Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

Francaviglia, Stefano; Ruffoni, Lorenzo

doi:10.1007/s10711-021-00601-6

Cited by 5 publications

(2 citation statements)

References 21 publications

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“…Differentials with zero reduced exponents at all punctures correspond to parabolic projective structures (see [Kra69; Kra71b; Kra71a; DD17; HD19]). Differentials with integer non-zero reduced exponents and trivial holonomy at the punctures (apparent singularities) correspond to branched projective structures (see [Man72;CDF14a;Cal+19;FR21]). The next lemma implies that for structures in P (Σ) the absolute value of the exponent at a puncture coincides with the value of the index, as defined in §3.4.…”

Section: The Complex Analytic Point Of Viewmentioning

confidence: 99%

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

Ballas¹,

Bowers²,

Casella³

et al. 2021

Preprint

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Suppose a relatively elliptic representation ρ of the fundamental group of the thrice-punctured sphere S is given. We prove that all projective structures on S with holonomy ρ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of ρ. In the process, we show that (on a general surface Σ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Möbius completion, and in terms of certain meromorphic quadratic differentials. Contents 1. Introduction 1 2. Basics on complex projective geometry 7 2.1. Configurations of circles 7 2.2. Elliptic Möbius transformations 9 2.3. Triangular immersions 12 3. Tame and relatively elliptic CP 1 -structures 21 3.1. Ends, framing, and grafting 22 3.2. The Möbius completion 26 3.3. Local properties of the developing map at an end 33 3.4. The index of a puncture 38 4. The complex analytic point of view 39 4.1. Local theory at regular singularities 39 4.2. Meromorphic projective structures 41 5. Structures on the thrice-punctured sphere 43 5.1. Triangular structures 44 5.2. Grafting Theorems A and B 49 Appendix A. Tables of atomic triangular immersions 53 References 56

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Section: The Complex Analytic Point Of Viewmentioning

confidence: 99%

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

Ballas¹,

Bowers²,

Casella³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For instance, Francaviglia-Ruffoni in [FR21] studied a certain locus of hyperelliptic branched projective structures in the context of the classical Riemann-Hilbert problem for sl 2 -systems.…”

Section: Introductionmentioning

confidence: 99%