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: Loftin, John C. Short title: Affine spheres and convex ℝℙ^{n}-manifolds Control number: 1 828 223 Primary classification: 53A Secondary classification(s): Review text:

The present paper relates affine differential geometry to the study of (flat) real projective structures on manifolds.

An ℝℙ^{n}-manifold is a manifold M with a distinguished atlas of coordinate charts to ℝℙ^{n} with coordinate changes in the projective group PGL(n+1, ℝ). Such a structure determines a projective equivalence class of projectively flat connections. Apart from exceptional cases (constant curvature Riemannian metrics) such a projectively flat connection will not be the Levi-Civita connection for a Riemannian structure. However, in some cases, one can find a canonical Riemannian metric on M with strong analytic properties related to the ℝℙ^{n}-structure.

An ℝℙ^{n}-manifold is convex if it is equivalent to the quotient Ω/Γ where Ω ⊂ ℝℙ^{n} is a properly convex domain and Γ ⊂ PGL(n + 1, ℝ) is a discrete subgroup acting properly on Ω. The present paper identifies convex ℝℙ^{n}- structures with structures arising in affine differential geometry and applies this correspondence to the moduli of ℝℙ^{n}-structures on surfaces when n = 2.

The basic object is an affine sphere, classically studied by Blaschke and others in the early twentieth century. An affine sphere is a strictly convex hypersurface in affine space satisfying a differential equation of Monge-Ampère type analogous to the Kähler-Einstein condition for Hermitian metrics on complex manifolds. The basic existence, uniqueness and regularity of solutions of this equation are due to Calabi, Loewner-Nirenberg, Cheng and Yau and others. In particular Cheng and Yau proved that every convex domain Ω ⊂ ℝℙ^{n} determines an affine sphere asymptotic to the cone over Ω in ℝ^{n+1} (Complete affine hyperspheres I. The completeness of affine metrics, Comm. Pure Appl. Math. 33 (1986), 839–866). (This settles a conjecture of Calabi.) Using this correspondence, the author proves (Theorem 1) that every convex ℝℙ^{n}-manifold carries a canonical complete Riemannian metric h and two canonical projectively flat affine connections ∇ and ¯∇. The projectively flat connection ∇ is due to Blaschke, and the average ˆ∇ = (∇ + ¯∇)/2 is the Levi-Civita connection of h. Furthermore ∇ and ¯∇ correspond to dual ℝℙ^{n}-structures, that is modelled on dual convex domains Ω, Ω̄ in dual projective spaces. The two connections agree precisely when Ω is the interior of a hyperquadric, in which case the ℝℙ^{n}-structure is a hyperbolic structure, via the Klein-Beltrami projective model of hyperbolic geometry.

In dimension n = 2, the reviewer has defined (Convex real projective structures on compact surfaces, J. Diff. Geo. 31 (1990), 791–845) a deformation space G(S) parametrizing equivalence classes of convex ℝℙ^{n}-structures on a compact surface S, and proved it to be diffeomorphic to ℝ^{16g−16} if S is closed orientable, and of genus g > 1. Using the analytic machinery, this paper contains a sharp analytic proof of this result.

With the conformal structure induced by the canonical Riemannian metric h, the convex ℝℙ^{n}-manifold M becomes a Riemann surface Σ. Following C.-P. Wang (Some examples of complete hyperbolic affine 2-spheres in ℝ^{3}, Global Differential Geometry and Global Analysis, Lecture Notes in Math. vol. 1481, Springer-Verlag, New York (1991) 271-280), the affine differential geometry determines a holomorphic cubic differential C ∈ H^{0}(Σ; K^{3}) (where K is the canonical bundle of Σ). Namely Pick form ˆ∇ − ∇ is a tensor field in

Hom(T Σ, End(T Σ)) ≅ T^{∗}Σ ⊗ (T^{∗}Σ ⊗ T Σ)

which corresponds to a holomorphic cubic differential. As explained to us by F. Labourie, dualizing the Pick form with respect to h produces a symmetric 3-tensor in Γ(S^{3}T^{∗}Σ). The holomorphic cubic differential C is then the (3,0)-Hodge component of the complexified 3-tensor.

Thus the deformation space G(S) of convex ℝℙ^{2}-structures on S identifies with the total space of the holomorphic ℂ^{5g−5}-bundle over the Teichmüller space T (S) whose fiber over a point σ ∈ T (S) is the vector space H^{0}(Σ; K^{3}) where Σ is the marked Riemann surface representing σ.

This approach closely relates to Hitchin’s description of a component ℭ of the deformation space

D = Hom(π_{1}(S), SL(3; ℝ))/SL(3; ℝ)

flat SL(3; ℝ)-bundles (Lie groups and Teichmüller space, Topology 31 (3), (1992), 449–473). For a fixed conformal structure on S, Hitchin finds a component of D which identifies with the complex vector space H^{0}(Σ, K^{2})⊕ H^{0}(Σ, K^{3}). Choi and the reviewer showed that ℭ contains the equivalence classes of holonomy representations ϕ : π_{1}(Σ) ⟶ SL(3; ℝ) of convex ℝℙ^{2}- structures, that is, ℭ equals G(S). (Convex real projective structures on closed surfaces are closed, Proc. A.M.S. 118 no.2 (1993), 657–661) Thus ϕ determines an element

Q ⊕ C ∈ H^{0}(Σ, K^{2}) ⊕ H^{0}(Σ, K^{3}).

The holomorphic quadratic differential Q is the Hopf differential of the (unique) ϕ-equivariant harmonic map of the universal covering space of Σ into the symmetric space of SL(3; ℝ). When Σ is the conformal structure induced by the affine metric h, then this map is holomorphic and the holomorphic cubic differential is the one induced by the Pick form above.