Twistor transforms of quaternionic functions and orthogonal complex structures

Journal of the European Mathematical Society

Volumn 16, Issue 11, 2014, Pages 2323-2353

  Gentili, Graziano ^a   Salamon, Simon ^b   Stoppato, Caterina ^a  

^a UNIVERSITY OF FLORENCE   (Italy)

^b KING'S COLLEGE LONDON   (United Kingdom)

Author keywords

Orthogonal complex structures; Quaternions; Regular quaternionic functions; Twistor methods

Indexed keywords

EID: 84911082250     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/488     Document Type: Article

References (30)

1. Bishop, E.: Conditions for the analyticity of certain sets. Michigan Math. J. 11, 289-304 (1964) Zbl 0143.30302 MR 0168801
2. Bisi, C., Gentili, G.: Möbius transformations and the Poincaré distance in the quaternionic setting. Indiana Univ. Math. J. 58, 2729-2764 (2009) Zbl 1193.30067 MR 2603766
3. Chen, J., Li, J.: Quaternionic maps between hyperkähler manifolds. J. Differential Geom. 55, 355-384 (2000) Zbl 1067.53035 MR 1847314
4. Colombo, F., Gentili, G., Sabadini, I., Struppa, D.: Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222, 1793-1808 (2009) Zbl 1179.30052 MR 2555912
5. Colombo, F., Sabadini, I., Sommen, F., Struppa, D. C.: Analysis of Dirac Systems and Computational Algebra. Progr. Math. Phys. 39, Birkhäuser Boston, Boston, MA (2004) Zbl 1064.30049 MR 2089988
6. Conway, J. B.: Functions of One Complex Variable. 2nd ed., Grad. Texts in Math. 11, Springer, New York (1978) Zbl 0277.30001 MR 0503901
7. Cullen, C. G.: An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32, 139-148 (1965) Zbl 0173.09001 MR 0173012
8. de Bartolomeis, P., Nannicini, A.: Introduction to the differential geometry of twistor spaces. In: Geometric Theory of Singular Phenomena in P.D.E., Symposia Math. 38, Cambridge Univ. Press, 91-160 (1998) Zbl 0921.53035 MR 1702089
9. Dolgachev, I. V.: Classical Algebraic Geometry. A Modern View. Cambridge Univ. Press, Cambridge (2012) Zbl 1252.14001 MR 2964027
10. Edge, W.: The Theory of Ruled Surfaces. Cambridge Univ. Press (1931) JFM 57.0842.05
11. Eells, J., Salamon, S.: Twistorial constructions of harmonic maps of surfaces into four-manifolds. Ann. Scuola Norm. Sup. Pisa 12, 589-640 (1985) Zbl 0627.58019 MR 0848842
12. Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J. 56, 655-667 (2008) Zbl 1184.30048 MR 2490652
13. Gentili, G., Stoppato, C.: The open mapping theorem for regular quaternionic functions. Ann. Scuola Norm. Sup. Pisa 8, 805-815 (2009) Zbl 1201.30067 MR 2647912
14. Gentili, G., Stoppato, C.: The zero sets of slice regular functions and the open mapping theorem. In: Hypercomplex Analysis and Applications, I. Sabadini and F. Sommen (eds.), Trends in Math., Birkhäuser, Basel, 95-107 (2011) Zbl 1234.30039 MR 3026135
15. Gentili, G., Struppa, D. C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216, 279-301 (2007) Zbl 1124.30015 MR 2353257
16. Gentili, G., Struppa, D. C.: On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math. 76, 15-25 (2008) Zbl 1194.30054 MR 2465984
17. Gentili, G., Struppa, D. C., Vlacci, F.: The fundamental theorem of algebra for Hamilton and Cayley numbers. Math. Z. 259, 895-902 (2008) Zbl 1144.30004 MR 24037471995
18. Joyce, D.: Hypercomplex algebraic geometry. Quart. J. Math. Oxford 49, 129-162 (1998) Zbl 0924.14002 MR 1634730
19. Mumford, D.: Algebraic Geometry. I. Classics in Math., Springer, Berlin (1995) Zbl 0821.14001 MR 1344216
20. Polo-Blanco, I., van der Put, M., Top, J.: Ruled quartic surfaces, models and classification. Geom. Dedicata 150, 151-180 (2011) Zbl 1226.14048 MR 2753703
21. Quillen, D.: Quaternionic algebra and sheaves on the Riemann sphere. Quart. J. Math. Oxford 49, 163-198 (1998) Zbl 0921.14003 MR 1634734
22. Salamon, S.: Harmonic and holomorphic maps. In: Geometry Seminar Luigi Bianchi II - 1984, Lecture Notes in Math. 1164, Springer, Berlin, 161-224 (1985) Zbl 0591.53031 MR 0829230
23. Salamon, S. M.: Quaternionic manifolds. In: Symposia Matematica, Vol. 26 (Rome, 1980), Academic Press, London, 139-151 (1982) Zbl 0534.53030 MR 0663029
24. 4. Math. Ann. 343, 853-899 (2009); see also arXiv:0704.3422v1 Zbl 1167.32017 MR 2471604
25. Shapiro, G.: On discrete differential geometry in twistor space. J. Geom. Phys. 68, 81-102 (2013) Zbl 1283.53013 MR 3035117
26. Stoppato, C.: Regular Moebius transformations of the space of quaternions. Ann. Global Anal. Geom. 39, 387-401 (2011) Zbl 1214.30044 MR 2776769
27. Stoppato, C.: A new series expansion for slice regular functions. Adv. Math. 231, 1401-1416 (2012) Zbl 1262.30059 MR 2964609
28. Stoppato, C.: Singularities of slice regular functions. Math. Nachr. 285, 1274-1293 (2012) Zbl 1253.30076 MR 2955794
29. Sudbery, A.: Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. 85, 199-224 (1979) Zbl 0399.30038 MR 0516081
30. Wood, J. C.: Harmonic morphisms and Hermitian structures on Einstein 4-manifolds. Int. J. Math. 3, 415-439 (1992) Zbl 0763.53051 MR 1163734