We show that a properly convex projective structure
on a closed
oriented surface of negative Euler characteristic arises from a Weyl connection if and
only if
is
hyperbolic. We phrase the problem as a nonlinear PDE for a Beltrami differential by using
that
admits
a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning
this nonlinear PDE into a transport equation, we obtain our result by applying methods
from geometric inverse problems. In particular, we use an extension of a remarkable
-energy
identity known as Pestov’s identity to prove a vanishing theorem for the relevant
transport equation.
Keywords
convex projective structures, Weyl connections, transport
equations, energy identity
