This is a review text file submitted electronically to MR. Reviewer: W.M.Goldman Reviewer number: 014521 Address:

Math Dept. University of Maryland,College Park, Maryland 20742 Author: Tanigawa, Harumi Short title: Divergence of Projective Structures and Lengths of Measured Laminations Control number: 1 695 198 Primary classification: 32G Secondary classification(s): 30F10 Review text:

This paper discusses the relationship between two parametrizations of complex projective structures (ℂP^{1}-structures) on a closed surface S. Classically, ℂP^{1}-structures are parametrized by elements ([M], q) of the total space of the cotangent bundle π : T^{∗}𝔗(S) ⟶ 𝔗(S) of Teichmüller space 𝔗(S), whereby the point [M] = π(([M], q)) in Teichmüller space records the complex structure underlying the ℂP^{1}-structure. The fiber is the vector space Q(M) of holomorphic quadratic differentials q ∈ Q(M) on the Riemann surface M which may be identified as tangent covectors in T_{[M]}^{∗}𝔗(S). The cotangent vector q arises as the Schwarzian differential of the developing map (regarded as an π_{1}(M)-equivariant meromorphic function defined on the universsal covering M̃ ⟶ M).

The second parametrization is due to Thurston and is much more recent. For this a ℂP^{1}-structure is realized as a pleated section of a flat H^{3}-bundle over S, in other words, as a pleated map from the universal covering surface S̃ into H^{3} which is equivariant with respect to a homomorphism ϕ : π_{1}(M) ⟶ PSL(2, ℂ). A pleated map is a piecewise totally geodesic embedding, which is bent along a geodesic lamination. The intrinsic geometry on the pleated surface is hyperbolic geometry, and is recorded by a (probably different) point [M′] in 𝔗(S). (Here 𝔗(S) is regarded as the space of hyperbolic structures rather than complex structures.) The extrinsic geometry is recorded by an invariant transverse measure to this lamination. Therefore ℂP^{1}-structures are parametrized by the Cartesian product 𝔗(S) × ML(S) where ML(S) is Thurston’s space of measured geodesic laminations.

Monodromy defines a map

mon : ℂP^{1}(S) ⟶ Hom(π_{1}(S), PSL(2, ℂ))/ PSL(2, ℂ)

from the moduli space ℂP^{1}(S) of ℂP^{1}-structures on S. In general this map is not proper. The author geometrically proves Kapovich’s result (On monodromy of complex projective structures, Inv. Math. 119 (1995), 243– 265) that, for a fixed Riemann surface M, the restriction mon |_{Q(M)} is proper (although mon is not proper. More generally she proves that for any compact K ⊂ 𝔗(S), the restriction mon |_{π^{−1}(K)} is proper.

The monodromy map is more directly related to (and more eaily computable in) Thurston’s parametrization using bending than the classical parametrization in terms of quadratic differentials. The proofs use the author’s previous work (Grafting, harmonic maps and projective structures on surfaces, J. Diff. Geo. 47 (1997), 339–319) to relate extremal length in the hyperbolic structure underlying the bent structure to the length of laminations. An interesting new concept is the extension of the length function associated to a measured geodesic lamination μ to a function ℓ_{μ} on the subset of Hom(π_{1}(S), PSL(2, ℂ))/ PSL(2, ℂ) consisting of nonelmentary representations, which is proved to be locally bounded.