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**A complete complex hypersurface in the ball of \(\mathbb{C}^N\).**
*(English)*
Zbl 1333.32018

The article presents a comprehensive answer to the question whether the open unit ball \(\mathbb B_N\) in \(\mathbb C^N\) admits connected \(k\)-dimensional complete closed complex submanifolds \(M_k\) for \(1\leq k<N\). Completeness of \(M_k\) means that \(\sup\{|p(t)|\,\,0\leq t<1\}<1\) for every path \(p:[0,1)\rightarrow M_k\) with finite length. The case \(N=2\) was solved by A. Alarcón and F. J. López [“Complete bounded embedded complex curves in \(C^2\)”, Preprint, arXiv:1305.2118] who proved that any convex domain of \(\mathbb C^2\) carries properly embedded complete complex curves. The affirmative answer to the above question results from the following existence theorem for a specific class of holomorphic functions on \(\mathbb B_N\): For every \(N\geq 2\) there exists a holomorphic function \(f\) on \(\mathbb B_N\) with the property that \(\sup\{|\text{ Re} f(q(t))|\,\,t\in [0,1)\}=\infty\) for every path \(q:[0,1)\rightarrow\mathbb B_N\) of finite length with \(\lim_{t\rightarrow 1}|q(t)|=1\). Therefore any connected component of a fiber \(f^{-1}(c)\), for \(c\) a regular value of \(f\), is an example for \(M_{N-1}\), and the existence of \(M_k\), \(1\leq k<N-1\), follows in the same way via the natural embedding \(\mathbb B_{k+1}\subset\mathbb B_N\). Any regular fiber of the real pluriharmonic function Re\((f)\) yields a complete closed real hypersurface of \(\mathbb B_N\). The existence theorem is deduced from a theorem in convex geometry whose proof is the heart of the paper. It states that there is an exhaustion of the open unit ball \(\mathbb B\) in \(\mathbb R^M\), \(M\geq 2\), by a sequence \((P_n)_{n\in\mathbb N}\) of convex polytopes, \(P_n\subset\) Int\( (P_{n+1}) \), \(n\in \mathbb N\), with the property that \(\sum_{n=1}^\infty |w_{n+1}-w_n|=\infty\) for every choice of \(w_n\in\) skel\((P_n)\), \(n\in\mathbb N\).

Reviewer: Eberhard Oeljeklaus (Bremen)

### MSC:

32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |

32F99 | Geometric convexity in several complex variables |

53C40 | Global submanifolds |

### Keywords:

bounded complete holomorphic embeddings; complete complex manifolds; complete holomorphic immersions
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\textit{J. Globevnik}, Ann. Math. (2) 182, No. 3, 1067--1091 (2015; Zbl 1333.32018)

### References:

[1] | A. Alarcón and F. J. López, ”Null curves in \(\mathbbC^3\) and Calabi-Yau conjectures,” Math. Ann., vol. 355, iss. 2, pp. 429-455, 2013. · Zbl 1269.53061 |

[2] | A. Alarcón and F. J. López, Complete bounded complex curves in \(\mathbbC^2\). |

[3] | 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. · Zbl 1288.32014 |

[4] | A. Brøndsted, An Introduction to Convex Polytopes, New York: Springer-Verlag, 1983, vol. 90. · Zbl 0509.52001 |

[5] | J. H. Conway and N. A. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Verlag, 1988, vol. 290. · Zbl 0634.52002 |

[6] | J. Globevnik and E. L. Stout, ”Holomorphic functions with highly noncontinuable boundary behavior,” J. Analyse Math., vol. 41, pp. 211-216, 1982. · Zbl 0564.32009 |

[7] | P. W. Jones, ”A complete bounded complex submanifold of \({\mathbf C}^3\),” Proc. Amer. Math. Soc., vol. 76, iss. 2, pp. 305-306, 1979. · Zbl 0418.32006 |

[8] | F. Martin, M. Umehara, and K. Yamada, ”Complete bounded holomorphic curves immersed in \(\mathbb C^2\) with arbitrary genus,” Proc. Amer. Math. Soc., vol. 137, iss. 10, pp. 3437-3450, 2009. · Zbl 1177.53056 |

[9] | P. Yang, ”Curvatures of complex submanifolds of \({\mathbf C}^n\),” J. Differential Geom., vol. 12, iss. 4, pp. 499-511 (1978), 1977. · Zbl 0355.53035 |

[10] | P. Yang, ”Curvature of complex submanifolds of \(C^n\),” in Several Complex Variables, Providence, R.I.: Amer. Math. Soc., 1977, pp. 135-137. · Zbl 0409.53043 |

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