# A complete complex hypersurface in the ball of $\mathbb{C}^N$

### Abstract

In 1977, P. Yang asked whether there exist complete immersed complex submanifolds $\varphi \colon M^k\rightarrow \mathbb{C}^N$ with bounded image. A positive answer is known for holomorphic curves $(k=1)$ and partial answers are known for the case when $k>1$. The principal result of the present paper is a construction of a holomorphic function on the open unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. A consequence is the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$. This gives a positive answer to Yang’s question in all dimensions $k, N, 1\leq k<N$, by providing properly embedded complete complex manifolds.

• [AL1] A. Alarcón and F. J. López, "Null curves in $\mathbb{C}^3$ and Calabi-Yau conjectures," Math. Ann., vol. 355, iss. 2, pp. 429-455, 2013.
@article{AL1, mrkey = {3010135},
author = {Alarc{ó}n, Antonio and L{ó}pez, Francisco J.},
title = {Null curves in {$\mathbb{C}\sp 3$} and {C}alabi-{Y}au conjectures},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {355},
year = {2013},
number = {2},
pages = {429--455},
issn = {0025-5831},
coden = {MAANA},
mrclass = {53C42 (32H02 53A10)},
mrnumber = {3010135},
mrreviewer = {Fei-Tsen Liang},
doi = {10.1007/s00208-012-0790-4},
zblnumber = {1269.53061},
}
• [AL2] A. Alarcón and F. J. López, Complete bounded complex curves in $\mathbb{C}^2$.
@misc{AL2,
author = {Alarc{ó}n, Antonio and L{ó}pez, Francisco J.},
title = {Complete bounded complex curves in {$\mathbb{C}^2$}},
note = {to appear in {\em J. Europ. Math. Soc.}},
arxiv = {1305.2118v2},
sortyear = {2015},
}
• [AF] A. Alarcón and F. Forstnerivc, "Every bordered Riemann surface is a complete proper curve in a ball," Math. Ann., vol. 357, iss. 3, pp. 1049-1070, 2013.
@article{AF, mrkey = {3118624},
author = {Alarc{ó}n, Antonio and Forstneri{\v{c}},
Franc},
title = {Every bordered {R}iemann surface is a complete proper curve in a ball},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {357},
year = {2013},
number = {3},
pages = {1049--1070},
issn = {0025-5831},
mrclass = {32B15 (32H35 32Q40 53C42)},
mrnumber = {3118624},
mrreviewer = {Veselin T. Videv},
doi = {10.1007/s00208-013-0931-4},
zblnumber = {1288.32014},
}
• [B] A. Brøndsted, An Introduction to Convex Polytopes, New York: Springer-Verlag, 1983, vol. 90.
@book{B, mrkey = {0683612},
author = {Brøndsted, Arne},
title = {An Introduction to Convex Polytopes},
series = {Grad. Texts in Math.},
volume = {90},
publisher = {Springer-Verlag},
year = {1983},
pages = {viii+160},
isbn = {0-387-90722-X},
mrclass = {52A25 (05B30 52-01)},
mrnumber = {0683612},
mrreviewer = {D. Barnette},
zblnumber = {0509.52001},
}
• [CS] J. H. Conway and N. A. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Verlag, 1988, vol. 290.
@book{CS, mrkey = {0920369},
author = {Conway, J. H. and Sloane, NJ A.},
title = {Sphere Packings, Lattices and Groups},
series = {Grundl. Math. Wissen.},
volume = {290},
publisher = {Springer-Verlag},
year = {1988},
pages = {xxviii+663},
isbn = {0-387-96617-X},
mrclass = {11H31 (05B40 11H06 20E32 52A43 52A45 94C30)},
mrnumber = {0920369},
mrreviewer = {J. M. Wills},
doi = {10.1007/978-1-4757-2016-7},
zblnumber = {0634.52002},
}
• [GS] J. Globevnik and E. L. Stout, "Holomorphic functions with highly noncontinuable boundary behavior," J. Analyse Math., vol. 41, pp. 211-216, 1982.
@article{GS, mrkey = {0687952},
author = {Globevnik, Josip and Stout, Edgar Lee},
title = {Holomorphic functions with highly noncontinuable boundary behavior},
journal = {J. Analyse Math.},
fjournal = {Journal d'Analyse Mathématique},
volume = {41},
year = {1982},
pages = {211--216},
issn = {0021-7670},
coden = {JOAMAV},
mrclass = {32D15 (32A40)},
mrnumber = {0687952},
mrreviewer = {Harold P. Boas},
doi = {10.1007/BF02803401},
zblnumber = {0564.32009},
}
• [J] P. W. Jones, "A complete bounded complex submanifold of ${\bf C}^{3}$," Proc. Amer. Math. Soc., vol. 76, iss. 2, pp. 305-306, 1979.
@article{J, mrkey = {0537094},
author = {Jones, Peter W.},
title = {A complete bounded complex submanifold of {${\bf C}\sp{3}$}},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {76},
year = {1979},
number = {2},
pages = {305--306},
issn = {0002-9939},
coden = {PAMYAR},
mrclass = {32A35 (30D55 53B25)},
mrnumber = {0537094},
mrreviewer = {Sun Yung A. Chang},
doi = {10.2307/2043009},
zblnumber = {0418.32006},
}
• [MUY] F. Martin, M. Umehara, and K. Yamada, "Complete bounded holomorphic curves immersed in $\Bbb C^2$ with arbitrary genus," Proc. Amer. Math. Soc., vol. 137, iss. 10, pp. 3437-3450, 2009.
@article{MUY, mrkey = {2515413},
author = {Martin, Francisco and Umehara, Masaaki and Yamada, Kotaro},
title = {Complete bounded holomorphic curves immersed in {$\Bbb C\sp 2$} with arbitrary genus},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {137},
year = {2009},
number = {10},
pages = {3437--3450},
issn = {0002-9939},
coden = {PAMYAR},
mrclass = {53A10 (32H02)},
mrnumber = {2515413},
mrreviewer = {Antonio Alarc{ó}n},
doi = {10.1090/S0002-9939-09-09953-5},
zblnumber = {1177.53056},
}
• [Y1] P. Yang, "Curvatures of complex submanifolds of ${\bf C}^{n}$," J. Differential Geom., vol. 12, iss. 4, pp. 499-511 (1978), 1977.
@article{Y1, mrkey = {0512921},
author = {Yang, Paul},
title = {Curvatures of complex submanifolds of {${\bf C}\sp{n}$}},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {12},
year = {1977},
number = {4},
pages = {499--511 (1978)},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {53C40},
mrnumber = {0512921},
mrreviewer = {Yoshiaki Maeda},
url = {http://projecteuclid.org/euclid.jdg/1214434221},
zblnumber = {0355.53035},
}
• [Y2] P. Yang, "Curvature of complex submanifolds of $C^{n}$," in Several Complex Variables, Providence, R.I.: Amer. Math. Soc., 1977, pp. 135-137.
@incollection{Y2, mrkey = {0450606},
author = {Yang, Paul},
title = {Curvature of complex submanifolds of {$C\sp{n}$}},
booktitle = {Several Complex Variables},
venue = {{P}roc. {S}ympos. {P}ure {M}ath., {V}ol. {XXX},
{P}art 2, {W}illiams {C}oll., {W}illiamstown, {M}ass., 1975},
pages = {135--137},
publisher = {Amer. Math. Soc.},