Quadratic Differentials

Strebel, Kurt

doi:10.1007/978-3-662-02414-0_2

Cited by 249 publications

(459 citation statements)

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“…The second school and the one this paper belongs to is studying holomorphic vector fields in their own right (classification) in, for example, Brickman and Thomas [4] and Douady et al [8], and in the study of quadratic differentials in Jenkins [11] and Strebel [18]. Quadratic differentials and holomorphic vector fields can both be viewed as foliations with singularities.…”

Section: History/motivationmentioning

confidence: 99%

“…Partial results for the global classification of complex polynomial vector fields go back to the classification of quadratic differentials having poles of order $ 2 [11,18]. The main case where classification of the global structure of quadratic differentials that applies to holomorphic vector fields is only a classification for given quadratic differentials.…”

Section: History/motivationmentioning

confidence: 99%

“…The main case where classification of the global structure of quadratic differentials that applies to holomorphic vector fields is only a classification for given quadratic differentials. When it comes to proving existence of a quadratic differential with prescribed combinatorial and analytic properties (for instance, the so-called 'moduli problem' for quadradic differentials with closed trajectories [18]), the full classification of vector fields is excluded since these theorems are only applicable to a special subclass of vector fields, or they assume a finite area condition, which holomorphic vector fields with at least one zero never satisfy. Muciño-Raymundo [13] studies vector fields but also only covers a specific class.…”

Section: History/motivationmentioning

confidence: 99%

See 2 more Smart Citations

Classification of complex polynomial vector fields in one complex variable

Branner

Dias

2010

Journal of Difference Equations and Applications

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Section: History/motivationmentioning

confidence: 99%

Section: History/motivationmentioning

confidence: 99%

Section: History/motivationmentioning

confidence: 99%

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Classification of complex polynomial vector fields in one complex variable

Branner

Dias

2010

Journal of Difference Equations and Applications

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“…Geometry of flat surfaces. We introduce the geometry of flat surfaces and refer to [Min92] and [Str84].…”

Section: Klaus Dankwartmentioning

confidence: 99%

Typical geodesics on flat surfaces

Dankwart¹

2011

Conform. Geom. Dyn.

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Abstract. We investigate typical behavior of geodesics on a closed flat surface S of genus g ≥ 2. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same conformal class. This quotient is asymptotically constant F a.e. We show that F is bounded from below by the inverse of the volume entropy e(S). Moreover, we construct a geodesic flow together with a measure on S which is induced by the Hausdorff measure of the Gromov boundary of the universal cover. Denote by e(S) the volume entropy of S and let c be a compact geodesic arc which connects singularities. We show that a typical geodesic passes through c with frequency that is comparable to exp(−e(S)l(c)). Thus a typical bi-infinite geodesic contains infinitely many singularities, and each geodesic between singularities c appears infinitely often with a frequency proportional to exp(−e(S)l(c)).

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“…Following ideas of Thurston, Strebel [27], Bowditch and Epstein [3] and Penner [24] constructed a triangulation of the decorated Teichmüller space of a punctured Riemann surface S which is equivariant under the action of the unbordered mapping class group. The quotient space, in which a point is an isomorphism class of metric fat graphs, gives a model for the corresponding decorated moduli space.…”

Section: Introductionmentioning

confidence: 99%

The unstable integral homology of the mapping class groups of a surface with boundary

Godin

2006

Math. Ann.

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Abstract. We construct a graph complex calculating the integral homology of the bordered mapping class groups. We compute the homology of the bordered mapping class groups of the surfaces S 1,1 , S 1,2 and S 2,1 . Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups. We compute the homology of the punctured mapping class groups associated to the surfaces S 1,1 and S 2,1 . Finally, we use Miller's operad to get the first Kudo-Araki and Browder operations on our graph complex. We also consider an unstable version of the higher Kudo-Araki-Dyer-Lashoff operations.

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Quadratic Differentials

Cited by 249 publications

References 0 publications

Classification of complex polynomial vector fields in one complex variable

Classification of complex polynomial vector fields in one complex variable

Typical geodesics on flat surfaces

The unstable integral homology of the mapping class groups of a surface with boundary

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