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).

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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.

Perspectives

While projective geometry is not associated to a nice metric on the surface, the hyperbolicity requirement provides some metric tools that work "almost everywhere" (on the surface, and on the moduli space). Trying to make them work everywhere (on the surface, and on the moduli space) requires some new ideas.

Dr. Lorenzo Ruffoni
Tufts University

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This page is a summary of: Bubbling complex projective structures with quasi-Fuchsian holonomy, Journal of Topology and Analysis, September 2019, World Scientific Pub Co Pte Lt, DOI: 10.1142/s1793525320500326.
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