1,185 research outputs found
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
Asymptotics of Hitchin's metric on the Hitchin section
We consider Hitchin's hyperk\"ahler metric on the moduli space
of degree zero -Higgs bundles over a compact
Riemann surface. It has been conjectured that, when one goes to infinity along
a generic ray in , converges to an explicit "semiflat" metric
, with an exponential rate of convergence. We show that this
is indeed the case for the restriction of to the tangent bundle of the
Hitchin section .Comment: 22 pages, 1 figure. v2: Minor revision
Slicing, skinning, and grafting
We prove that a Bers slice is never algebraic, meaning that its Zariski
closure in the character variety has strictly larger dimension. A corollary is
that skinning maps are never constant.
The proof uses grafting and the theory of complex projective structures.Comment: 11 pages, 1 figure, to appear in American Journal of Mathematic
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