×

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\((P_n)\), \(n\in\mathbb N\).

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32F99 Geometric convexity in several complex variables
53C40 Global submanifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

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 · doi:10.1007/s00208-012-0790-4
[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 · doi:10.1007/s00208-013-0931-4
[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 · doi:10.1007/978-1-4757-2016-7
[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 · doi:10.1007/BF02803401
[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 · doi:10.2307/2043009
[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 · doi:10.1090/S0002-9939-09-09953-5
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.