Limits of Blaschke metrics

Abstract

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

35 pages, 3 figures, referee's suggestions added. To appear in Duke Math. J

References in corpus (3)

• Length spectral rigidity of non-positively curved surfaces
• A structure theorem for geodesic currents and length spectrum compactifications
• Convex RP^2 Structures and Cubic Differentials under Neck Separation

Cited by in corpus (3)

• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces II: naturality
• Length spectrum compactification of the ${\mathrm{SO}}_0(2,3)$-Hitchin component
• Boundary of the Gothen components