Characterization of properly convex branched projective structures

Determine necessary and sufficient conditions for a branched real projective structure on a closed surface to be properly convex.

Background

The paper characterizes a class of properly convex branched structures with purely Hitchin holonomy and explains that representations with odd Stiefel–Whitney number cannot arise from branched covers of genuine convex structures. It then explicitly identifies the general criterion for proper convexity of branched projective structures as unresolved.

References

The question of determining necessary and sufficient conditions for a branched projective structure to be properly convex therefore remains open.

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 6, Subsection “Properly convex branched projective structures”