Geometry & Topology Volume 25, issue 7 (2021)

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Holomorphic Legendrian curves in $\mathbb{CP}^3$ and superminimal surfaces in $\mathbb{S}^4$

Antonio Alarcón, Franc Forstnerič and Finnur Lárusson

Geometry & Topology 25 (2021) 3507–3553

DOI: 10.2140/gt.2021.25.3507

Bibliography

1	A Alarcón, Complete complex hypersurfaces in the ball come in foliations, preprint (2018) arXiv:1802.02004
2	A Alarcón, I Castro-Infantes, Complete minimal surfaces densely lying in arbitrary domains of ℝⁿ, Geom. Topol. 22 (2018) 571 MR3720350
3	A Alarcón, I Castro-Infantes, Interpolation by conformal minimal surfaces and directed holomorphic curves, Anal. PDE 12 (2019) 561 MR3861901
4	A Alarcón, B Drinovec Drnovšek, F Forstnerič, F J López, Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves, Proc. Lond. Math. Soc. 111 (2015) 851 MR3407187
5	A Alarcón, F Forstnerič, Every bordered Riemann surface is a complete proper curve in a ball, Math. Ann. 357 (2013) 1049 MR3118624
6	A Alarcón, F Forstnerič, Null curves and directed immersions of open Riemann surfaces, Invent. Math. 196 (2014) 733 MR3211044
7	A Alarcón, F Forstnerič, Darboux charts around holomorphic Legendrian curves and applications, Int. Math. Res. Not. 2019 (2019) 893 MR3910477
8	A Alarcón, F Forstnerič, New complex analytic methods in the theory of minimal surfaces: a survey, J. Aust. Math. Soc. 106 (2019) 287 MR3951461
9	A Alarcón, F Forstnerič, A foliation of the ball by complete holomorphic discs, Math. Z. 296 (2020) 169 MR4140736
10	A Alarcón, F Forstnerič, The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends, Rev. Mat. Iberoam. 37 (2021) 1399 MR4269402
11	A Alarcón, F Forstnerič, F J López, Embedded minimal surfaces in ℝⁿ, Math. Z. 283 (2016) 1 MR3489056
12	A Alarcón, F Forstnerič, F J López, Holomorphic Legendrian curves, Compos. Math. 153 (2017) 1945 MR3705282
13	A Alarcón, F Forstnerič, F J López, New complex analytic methods in the study of nonorientable minimal surfaces in ℝⁿ, 1283, Amer. Math. Soc. (2020) MR4078111
14	A Alarcón, F Forstnerič, F J López, Minimal surfaces from a complex analytic viewpoint, Springer (2021) MR4237295
15	A Alarcón, J Globevnik, Complete embedded complex curves in the ball of ℂ² can have any topology, Anal. PDE 10 (2017) 1987 MR3694012
16	A Alarcón, J Globevnik, F J López, A construction of complete complex hypersurfaces in the ball with control on the topology, J. Reine Angew. Math. 751 (2019) 289 MR3956697
17	A Alarcón, F J López, Minimal surfaces in ℝ³ properly projecting into ℝ², J. Differential Geom. 90 (2012) 351 MR2916039
18	A Alarcón, F J López, Complete bounded embedded complex curves in ℂ², J. Eur. Math. Soc. 18 (2016) 1675 MR3519537
19	H Behnke, K Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann. 120 (1949) 430 MR29997
20	J Bolton, L M Woodward, Higher singularities and the twistor fibration π: ℂP³ → S⁴, Geom. Dedicata 80 (2000) 231 MR1762511
21	R L Bryant, Conformal and minimal immersions of compact surfaces into the 4–sphere, J. Differential Geom. 17 (1982) 455 MR679067
22	M P do Carmo, Riemannian geometry, Birkhäuser (1992) MR1138207
23	J Eells Jr, S Kobayashi, editors, Proceedings of the United States–Japan seminar in differential geometry, Nippon Hyoronsha (1966) MR0212686
24	J E Fornæss, F Forstnerič, E F Wold, Holomorphic approximation : the legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan, from: "Advancements in complex analysis" (editors D Breaz, M T Rassias), Springer (2020) 133
25	F Forstnerič, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, 56, Springer (2017) MR3700709
26	F Forstnerič, The Calabi–Yau property of superminimal surfaces in self-dual Einstein four-manifolds, J. Geom. Anal. 31 (2021) 4754
27	F Forstnerič, Immersions of open Riemann surfaces into the Riemann sphere, Izv. Math. 85 (2021) 562
28	F Forstnerič, F Lárusson, Holomorphic Legendrian curves in projectivised cotangent bundles, preprint (2018) arXiv:1809.09391
29	F Forstnerič, F Lárusson, The Oka principle for holomorphic Legendrian curves in ℂ²ⁿ⁺¹, Math. Z. 288 (2018) 643 MR3774429
30	T Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984) 257 MR777909
31	T Friedrich, On superminimal surfaces, Arch. Math. (Brno) 33 (1997) 41 MR1464300
32	J Globevnik, A complete complex hypersurface in the ball of ℂ^N, Ann. of Math. 182 (2015) 1067 MR3418534
33	M Goresky, R MacPherson, Stratified Morse theory, 14, Springer (1988) MR932724
34	R C Gunning, R Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967) 103 MR223560
35	Z X He, O Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993) 369 MR1207210
36	L P d M Jorge, F Xavier, A complete minimal surface in ℝ³ between two parallel planes, Ann. of Math. 112 (1980) 203 MR584079
37	S L Kleiman, The transversality of a general translate, Compos. Math. 28 (1974) 287 MR360616
38	C LeBrun, Fano manifolds, contact structures, and quaternionic geometry, Int. J. Math. 6 (1995) 419 MR1327157
39	C LeBrun, S Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994) 109 MR1288469
40	N Nadirashvili, Hadamard’s and Calabi–Yau’s conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996) 457 MR1419004
41	K Oka, Sur les fonctions analytiques de plusieurs variables, III : Deuxième problème de Cousin, J. Sci. Hiroshima Univ. 9 (1939) 7
42	R Penrose, Twistor theory as an approach to fundamental physics, from: "Foundations of mathematics and physics one century after Hilbert" (editor J Kouneiher), Springer (2018) 253 MR3822555
43	R Penrose, M A H MacCallum, Twistor theory: an approach to the quantisation of fields and space-time, Phys. Rep. 6C (1973) 241 MR475660
44	H L Royden, Function theory on compact Riemann surfaces, J. Anal. Math. 18 (1967) 295 MR214757
45	H Whitney, Differentiable manifolds, Ann. of Math. 37 (1936) 645 MR1503303
46	J C Wood, Twistor constructions for harmonic maps, from: "Differential geometry and differential equations" (editors C H Gu, M Berger, R L Bryant), Lecture Notes in Math. 1255, Springer (1987) 130 MR895404
47	P Yang, Curvatures of complex submanifolds of ℂⁿ, J. Differential Geom. 12 (1977) 499 MR512921
48	S T Yau, Review of geometry and analysis, from: "Mathematics: frontiers and perspectives" (editors V Arnold, M Atiyah, P Lax, B Mazur), Amer. Math. Soc. (2000) 353 MR1754787