Geometry & Topology

Approximation theory for nonorientable minimal surfaces and applications

Antonio Alarcón and Francisco J López

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a version of the classical Runge and Mergelyan uniform approximation theorems for nonorientable minimal surfaces in Euclidean 3–space 3. Then we obtain some geometric applications. Among them, we emphasize the following ones:

  • A Gunning–Narasimhan-type theorem for nonorientable conformal surfaces.
  • An existence theorem for nonorientable minimal surfaces in 3 with arbitrary conformal structure, properly projecting into a plane.
  • An existence result for nonorientable minimal surfaces in 3 with arbitrary conformal structure and Gauss map omitting one projective direction.

Article information

Geom. Topol., Volume 19, Number 2 (2015), 1015-1062.

Received: 2 October 2013
Revised: 29 May 2014
Accepted: 6 July 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49Q05: Minimal surfaces [See also 53A10, 58E12]
Secondary: 30E10: Approximation in the complex domain

nonorientable minimal surfaces uniform approximation


Alarcón, Antonio; López, Francisco J. Approximation theory for nonorientable minimal surfaces and applications. Geom. Topol. 19 (2015), no. 2, 1015--1062. doi:10.2140/gt.2015.19.1015.

Export citation


  • A Alarcón, I Fernández, Complete minimal surfaces in $\mathbb R\sp 3$ with a prescribed coordinate function, Differential Geom. Appl. 29 (2011) S9–S15
  • A Alarcón, I Fernández, F J López, Complete minimal surfaces and harmonic functions, Comment. Math. Helv. 87 (2012) 891–904
  • A Alarcón, I Fernández, F J López, Harmonic mappings and conformal minimal immersions of Riemann surfaces into $\mathbb{R}\sp {\rm N}$, Calc. Var. PDE 47 (2013) 227–242
  • A Alarcón, F Forstnerič, Null curves and directed immersions of open Riemann surfaces, Invent. Math. 196 (2014) 733–771
  • A Alarcón, F J López, Gauss map of nonorientable minimal surfaces in $\mathbb{R}\sp n$, in preparation
  • A Alarcón, F J López, Minimal surfaces in $\mathbb R\sp 3$ properly projecting into $\mathbb{R}\sp 2$, J. Differential Geom. 90 (2012) 351–381
  • A Alarcón, F J López, Compact complete null curves in complex $3$–space, Israel J. Math. 195 (2013) 97–122
  • A Alarcón, F J López, Null curves in $\mathbb{C}\sp 3$ and Calabi–Yau conjectures, Math. Ann. 355 (2013) 429–455
  • A Alarcón, F J López, Complete nonorientable minimal surfaces in $\mathbb {R}^3$ and asymptotic behavior, Anal. Geom. Metr. Spaces 2 (2014) 214–234
  • A Alarcón, F J López, Properness of associated minimal surfaces, Trans. Amer. Math. Soc. 366 (2014) 5139–5154
  • E Bishop, Subalgebras of functions on a Riemann surface, Pacific J. Math. 8 (1958) 29–50
  • J Douglas, One-sided minimal surfaces with a given boundary, Trans. Amer. Math. Soc. 34 (1932) 731–756
  • H M Farkas, I Kra, Riemann surfaces, 2nd edition, Graduate Texts in Math. 71, Springer (1992)
  • L Ferrer, F Martín, W H Meeks, III, Existence of proper minimal surfaces of arbitrary topological type, Adv. Math. 231 (2012) 378–413
  • O Forster, Lectures on Riemann surfaces, Graduate Texts in Math. 81, Springer (1981)
  • F Forstnerič, Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. 56, Springer, Heidelberg (2011)
  • H Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan 40 (1988) 235–247
  • R C Gunning, R Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967) 103–108
  • L H örmander, An introduction to complex analysis in several variables, 3rd edition, North-Holland Math. Library 7, North-Holland, Amsterdam (1990)
  • L P d M Jorge, F Xavier, A complete minimal surface in $\mathbb{R}\sp{3}$ between two parallel planes, Ann. of Math. 112 (1980) 203–206
  • Y Kusunoki, Y Sainouchi, Holomorphic differentials on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971) 181–194
  • S Lie, Beiträge zur Theorie der Minimalflächen, Math. Ann. 14 (1878) 331–416
  • F J López, A nonorientable complete minimal surface in $\mathbb{R}\sp 3$ between two parallel planes, Proc. Amer. Math. Soc. 103 (1988) 913–917
  • F J López, F Martin, S Morales, Complete nonorientable minimal surfaces in a ball of $\mathbb R\sp 3$, Trans. Amer. Math. Soc. 358 (2006) 3807–3820
  • F J López, A Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991) 293–300
  • G Martens, Minimale Blätterzahl bei Überlagerungen Kleinscher Flächen über der projektiven Ebene, Arch. Math. $($Basel$)$ 30 (1978) 481–486
  • W H Meeks, III, The classification of complete minimal surfaces in $\mathbb{ R}\sp{3}$ with total curvature greater than $-8\pi $, Duke Math. J. 48 (1981) 523–535
  • S N Mergelyan, On the representation of functions by series of polynomials on closed sets, Doklady Akad. Nauk SSSR 78 (1951) 405–408
  • S Morales, On the existence of a proper minimal surface in $\mathbb{R}\sp 3$ with a conformal type of disk, Geom. Funct. Anal. 13 (2003) 1281–1301
  • N Nadirashvili, Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996) 457–465
  • R Osserman, A survey of minimal surfaces, 2nd edition, Dover Publ., New York (1986)
  • C Runge, Zur Theorie der Analytischen Functionen, Acta Math. 6 (1885) 245–248
  • R Schoen, S T Yau, Lectures on harmonic maps, Conf. Proc. and Lecture Notes in Geom. and Topol. II, International Press, Cambridge, MA (1997)