Proper superminimal surfaces of given conformal types in the hyperbolic four-space

Forstnerič, Franc

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 1, pp. 145-172

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline{M}\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $M\to H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^4$.

Classification : 53A10, 53C28, 32E30, 37J55
Keywords: superminimal surface, hyperbolic space, twistor space, complex contact manifold, holomorphic Legendrian curve

[1] Alarcón, Antonio; Drnovšek, Barbara Drinovec; Forstnerič, Franc; López, Francisco J. Minimal surfaces in minimally convex domains, Trans. Am. Math. Soc., Volume 371 (2019) no. 3, pp. 1735-1770 | MR | DOI | Zbl

[2] Alarcón, Antonio; Forstnerič, Franc Every bordered Riemann surface is a complete proper curve in a ball, Math. Ann., Volume 357 (2013) no. 3, pp. 1049-1070 | DOI | Zbl | MR

[3] Alarcón, Antonio; Forstnerič, Franc Null curves and directed immersions of open Riemann surfaces, Invent. Math., Volume 196 (2014) no. 3, pp. 733-771 | Zbl | MR | DOI

[4] Alarcón, Antonio; Forstnerič, Franc Darboux charts around holomorphic Legendrian curves and applications, Int. Math. Res. Not., Volume 2019 (2019) no. 3, pp. 893-922 | DOI | Zbl | MR

[5] Alarcón, Antonio; Forstnerič, Franc New complex analytic methods in the theory of minimal surfaces: a survey, J. Aust. Math. Soc., Volume 106 (2019) no. 3, pp. 287-341 | Zbl | MR | DOI

[6] Alarcón, Antonio; Forstnerič, Franc The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends, Rev. Mat. Iberoam., Volume 37 (2021) no. 4, pp. 1399-1412 | Zbl | MR | DOI

[7] Alarcón, Antonio; Forstnerič, Franc; Lárusson, Finnur Holomorphic Legendrian curves in ${\mathrm{ℂ\mathbb{P}}}^3$ and superminimal surfaces in ${𝕊}^4$, Geom. Topol., Volume 25 (2021) no. 7, pp. 3507-3553 | Zbl | MR | DOI

[8] Alarcón, Antonio; Forstnerič, Franc; López, Francisco J. Holomorphic Legendrian curves, Compos. Math., Volume 153 (2017) no. 9, pp. 1945-1986 | Zbl | MR | DOI

[9] Anderson, Michael T. Complete minimal varieties in hyperbolic space, Invent. Math., Volume 69 (1982) no. 3, pp. 477-494 | Zbl | MR | DOI

[10] Atiyah, Michael F.; Hitchin, Nigel J.; Singer, Isadore M. Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. Lond., Ser. A, Volume 362 (1978) no. 1711, pp. 425-461 | Zbl | MR

[11] Biquard, Olivier Métriques autoduales sur la boule, Invent. Math., Volume 148 (2002) no. 3, pp. 545-607 | Zbl | DOI

[12] Bryant, Robert L. Conformal and minimal immersions of compact surfaces into the $4$-sphere, J. Differ. Geom., Volume 17 (1982) no. 3, pp. 455-473 | Zbl | MR

[13] do Carmo, Manfredo Perdigão Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser, 1992 (translated from the second Portuguese edition by Francis Flaherty) | MR

[14] Černe, Miran; Flores, Manuel Generalized Ahlfors functions, Trans. Am. Math. Soc., Volume 359 (2007) no. 2, pp. 671-686 | Zbl | MR | DOI

[15] Drinovec Drnovšek, Barbara; Forstnerič, Franc Holomorphic curves in complex spaces, Duke Math. J., Volume 139 (2007) no. 2, pp. 203-253 | MR | Zbl

[16] Drnovšek, Barbara Drinovec; Forstnerič, Franc The Poletsky–Rosay theorem on singular complex spaces, Indiana Univ. Math. J., Volume 61 (2012) no. 4, pp. 1407-1423 | Zbl | MR | DOI

[17] Eells, James; Salamon, Simon Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 12 (1985) no. 4, pp. 589-640 | Zbl | Numdam | MR

[18] Forstnerič, Franc Stein manifolds and holomorphic mappings (The homotopy principle in complex analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 56, Springer, 2017 | DOI

[19] Forstnerič, Franc The Calabi–Yau property of superminimal surfaces in self-dual Einstein four-manifolds, J. Geom. Anal., Volume 31 (2021) no. 5, pp. 4754-4780 | Zbl | MR | DOI

[20] Forstnerič, Franc Mergelyan approximation theorem for holomorphic Legendrian curves, Anal. PDE, Volume 15 (2022) no. 4, pp. 983-1010 | Zbl | MR | DOI

[21] Forstnerič, Franc; Globevnik, Josip Discs in pseudoconvex domains, Comment. Math. Helv., Volume 67 (1992) no. 1, pp. 129-145 | Zbl | MR | DOI

[22] Friedrich, Thomas On surfaces in four-spaces, Ann. Global Anal. Geom., Volume 2 (1984) no. 3, pp. 257-287 | DOI | Zbl | MR

[23] Friedrich, Thomas On superminimal surfaces, Arch. Math., Brno, Volume 33 (1997) no. 1-2, pp. 41-56 | Zbl | MR

[24] Gauduchon, Paul La correspondance de Bryant, Theory of minimal submanifolds and applications (Astérisque), Volume 154-155, Société Mathématique de France, 1987, pp. 181-208 | Numdam

[25] Gauduchon, Paul Les immersions super-minimales d’une surface compacte dans une variété riemannienne orientée de dimension 4, Theory of minimal submanifolds and applications (Astérisque), Volume 154-155, Société Mathématique de France, 1987, pp. 151-180 | Numdam

[26] Goresky, Mark; MacPherson, Robert Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 14, Springer, 1988 | DOI

[27] Graham, C. Robin; Lee, John M. Einstein metrics with prescribed conformal infinity on the ball, Adv. Math., Volume 87 (1991) no. 2, pp. 186-225 | MR | Zbl | DOI

[28] Hitchin, Nigel J. Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differ. Geom., Volume 42 (1995) no. 1, pp. 30-112 | Zbl | MR

[29] Kleiman, Steven L. The transversality of a general translate, Compos. Math., Volume 28 (1974), pp. 287-297 | Zbl | Numdam | MR

[30] Kommerell, Karl Die Krümmung der zweidimensionalen Gebilde im ebenen Raum von vier Dimensionen, Tübingen, Volume 53 (1897), 8 | Zbl

[31] Lawson, H. Blaine jun. Surfaces minimales et la construction de Calabi–Penrose, Séminaire Bourbaki 1983/84 (Astérisque), Volume 1983, Société Mathématique de France, 1985, pp. 197-211 | Zbl | Numdam | MR

[32] Lawson, H. Blaine jun. La classification des $2$-sphères minimales dans l’espace projectif complexe, Theory of minimal submanifolds and applications (Astérisque), Volume 154-155, Société Mathématique de France, 1987, pp. 131-149 | Numdam

[33] LeBrun, Claude; Salamon, Simon Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., Volume 118 (1994) no. 1, pp. 109-132 | Zbl | DOI

[34] Penrose, Roger Twistor algebra, J. Math. Phys., Volume 8 (1967), pp. 345-366 | MR | Zbl | DOI

[35] Poincaré, Henri Sur la dynamique de l’électron, Rend. Circ. Mat. Palermo, Volume 21 (1906), pp. 129-175 | Zbl | DOI

[36] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, 1994

[37] Salamon, Simon Topics in four-dimensional Riemannian geometry, Geometry seminar “Luigi Bianchi” (Pisa, 1982) (Lecture Notes in Mathematics), Volume 1022, Springer, 1982, pp. 33-124 | Zbl | MR | DOI

[38] Stout, Edgar L. Bounded holomorphic functions on finite Riemann surfaces, Trans. Am. Math. Soc., Volume 120 (1965), pp. 255-285 | Zbl

[39] Yau, Shing-Tung Review of geometry and analysis, Mathematics: Frontiers and perspectives, American Mathematical Society, 2000, pp. 353-401

[40] Yau, Shing-Tung Review of geometry and analysis, Asian J. Math., Volume 4 (2000) no. 1, pp. 235-278 | MR