Flat metrics, cubic differentials and limits of projective holonomies. (English) Zbl 1145.57014
The author and F. Labourie have independently shown that a convex real projective structure on an oriented surface (of genus \(\geq 2\)) corresponds precisely to a pair consisting of a conformal structure on the surface and a holomorphic cubic differential. Now fix the conformal structure and allow the cubic differential \(\lambda U_0\) to vary as \(\lambda \to \infty\).
In this paper, the limiting holonomy of convex real projective structures is then determined along certain paths. An application to Kim’s compactification of the deformation space of convex real projective structures is given.
In this paper, the limiting holonomy of convex real projective structures is then determined along certain paths. An application to Kim’s compactification of the deformation space of convex real projective structures is given.
Reviewer: John F. Oprea (Cleveland)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 53A15 | Affine differential geometry |
References:
| [1] | Bestvina M. (1988). Degenerations of the hyperbolic space. Duke Math. J 56(1): 143–161 · Zbl 0652.57009 · doi:10.1215/S0012-7094-88-05607-4 |
| [2] | Calabi E. (1972). Complete affine hyperspheres I. Instituto Nazionale di Alta Matematica Symposia Mathematica 10: 19–38 · Zbl 0252.53008 |
| [3] | Cheng S.-Y. and Yau S.-T. (1977). On the regularity of the Monge-Ampère equation det \((\partial^2u/\partial x^i\partial x^j)=F(x,u)\) Commun. Pure Appl. Math. 30: 41–68 · Zbl 0347.35019 · doi:10.1002/cpa.3160300104 |
| [4] | Cheng S.-Y. and Yau S.-T. (1986). Complete affine hyperspheres. part I. The completeness of affine metrics. Commun. Pure Appl. Math. 39(6): 839–866 · Zbl 0623.53002 · doi:10.1002/cpa.3160390606 |
| [5] | Fathi, A., Laudenbach, F., Poénaru, V. (eds.): Travaux de Thurston sur les surfaces. Astérisque, vol. 66. Société Mathématique de France, Paris, Séminaire Orsay, With an English summary (1979) |
| [6] | Goldman W.M. (1990). Convex real projective structures on compact surfaces. J. Differential Geom. 31: 791–845 · Zbl 0711.53033 |
| [7] | Han, Z.-C.: Remarks on the geometric behavior of harmonic maps between surfaces. In Elliptic and Parabolic Methods in Geometry, pp 57–66. (Minneapolis, MN, 1994) A K Peters, Wellesley, MA (1996) · Zbl 0871.58025 |
| [8] | Kac, V.G., Vinberg, È. B.: Quasi-homogeneous cones. Math. Notes 1, 231–235 (1967) Translated from Mat. Zametki 1, 347–354 (1967) · Zbl 0163.16902 |
| [9] | Kim I. (2001). Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds. J. Differential Geom. 58(2): 189–218 · Zbl 1076.53053 |
| [10] | Kim I. (2005). Compactification of strictly convex real projective structures. Geom. Dedicata 113: 185–195 · Zbl 1087.51008 · doi:10.1007/s10711-005-0550-7 |
| [11] | Labourie, F.: Flat projective structures on surfaces and cubic holomorphic differentials. Preprint. · Zbl 1158.32006 |
| [12] | Labourie, F.: In Proceedings of the GARC Conference in Differential Geometry, Seoul National University, Fall 1997 (1997) |
| [13] | Loftin J.C. (2001). Affine spheres and convex \({\mathbb{RP}}^n\) manifoldsAm. J. Math. 123(2): 255–274 · Zbl 0997.53010 · doi:10.1353/ajm.2001.0011 |
| [14] | Loftin, J.C.: The compactification of the moduli space of \({\mathbb{RP}}^2\) surfaces, I. J. Differential Geom. 68(2), 223–276 (2004) math.DG/0311052 · Zbl 1085.14024 |
| [15] | Minsky Y.N. (1992). Harmonic maps, length and energy in Teichmüller space. J. Differential Geom. 35(1): 151–217 · Zbl 0763.53042 |
| [16] | Morgan J.W. and Shalen P.B. (1984). Valuations, trees and degenerations of hyperbolic structures. I. Ann. Math. (2) 120(3): 401–476 · Zbl 0583.57005 · doi:10.2307/1971082 |
| [17] | Parreau, A.: Dégénérescence de sous-groupes discrets des groupes de Lie semisimples et actions de groupes sur des immeubles affines. Ph.D. thesis, Université de Paris-sud (2000) |
| [18] | Paulin F. (1988). Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94(1): 53–80 · Zbl 0673.57034 · doi:10.1007/BF01394344 |
| [19] | Paulin F. (1997). Dégénérescence de sous-groupes discrets des groupes de Lie semi-simples. C. R. Acad. Sci. Paris Sér. I Math. 324(11): 1217–1220 · Zbl 0877.22005 |
| [20] | Wang, C.-P.: Some examples of complete hyperbolic affine 2-spheres in \({\mathbb{R}}^3\) . In Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 1481, pp 272–280. Springer-Verlag (1991) |
| [21] | Wolf M. (1989). The Teichmüller theory of harmonic maps. J. Differential Geom. 29(2): 449–479 · Zbl 0655.58009 |
| [22] | Wolf M. (1995). Harmonic maps from surfaces to R-trees. Math. Z. 218(4): 577–593 · Zbl 0827.58011 · doi:10.1007/BF02571924 |
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