## 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:

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