Prescribed holonomy for projective structures on compact surfaces

Lorentz spacetimes of constant curvature

“…Projective structures on tori are still classified by the cotangent bundle of Teichmüller space, but the dimension is 2 rather than 6g −6 and it is more natural to think of two affine structures as differing by a holomorphic differential than by a holomorphic quadratic differential. It is known that every homomorphism of the fundamental group of a closed surface of genus g to SL 2 C with image which is not conjugate into SU(2) nor into the stabilizer of a point or a pair of points in CP 1 is the holonomy of a projective structure [29], and that given one projective structure there are infinitely many others with the same holonomy, but the map from the space of projective structures to the space of holonomy representations is not a covering map [30,31].…”

“…Suppose we are given a closed spacelike surface with a neighbourhood modelled on de Sitter space. Gallo [29] actually shows that given a homomorphism from the fundamental group of a surface to SL 2 C with image not conjugate into SU(2) nor into the stabilizer of a point or pair of points in CP 1 , there is a decomposition of the surface into pairs of pants, such that the holonomy of any pair of pants is quasifuchsian. After an arbitrarily small change in the holonomy of each of the boundary circles, the eigenvalue re iθ of the holonomy at a fixed point on CP 1 will have θ a rational multiple of π.…”

Notes on a paper of Mess

“…Mess cites Gallo's announcement [52] for this, though the complete proof was not made available until several years later, in joint work with Marden and M. Kapovich [54]. Gallo's proposed proof of Theorem N.6.2 is based on the existence of a pants decomposition of S with the property that the holonomy of each pair of pants is quasi-Fuchsian.…”

The symplectic nature of the space of projective connections on Riemann surfaces

“…(For details, refer to [2], [17] and [26,Chap. For more information on the mapping F, the reader is referred to [5], [7], [12], [14] and [26, pp. See also Sect.…”