Geodesic
Complex Structures and Conformal Geometry
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 199-224.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

A characterization of certain complex structures on conformally-flat domains in real dimension 4 is carried out in the context of Hermitian geometry and twistor spaces. The presentation is motivated by some classical surface theory, whilst the problem itself leads to a refined classification of quadrics in complex projective 3-space. The main results are sandwiched between general facts in real dimension 2n and some concluding examples in real dimension 6. 
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Salamon, Simon. Complex Structures and Conformal Geometry. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 199-224. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a9/

[1] V. Apostolov - P. Gauduchon - G. Grantcharov, Bi-Hermitian structures on complex surfaces, Proc. London Math. Soc. (3), 79, 2 (1999), 414-428. | DOI | MR | Zbl

[2] V. Apostolov - G. Grantcharov - S. Ivanov, Orthogonal complex structures on certain Riemannian 6-manifolds, Differ. Geom. Appl., 11 (1999), 279-296. | DOI | MR | Zbl

[3] V. Apostolov - M. Gualtieri, Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, Comm. Math. Phys., 271 (2007), 561-575. | DOI | MR | Zbl

[4] M. F. Atiyah - N. J. Hitchin - I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362, 1711 (1978), 425-461. | DOI | MR | Zbl

[5] M. F. Atiyah - R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys., 55 (1977), 117-124. | MR | Zbl

[6] P. Baird - J. C. Wood, Harmonic Morphisms between Riemannian Manifolds, volume 29 of London Mathematical Society Monographs, Oxford University Press (Oxford, 2003). | DOI | MR | Zbl

[7] L. Bérard Bergery - T. Ochiai, On some generalizations of the construction of twistor spaces. In Global Riemannian geometry (Durham, 1983) (Ellis Horwood, Chichester, 1984), 52-59. | MR

[8] A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987. | DOI | MR

[9] E. Bishop, Conditions for the analyticity of certain sets, Michigan Math. J., 11 (1964), 289-304. | MR | Zbl

[10] L. Borisov - S. Salamon - J. Viaclovsky, Orthogonal complex structures in Euclidean spaces, In preparation.

[11] R. L. Bryant, Submanifolds and special structures on the octonians, J. Differ. Geom., 17 (1982), 185-232. | MR | Zbl

[12] R. L. Bryant, Lie groups and twistor spaces, Duke Math. J., 52 (1985), 223-261. | DOI | MR | Zbl

[13] F. E. Burstall - J. H. Rawnsley, Twistor Theory for Riemannian Symmetric Spaces, Lecture Notes Math. 1424 (Springer-Verlag Berlin, 1990). | DOI | MR | Zbl

[14] E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc., 87 (1958), 407-438. | DOI | MR | Zbl

[15] É. Cartan, The Theory of Spinors, Dover Publications Inc. (New York, 1981). With a foreword by Raymond Streater, A reprint of the 1966 English translation, Dover Books on Advanced Mathematics. | MR | Zbl

[16] S. S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc., 6 (1955), 771-782. | DOI | MR | Zbl

[17] S. K. Donaldson - J. Fine, Toric anti-self-dual 4-manifolds via complex geometry, Math. Ann., 336 (2006), 281-309. | DOI | MR | Zbl

[18] J. Eells - S. Salamon, Twistorial constructions of harmonic maps of surfaces into four-manifolds, Ann. Sc. Norm. Sup. Pisa, 12 (1985), 589-640. | fulltext EuDML | MR | Zbl

[19] L. C. Evans - R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992). | MR | Zbl

[20] A. Fujiki - M. Pontecorvo, On Hermitian geometry of complex surfaces. In Complex, contact and symmetric manifolds, Progr. Math. 234 (Birkhäuser, Boston, 2005), 153-163. | DOI | MR | Zbl

[21] P. Grassberger, On the Hausdorff Dimension of Fractal Attractors, J. Stat. Phys, 26 (1981), 173-179. | DOI | MR

[22] A. Gray - E. Abbena - S. Salamon, Modern Differential Geometry of Curves and Surfaces, with Mathematica, CRC Press, Taylor and Francis (2006). | MR | Zbl

[23] M. Gualtieri, Generalized complex geometry. arXiv:math/0703298. | DOI | MR | Zbl

[24] R. Hilborn, Chaos and Nonlinear Dynamics. An Introduction for Scientists and Engineers, Oxford University Press (New York, 1994). | MR

[25] N. Hitchin, Bihermitian metrics on Del Pezzo surfaces, J. Symplectic Geom., 5, 1 (2007), 1-8. | MR | Zbl

[26] D. Joyce, The hypercomplex quotient and the quaternionic quotient, Math. Ann., 290 (1991), 323-340. | fulltext EuDML | DOI | MR | Zbl

[27] P. Kobak, Explicit doubly-Hermitian metrics, Differential Geom. Appl., 10 (1999), 179-185. | DOI | MR | Zbl

[28] C. Lebrun - Y. S. Poon, Self-dual manifolds with symmetry. In Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (Providence, RI, 1993), 365-377. | MR | Zbl

[29] C. R. Lebrun, Explicit self-dual metrics on $\mathbb{CP}^{2}\# \cdots \# \mathbb{CP}^{2}$, J. Differ. Geom., 34 (1991), 223-253. | MR | Zbl

[30] J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195. | MR | Zbl

[31] D. Mumford, Algebraic Geometry. I, Classics in Mathematics (Springer-Verlag, Berlin, 1995). Complex projective varieties. | MR

[32] A. Newlander - L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (2) (1957), 391-404. | DOI | MR | Zbl

[33] N. R. O'Brian - J. R. Rawnsley, Twistor spaces, Ann. Global Anal. Geom., 3 (1985), 29-58. | DOI | MR

[34] M. Pontecorvo, Uniformization of conformally flat Hermitian surfaces, Differential Geom. Appl., 2, 3 (1992), 295-305. | DOI | MR | Zbl

[35] Y. S. Poon, Compact self-dual manifolds with positive scalar curvature, J. Differ. Geom., 24 (1986), 97-132. | MR | Zbl

[36] S. Salamon - J. Viaclovsky, Orthogonal complex structures on domains of $\mathbb{R}^{4}$, to appear in Math. Ann. | DOI | MR | Zbl

[37] S. M. Salamon, Orthogonal complex structures. In Differential geometry and applications (Brno, 1995) (Masaryk Univ., Brno, 1996), 103-117. | MR | Zbl

[38] S. M. Salamon, Hermitian geometry. In Invitations to Geometry and Topology, Oxf. Grad. Texts Math. 7, (Oxford University Press, 2002), 233-291. | MR

[39] B. Shiffman, On the removal of singularities of analytic sets, Michigan Math. J., 15 (1968), 111-120. | MR | Zbl

[40] M. J. Slupinski, The twistor space of the conformal six sphere and vector bundles on quadrics, J. Geom. Phys., 19 (1996), 246-266. | DOI | MR | Zbl

[41] F. Tricerri - L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc., 267 (1981), 365-397. | DOI | MR | Zbl

[42] J. C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math., 3, 3 (1992), 415-439. | DOI | MR | Zbl