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

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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