## 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
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### 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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