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Flat Geometry on Riemann Surfaces and Teichm¨Uller Dynamics in Moduli Spaces

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Flat Geometry on Riemann Surfaces and Teichm¨Uller Dynamics in Moduli Spaces Flat geometry on Riemann surfaces and Teichm¨uller dynamics in moduli spaces Anton Zorich Obergurgl, December 15, 2008 From Billiards to Flat Surfaces 2 Billiardinapolygon ................................. ...............................3 Closedtrajectories................................. ................................4 Challenge.......................................... ..............................5 Motivationforstudyingbilliards ..................... ..................................6 Gasoftwomolecules.................................. .............................7 Unfoldingbilliardtrajectories ...................... ...................................8 Flatsurfaces....................................... ...............................9 Rationalpolygons................................... .............................. 10 Billiardinarectangle ............................... ............................... 11 Unfoldingrationalbilliards ......................... ................................. 13 Flatpretzel........................................ .............................. 14 Surfaceswhich are more flat thanthe others. ............................. 15 Very flat surfaces 16 Very flat surfaces: construction from a polygon. ............................... 17 Propertiesofveryflatsurfaces........................ ............................... 19 Conicalsingularity ................................. ............................... 20 Familiesofflatsurfaces.............................. .............................. 21 Familyofflattori.................................... .............................. 22 Holomorphic 1-forms versus very flat surfaces 23 Fromflattocomplexstructure .......................... ............................. 24 Fromcomplextoflatstructure .......................... ............................. 25 Concise geometro-analyticdictionary . ................................ 26 Flat surfaces and quadratic differentials. .................................. 27 Volumeelement...................................... ............................ 28 Groupaction........................................ ............................. 29 Masur—VeechTheorem ................................. .......................... 30 Hopeforamagicwand .................................. .......................... 31 RelationtotheTeichm¨ullermetric .. .. .. .. .. .. .. ................................ 32 Generic geodesics on very flat surfaces 33 Asymptoticcycleforatorus........................... .............................. 34 1 Asymptoticcycle for generalsurfaces . .. .. .. .. .. .. ............................... 35 Asymptotic flag: empirical description . ................................ 36 Lyapunovexponents .................................. ............................ 41 Saddle connections and closed geodesics 42 Countingofclosedgeodesics .......................... ............................. 43 Exactquadraticasymptotics.......................... ............................... 44 Phenomenon of multiple saddle connections. .............................. 45 Homologoussaddleconnections........................ ............................. 46 Billiards in rectangular polygons 49 L-shapedbilliard................................... ............................... 50 Rectangularpolygons ................................ ............................. 51 Billiards versus quadratic differentials . ................................... 52 Numberofgeneralizeddiagonals....................... .............................. 53 Naiveintuitiondoesnothelp.......................... ................................ 54 Billiardplayers .................................... ............................... 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 and P. Hooper: www.math.brown.edu/ res/Billiards/index.html) ∼ 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. x2 a m1 m2 x1 x2 x 0 a | a x1 Neglecting the sizes of the beads we can describe the configuration space of our system using coordinates 0 < x1 x2 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 x˜1 = √m1x1 (x˜2 = √m2x2 we get a new right triangle ∆ as a configuration space. x˜2 = √m2 x2 m1 m2 x1 x2 x 0 x˜1 = √m1 x1 Lemma In coordinates (˜x1, 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 pi π of π. Properties of qi 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 Unfolding rational billiards We can apply an analogous procedure to any rational billiard. Consider, for example, a triangle with angles π/8, 3π/8,π/2. It is easy to check that a generic trajectory in such billiard table has 16 directions (instead of 4 for a rectangle). Using 16 copies of the triangle we unfold the billiard into a regular octagon with opposite sides identified by parallel translations. 13 / 55 Flat surface of genus two Identifying the pair of horizontal sides and then the pair of vertical sides of a regular octagon we get a torus with a single square hole. Identifying two opposite sides of the hole we get a torus with two distinct holes. Identifying the holes we get a surface of genus two. 14 / 55 8 Surfaces which are more flat than the others Note that the flat metric on the resulting surface has trivial holonomy, since the identifications of the sides were performed by parallel translations. As before, a billiard trajectory is unfolded to a geodesic on the surface. But now geodesics resemble geodesics on the torus: they do not have self-intersections! We abandon rational billiards for a while and pass to a more systematic study of “very flat” surfaces. 15 / 55 Very flat surfaces 16 / 55 Very flat surfaces: construction from a polygon Consider a broken line constructed from vectors ~v1,...,~vk. ~v3 ~v2 ~v4 ~v1 ~v4 ~v1 ~v2 ~v3 and another one constructed from the same vectors taken in another order. If we are lucky enough the two broken lines do not intersect and form a polygon. 17 / 55 9 Very flat surfaces: construction from a polygon ~v3 ~v2 ~v ~v1 4 ~v4 ~v1 ~v2 ~v3 Identifying the corresponding pairs of sides by parallel translations we get a closed surface endowed with a flat metric. 18 / 55 Properties of very flat surfaces The flat metric is nonsingular outside of a finite number of conical singularities (inherited from the • vertices of the polygon). The flat metric has trivial holonomy, i.e. parallel transport along any closed path brings a tangent • vector to itself. In particular, all cone angles are integer multiples of 2π. • By convention, the choice of the vertical direction (“direction to the North”) will be considered as a • part of the “very flat structure”. For example, a surface obtained from a rotated polygon is considered as a different very flat surface. A conical singularity with the cone angle 2π N has N outgoing directions to the North. • · 19 / 55 10 Example: conical singularity with cone angle 6π Locally a neighborhood of a conical point looks like a “monkey saddle”. A neighborhood of a conical point with a cone angle 6π can be glued from six metric half discs.
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