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Null holomorphic curves in \(\mathbb C^3\) and applications to the conformal Calabi-Yau problem. (English) Zbl 1338.53085

Fornæss, John Erik (ed.) et al., Complex geometry and dynamics. The Abel symposium 2013, Trondheim, Norway, July 2–5, 2013. Cham: Springer (ISBN 978-3-319-20336-2/hbk; 978-3-319-20337-9/ebook). Abel Symposia 10, 101-121 (2015).
Summary: In this paper we survey some recent contributions by the authors [Math. Ann. 357, No. 3, 1049–1070 (2013; Zbl 1288.32014); Invent. Math. 196, No. 3, 733–771 (2014; Zbl 1297.32009); Math. Ann. 363, No. 3–4, 913–951 (2015; Zbl 1343.53053)] to the theory of null holomorphic curves in the complex Euclidean space \(\mathbb C^3\), as well as their applications to null holomorphic curves in the special linear group \(\mathrm{SL}_2(\mathbb C)\), minimal surfaces in the Euclidean space \(\mathbb R^3\), and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space \(\mathbb H^3\). The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.
For the entire collection see [Zbl 1336.32001].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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