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Bubbling complex projective structures with quasi-Fuchsian holonomy

Dipartimento di Matematica — Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

E-mail Address: lorenzo.ruffoni2@gmail.com

https://doi.org/10.1142/S1793525320500326Cited by:3 (Source: Crossref)

Abstract

For a given quasi-Fuchsian representation $ρ : π_{1} (S) \to {PSL}_{2} ℂ$ of the fundamental group of a closed surface $S$ of genus $g \geq 2$ , we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points can be obtained from some unbranched structure on $S$ with the same holonomy by bubbling, i.e. a suitable connected sum with a copy of $ℂ ℙ^{1}$ .

Keywords:

AMSC: 57M50, 20H10, 14H15

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Vol. 13, No. 03

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History

Received 7 December 2017

Revised 24 June 2019

Accepted 8 September 2019

Published: 17 October 2019

Keywords

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From hyperbolic surfaces to projective surfaces.

What is it about?

In pure mathematics we often start with an example, and then write a definition, to encapsulate the features of that example. But then it is hard to know how many other examples are covered under the same umbrella. In this paper we show that if a projective surfaces fits a certain hyperbolicity condition, then it is actually a hyperbolic surface (up to blowing some bubbles on it).

Why is it important?

Projective structures are a geometric tool for the study of differential equations and group representations on Riemann surfaces. Certain classes of equations require the presence of singularities (i.e. branch points), and our results provide a better understanding of the geometric origins of this singular behavior, under some hyperbolicity conditions.

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Lorenzo Ruffoni

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Bubbling complex projective structures with quasi-Fuchsian holonomy

Abstract

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