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Geodesics on flat surfaces. (English) Zbl 1113.32004

Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 121-146 (2006).
This is a survey article expanding the author’s lecture at the ICM in Madrid, 2006. The main objects of study are flat surfaces, these are closed surfaces endowed with a flat (Euclidean) metric with several cone singularities where all cone angles are multiples of \(2 \pi\), and the moduli spaces of such flat surfaces. This is a very appealing and interesting topic which ties together various problems in geometry, topology, dynamical systems and Teichmüller theory, and has been the subject of much study and research in recent years, both by the author and his co-workers, as well as several eminent experts in dynamical systems, Teichmüller Theory, low dimensional topology and complex analysis (the references give a fairly comprehensive bibliography of recent works in the field). We give a brief description of the topics covered.
Consider a closed orientable surface \(S\) of genus \(g \geq 1\). One can endow the surface with a singular flat (Euclidean) metric (normalized so that the area is one), together with a finite number of cone points \(P_1, \dots,P_m\) where the cone angle at \(P_k\) is \(2\pi(d_k+1)\), \(d_k\in {\mathbb N}\), and \(d_1+\cdots +d_m=2g-2\) by Gauss-Bonnet. These structures have trivial holonomy and one can further endow each structure with a distinguished direction (north). It turns out that the moduli space of such structures can be identified with the space \({\mathcal H}(d_1,\ldots,d_m)\) of pairs (Riemann surface, holomorphic \(1\)-form) of complex structures on \(S\) together with a holomorphic one-form on \(S\) (endowed with the complex structure) with \(m\) zeroes of degrees \(d_1,\dots,d_m\). These spaces (for various choices of partitions of \(2g-2\)) stratify the moduli space of flat structures on \(S\). There is a natural action of \(\text{ SL}(2, {\mathbb {R}})\) on each stratum, which restricts to a continuous action of the diagonal subgroup of \(\text{ SL}(2, {\mathbb R})\), this is called the Teichmüller geodesic flow on the stratum. The general philosophy expounded on in this survey is that considerable information about the geometry and dynamics of an individual flat surface \(S\) can be obtained by studying the orbit (and its closure) of \(S\) under the action of the Teichmüller geodesic flow, as well as the linear group \(\text{ SL}(2, {\mathbb R})\).
In section one of the survey, the author considers a fixed flat surface \(S\) and looks at geodesics on the surface going in generic directions. A fundamental theorem of S. Kerckhoff, H. Masur and J. Smillie states that the directional flow is uniquely ergodic in almost every direction, from which one can deduce that generically, the geodesic is asymptotic to a cycle \(c \in H_1(S, {\mathbb R})\). The author describes the way that the approximating cycles \(c_j\) deviate from the direction of the asymptotic cycle \(c\), and the connection of the resulting deviation spectrum with the top \(g\) Lyapunov exponents of the Teichmüller geodesic flow on the corresponding connected component of the stratum \({\mathcal H}(d_1,\dots,d_m)\), (Theorem 1) together with a sketch of the proof. This is based on works of the author, G. Forni, A. Avila and M. Viana, also using fundamental ideas of H. Masur and W. Veech employed in the proof of the ergodicity of the Teichmüller geodesic flow. Related results on the formula for the sum of the Lyapanov exponents by M. Kontsevich, G. Forni, R. Krikorian, I. Bouw and M. Möller are also mentioned.
The second section is concerned with closed regular geodesics and saddle connections (regular geodesic segments connecting two, not necessarily distinct, cone singularities) on a flat surface \(S\). The asymptotics of the corresponding counting functions \(N_{cg}(S,L)\) and \(N_{sc}(S,L)\) of equivalence classes of closed geodesics (two closed geodesics are equivalent if they bound a Euclidean cylinder on \(S\)), and saddle connections on \(S\) of length at most \(L\) are discussed, as well as the principal boundary of the connected components of the strata \({\mathcal H}(d_1,\dots,d_m)\), and the evaluation of the Siegel-Veech constants. Much of this section is based on work of the author with A. Eskin and H. Masur.
The final section is concerned with the problem of describing the classifying invariants of the connected components of the strata \({\mathcal H}(d_1,\dots,d_m)\), and is based on joint work of the author with M. Kontsevich. The two classifying invariants are the spin structure and hyperellipticity, and a classification theorem for Abelian differentials (Theorem 3) is given.
For the entire collection see [Zbl 1095.00006].

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
30F60 Teichmüller theory for Riemann surfaces
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