards/index.html ) and P Then, one can ask further questions: • Estimate the number N ( L ) of periodic trajectories of length bounded by L ≫ 1 when L → + ∞ . • Is the billiard flow ergodic for almost any triangle? 5 / 55 Motivation to study billiards: gas of two molecules in a one-dimensional cham- ber Consider two elastic beads (“molecules”) sliding along a rod. They are bounded from two sides by solid walls. All collisions are ideal — without loss of energy. x 2 a m 1 m 2 x x 1 x 2 0 a | a x 1 Neglecting the sizes of the beads we can describe the configuration space of our system using coordinates 0 < x 1 ≤ x 2 ≤ a of the beads, where a is the distance between the walls. This gives a right isosceles triangle. 6 / 55 4

Gas of two molecules Rescaling the coordinates by square roots of masses = √ m 1 x 1 � x 1 ˜ = √ m 2 x 2 x 2 ˜ we get a new right triangle ∆ as a configuration space. x 2 = √ m 2 x 2 ˜ m 1 m 2 x x 1 x 2 0 x 1 = √ m 1 x 1 ˜ Lemma In coordinates (˜ x 1 , ˜ x 2 ) trajectories of the system of two beads on a rod correspond to billiard trajectories in the triangle ∆ . 7 / 55 Unfolding billiard trajectories Identifying the boundary of two triangles we get a flat sphere. A billiard trajectory unfolds to a geodesic on this flat sphere. 8 / 55 5

Flat surfaces The surface of the cube represents a flat sphere with eight conical singularities. The metric does not have singularities on the edges. After parallel transport around a conical singularity a vector comes back pointing to a direction different from the initial one, so this flat metric has nontrivial holonomy . 9 / 55 Rational polygons A polygon Π is called rational if all the angles of Π are rational multiples p i q i π of π . Properties of billiards in rational polygons are known much better. Consider a model case of a rectangular billiard. As before instead of reflecting the trajectory we can reflect the billiard table. The trajectory unfolds to a straight line. Folding back the copies of the billiard table we project this line to the actual trajectory. 10 / 55 6

Billiard in a rectangle Fix a (generic) trajectory. At any moment the ball moves in one of four directions. They correspond to four copies of the billiard table; other copies can be obtained from these four by a parallel translation: D A A C B B A D A D B C C D D A 11 / 55 Billiard in a rectangle versus flow on a torus Identifying the equivalent patterns by a parallel translation we obtain a torus; the billiard trajectory unfolds to a “straight line” on the corresponding torus. 12 / 55 7