Flat Surfaces

By A. Zorich. 2006

572 citations recorded by the source corpus.

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to be isomorphic to the moduli spaces of holomorphic one-forms. One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmuller geodesic flow and under the linear group action. In particular, the Teichmuller geodesic flow plays the role of a time acceleration machine (renormalization procedure) which allows to study the asymptotic behavior of interval exchange transformations and of surface foliations. This long survey is an attempt to present some selected ideas, concepts and facts in Teichmuller dynamics in a playful way.

Subfields: Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/193755f7bf46d1b244bb5b2a96618e88bca78e7a

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In Mechanical Engineering

Citations in the corpus

Quasimodularity and large genus limits of Siegel-Veech constants (2016, methodology): “to [51] and [39] for the motivation and definition of Lyapunov exponents.”
On the moduli space of singular Euclidean surfaces (2007, background): “Its fibers describe a foliation whose leaves carry a geometric structure locally modelled on the complex projective space CP, see [36] for proofs of these facts and a discussion of other related geometric structures on Tg,n.”
Euler characteristics of Teichmüller curves in genus two (2006, background): “7 is an unpublished result of Kontsevich and Zorich, mentioned in [85].”
The field of definition of affine invariant submanifolds of the moduli space of abelian differentials (2012, methodology): “For an introduction to the field of translation surfaces, see the surveys by Masur and Tabachnikov [21] and Zorich [37].”
Spectral Networks (2012, unknown): “For example, in the case K = 2 the walls in Wθ are horizontal trajectories of a holomorphic quadratic differential eφ2, and the leading asymptotic behavior of a generic such trajectory is classified by a virtual “limit cycle” in H1(C,R), with deviations governed by certain Lyapunov exponents; see [2”
Cylinder deformations in orbit closures of translation surfaces (2013, methodology): “Background For an introduction to translation surfaces, see the surveys by Masur and Tabachnikov [14] and Zorich [30].”
A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle (2010, background): “For more details on square-tiled surfaces, see [52] and references therein.”