Show Summary Details
More options …

# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
See all formats and pricing
More options …
Volume 2019, Issue 753

# Embedded minimal surfaces of finite topology

William H. Meeks III
/ Joaquín Pérez
Published Online: 2017-03-16 | DOI: https://doi.org/10.1515/crelle-2017-0008

## Abstract

In this paper we prove that a complete, embedded minimal surface M in ${ℝ}^{3}$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion $\overline{M}$. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

## References

• [1]

J. Bernstein and C. Breiner, Conformal structure of minimal surfaces with finite topology, Comment. Math. Helv. 86 (2011), no. 2, 353–381.

• [2]

J. Bernstein and C. Breiner, Helicoid-like minimal disks and uniqueness, J. reine angew. Math. 655 (2011), 129–146.

• [3]

T. H. Colding and W. P. Minicozzi II, An excursion into geometric analysis, Surveys of differential geometry IX – Eigenvalues of Laplacian and other geometric operators, International Press, Somerville (2004), 83–146. Google Scholar

• [4]

T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. I: Estimates off the axis for disks, Ann. of Math. (2) 160 (2004), 27–68.

• [5]

T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. II: Multi-valued graphs in disks, Ann. of Math. (2) 160 (2004), 69–92.

• [6]

T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. III: Planar domains, Ann. of Math. (2) 160 (2004), 523–572.

• [7]

T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV: Locally simply-connected, Ann. of Math. (2) 160 (2004), 573–615.

• [8]

T. H. Colding and W. P. Minicozzi II, The Calabi–Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), 211–243.

• [9]

T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. V: Fixed genus, Ann. of Math. (2) 181 (2015), no. 1, 1–153.

• [10]

P. Collin, Topologie et courbure des surfaces minimales de ${ℝ}^{3}$, Ann. of Math. (2) 145 (1997), 1–31. Google Scholar

• [11]

L. Hauswirth, J. Pérez and P. Romon, Embedded minimal ends of finite type, Trans. Amer. Math. Soc. 353 (2001), 1335–1370.

• [12]

W. H. Meeks III and J. Pérez, The classical theory of minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 48 (2011), 325–407.

• [13]

W. H. Meeks III and J. Pérez, A survey on classical minimal surface theory, Univ. Lecture Ser. 60, American Mathematical Society, Providence 2012. Google Scholar

• [14]

W. H. Meeks III and J. Pérez, Finite type annular ends for harmonic functions, Math. Ann. (2016), 10.1007/s00208-016-1407-0.

• [15]

W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, in progress.

• [16]

W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, preprint (2015), https://arxiv.org/abs/1505.06761; to appear in J. Differential Geom.

• [17]

W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362.

• [18]

W. H. Meeks III, J. Pérez and A. Ros, Structure theorems for singular minimal laminations, preprint (2016), https://arxiv.org/abs/1602.03197v1.

• [19]

W. H. Meeks III and H. Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), 723–754. Google Scholar

• [20]

R. Osserman, Global properties of minimal surfaces in ${E}^{3}$ and ${E}^{n}$, Ann. of Math. (2) 80 (1964), no. 2, 340–364. Google Scholar

• [21]

H. Rosenberg, Minimal surfaces of finite type, Bull. Soc. Math. France 123 (1995), 351–354.

Revised: 2017-01-10

Published Online: 2017-03-16

Published in Print: 2019-08-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1309236

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-52368-P

This material is based upon work of William H. Meeks III for the NSF under Award No. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. The research of Joaquín Pérez was partially supported by MINECO/FEDER grant no. MTM2014-52368-P.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 753, Pages 159–191, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.