Einstein-like geometric structures on surfaces

Fox, Daniel J. F.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 499-585.

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein-Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein-Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Ampère metrics of Lorentzian and Riemannian signature and a Riemannian Einstein Kähler affine metric. A mean curvature zero space-like immersed Lagrangian submanifold of a para-Kähler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Classification : 53A15, 53C25, 53A30, 57N16, 57M50

[1] J. A. Aledo, J. M. Espinar and J. A. Gálvez, The Codazzi equation for surfaces, Adv. Math. 224 (2010), 2511–2530. | MR | Zbl

[2] S. Armstrong, Einstein connections and involutions via parabolic geometries, J. Geom. Phys. 60 (2010), 1424–1440. | MR | Zbl

[3] A. Banyaga, “The Structure of Classical Diffeomorphism Groups”, Mathematics and its Applications, Vol. 400, Kluwer Academic Publishers Group, Dordrecht, 1997. | MR | Zbl

[4] Y. Benoist, “Convexes divisibles. I”, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, 339–374. | MR | Zbl

[5] S. B. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1990), 1–17. | MR | Zbl

[6] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105–126. | MR | Zbl

[7] E. Calabi, On compact, Riemannian manifolds with constant curvature. I, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 155–180. | MR | Zbl

[8] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 223–230. | MR | Zbl

[9] E. Calabi, Complete affine hyperspheres. I, Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, 1972, 19–38. | MR | Zbl

[10] E. Calabi, Extremal Kähler metrics, In: “Seminar on Differential Geometry”, Ann. of Math. Stud., Vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, 259–290. | MR | Zbl

[11] E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982), 91–126. | MR | Zbl

[12] E. Calabi, Convex affine maximal surfaces, Results Math. 13 (1988), 199–223, reprinted in: “Affine Differentialgeometrie” [(Oberwolfach, 1986), 199–223, Tech. Univ. Berlin, Berlin, 1988]. | MR | Zbl

[13] D. M. J. Calderbank, Möbius structures and two-dimensional Einstein-Weyl geometry, J. Reine Angew. Math. 504 (1998), 37–53. | MR | Zbl

[14] D. M. J. Calderbank, The Faraday 2-form in Einstein-Weyl geometry, Math. Scand. 89 (2001), 97–116. | MR | Zbl

[15] D. M. J. Calderbank, Two dimensional Einstein-Weyl structures, Glasg. Math. J. 43 (2001), 419–424. | MR | Zbl

[16] A. Čap and A. R. Gover, Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), 1511–1548. | MR | Zbl

[17] A. Čap and J. Slovák, “Parabolic Geometries. I. Background and General Theory”, Mathematical Surveys and Monographs, Vol. 154, American Mathematical Society, Providence, RI, 2009. | MR | Zbl

[18] A. Čap and V. Žádník, Contact projective structures and chains, Geom. Dedicata 146 (2010), 67–83. | MR | Zbl

[19] X. Chen, P. Lu and G. Tian, A note on uniformization of Riemann surfaces by Ricci flow, Proc. Amer. Math. Soc. 134 (2006), 3391–3393. | MR | Zbl

[20] S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), 407–419. | MR | Zbl

[21] S.  Y.  Cheng  and  S.  T.  Yau, On the regularity of the Monge-Ampère equation $\det ({\partial }^{2}u/\partial x_i\partial x_j)=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), 41–68. | MR | Zbl

[22] S. Y. Cheng and S. T. Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980), 507–544. | MR | Zbl

[23] S. Y. Cheng and S. T. Yau, The real Monge-Ampère equation and affine flat structures, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) (Beijing), Science Press, 1982, 339–370. | MR | Zbl

[24] S. Y. Cheng and S. T. Yau, Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math. 39 (1986), 839–866. | MR | Zbl

[25] S. Choi and W. M. Goldman, The deformation spaces of convex ${\mathrm{ℝ\mathbb{P}}}^2$-structures on 2-orbifolds, Amer. J. Math. 127 (2005), 1019–1102. | MR | Zbl

[26] R. Courant and D. Hilbert, “Methods of mathematical physics. Vol. II: Partial Differential Equations”, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. | MR | Zbl

[27] P. Dazord, Sur la géométrie des sous-fibrés et des feuilletages lagrangiens, Ann. Sci. École Norm. Sup. (4) 14 (1981), 465–480. | EuDML | Numdam | MR | Zbl

[28] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory, J. Differential Geom. 3 (1969), 19–43. | MR | Zbl

[29] V. A. Fateev, E. Onofri and A. B. Zamolodchikov, Integrable deformations of the $\mathrm{O}\left(3\right)$ sigma model. The sausage model, Nuclear Phys. B 406 (1993), 521–565. | MR | Zbl

[30] D. J. F. Fox, Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations, arXiv:0909.1897. | MR

[31] D. J. F. Fox, Contact projective structures, Indiana Univ. Math. J. 54 (2005), 1547– 1598. | MR | Zbl

[32] D. J. F. Fox, Commutative, nonassociative algebras with non-degenerate, invariant Killing form, and the associated cubic polynomials, preprint, 2010.

[33] Ó. García-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994), 88–96. | MR | Zbl

[34] P. Gauduchon, La $1$-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), 495–518. | EuDML | MR | Zbl

[35] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1\times S^3$, J. Reine Angew. Math. 469 (1995), 1–50. | EuDML | MR | Zbl

[36] W. M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), 791–845. | MR | Zbl

[37] F. Herrlich and G. Schmithüsen, On the boundary of Teichmüller disks in Teichmüller and in Schottky space, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Zürich 11 (2007) 293–349. | MR | Zbl

[38] R. Hildebrand, The cross-ratio manifold: a model of centro-affine geometry, Int. Electron. J. Geom. 4 (2011), 32–62. | MR

[39] R. Hildebrand, Half-dimensional immersions in para-Kähler manifolds, Int. Electron. J. Geom. 4 (2011), 85–113. | MR

[40] Z. Hou, S. Deng, S. Kaneyuki and K. Nishiyama, Dipolarizations in semisimple Lie algebras and homogeneous para-Kähler manifolds, J. Lie Theory 9 (1999), 215–232. | EuDML | MR | Zbl

[41] J. Jost and Y. L. Xin, Some aspects of the global geometry of entire space-like submanifolds, Results Math. 40 (2001), 233–245. | MR | Zbl

[42] S. Kaneyuki, Compactification of parahermitian symmetric spaces and its applications. II. Stratifications and automorphism groups, J. Lie Theory 13 (2003), 535–563. | EuDML | MR | Zbl

[43] S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985), 81–98. | MR | Zbl

[44] J. L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. | MR | Zbl

[45] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), 293–311. | MR | Zbl

[46] S. Kobayashi, The first Chern class and holomorphic symmetric tensor fields, J. Math. Soc. Japan 32 (1980), 325–329. | MR | Zbl

[47] S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, Vol. II, Interscience Tracts in Pure and Applied Mathematics, n. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. | MR | Zbl

[48] M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, In: “Symplectic Geometry and Mirror Symmetry” (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 203–263. | MR | Zbl

[49] F. Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007), 1057–1099. | MR | Zbl

[50] J. Loftin and I. McIntosh, Minimal Lagrangian surfaces in ${\mathrm{ℂℍ}}^2$ and representations of surface groups into $SU(2,1)$, Geom. Dedicata 162 (2013), 67–93. | MR | Zbl

[51] J. C. Loftin, Affine spheres and convex ${\mathrm{ℝ\mathbb{P}}}^n$-manifolds, Amer. J. Math. 123 (2001), 255–274. | MR | Zbl

[52] J. C. Loftin, Riemannian metrics on locally projectively flat manifolds, Amer. J. Math. 124 (2002), 595–609. | MR | Zbl

[53] J. C. Loftin, The compactification of the moduli space of convex ${\mathrm{ℝ\mathbb{P}}}^2$ surfaces. I, J. Differential Geom. 68 (2004), 223–276. | MR | Zbl

[54] J. C. Loftin, Flat metrics, cubic differentials and limits of projective holonomies, Geom. Dedicata 128 (2007), 97–106. | MR | Zbl

[55] J.  C.  Loftin,  Survey on affine spheres,  In:  “Handbook of Geometric Analysis”, Vol. II, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 13 (2010), 161–192, arXiv:0809.1186. | MR | Zbl

[56] J. C. Loftin, S. T. Yau and E. Zaslow, Affine manifolds, SYZ geometry and the “Y” vertex, J. Differential Geom. 71 (2005), 129–158. | MR | Zbl

[57] J. C. Loftin, S. T. Yau and E. Zaslow, Erratum to affine manifolds, SYZ geometry and the “Y” vertex, Loftin’s web page, 2008. | Zbl

[58] H. Masur and S. Tabachnikov, Rational billiards and flat structures, In: “Handbook of Dynamical Systems”, Vol. 1A, North-Holland, Amsterdam, 2002, 1015–1089. | MR | Zbl

[59] I. McIntosh, Special Lagrangian cones in ${ℂ}^3$ and primitive harmonic maps, J. London Math. Soc. (2) 67 (2003), 769–789. | MR | Zbl

[60] N. Mok and S. T. Yau, Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, In: “The Mathematical Heritage” of Henri Poincaré, Part 1 (Bloomington, Ind., 1980), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI 39 (1983) 41–59. | MR | Zbl

[61] M. Noguchi, Yang-Mills-Higgs theory on a compact Riemann surface, J. Math. Phys. 28 (1987), 2343–2346. | MR | Zbl

[62] K. Nomizu and F. Podestà, On affine Kähler structures, Bull. Soc. Math. Belg. Sér. B 41 (1989), 275–282. | MR | Zbl

[63] K. Nomizu and T. Sasaki, “Affine Differential Geometry”, Cambridge Tracts in Mathematics, Vol. 111, Cambridge University Press, Cambridge, 1994. | MR | Zbl

[64] R. Schoen and S. T. Yau, “Lectures on Differential Geometry”, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. | MR | Zbl

[65] H. Shima and K. Yagi, Geometry of Hessian manifolds, Differential Geom. Appl. 7 (1997), 277–290. | MR | Zbl

[66] C. H. Taubes, Minimal surfaces in germs of hyperbolic 3-manifolds, Proceedings of the Casson Fest, Geom. Topol. Monogr., Geom. Topol. Publ., Coventry 7 (2004), 69–100. | MR | Zbl

[67] T. Y. Thomas, “The differential invariants of generalized spaces”, Chelsea Publishing Company, New York, 1934. | Zbl

[68] K. P. Tod, Compact $3$-dimensional Einstein-Weyl structures, J. London Math. Soc. (2) 45 (1992), 341–351. | MR | Zbl

[69] M. Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), 79–94. | MR | Zbl

[70] M. Troyanov, On the moduli space of singular Euclidean surfaces, In: “Handbook of Teichmüller Theory”, Vol. I, IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Zürich 11 (2007), 507–540. | MR | Zbl

[71] N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, In: “Handbook of Geometric Analysis”, n. 1, Adv. Lect. Math. (ALM), Vol. 7, Int. Press, Somerville, MA, 2008, 467–524. | MR | Zbl

[72] M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut. 19 (2006), 7–100. | EuDML | MR | Zbl

[73] A. Vitter, Affine structures on compact complex manifolds, Invent. Math. 17 (1972), 231–244. | EuDML | MR | Zbl

[74] L. Vrancken, Centroaffine differential geometry and its relations to horizontal submanifolds, In: “PDEs, Submanifolds and Affine Differential Geometry” (Warsaw, 2000), Banach Center Publ., Polish Acad. Sci., Warsaw 57 (2002), 21–28. | EuDML | MR | Zbl

[75] C. P. Wang, Some examples of complete hyperbolic affine $2$-spheres in ${ℝ}^3$, In: “Global Differential Geometry and Global Analysis” (Berlin, 1990), Lecture Notes in Math., Vol. 1481, Springer, Berlin, 1991, 271–280. | MR | Zbl

[76] H. Weyl, “The classical groups. Their invariants and representations”, Princeton University Press, Princeton, N.J., 1939. | MR | Zbl

[77] M. Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449–479. | MR | Zbl