A foliation of the ball by complete holomorphic discs
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Mathematische Zeitschrift
A foliation of the ball by complete holomorphic discs Antonio Alarcón1 · Franc Forstneriˇc2,3 Received: 24 May 2019 / Accepted: 28 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We show that the open unit ball Bn of Cn (n > 1) admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs. Keywords Riemann surface · Holomorphic disc · Foliation · Complete Riemannian manifold Mathematics Subject Classification 32B15 · 32H02 · 32M17 · 53C12
1 Introduction An open connected submanifold M of a Euclidean space is said to be complete if every divergent path in M has infinite Euclidean length; equivalently, if the restriction of the Euclidean metric ds 2 to M is a complete Riemannian metric on M. Recall that a path γ : [0, 1) → M is called divergent if γ (t) leaves any compact subset of M as t → 1. For n > 1, we denote by Bn the open unit ball of Cn . In this paper, we prove the following result. Theorem 1 For any integer n > 1 there exists a nonsingular holomorphic foliation F of Bn all of whose leaves are complete properly embedded holomorphic discs in Bn . Theorem 1 seems to be the first result which provides control of the topology of all leaves in a nonsingular holomorphic foliation of the ball by complete leaves; in our examples, all leaves are the simplest possible ones, namely, discs. Our proof easily adapts to show that we can ensure completeness of leaves in any given Riemannian metric on the ball, and not only
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Antonio Alarcón [email protected] Franc Forstneriˇc [email protected]
1
Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain
2
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
3
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
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A. Alarcón, F. Forstneriˇc
the Euclidean metric. We do not know whether a comparable result holds for leaves with prescribed but nontrivial topology. See also Remark 1 for a generalization of Theorem 1 to bounded pseudoconvex Runge domains in Cn for n > 1. Before proceeding, we place our result in the context of what is known. The question whether there exist bounded (relatively compact) complete complex submanifolds of Cn for n > 1 was raised by Yang [19] in 1977. The first examples were found in 1979 by Jones [17] who showed that the disc D = {z ∈ C : |z| < 1} admits a complete bounded holomorphic immersion into C2 , embedding into C3 , and proper embedding into the ball of C4 . Interest in this subject has recently been revived due to new construction methods. It was shown that there are complete properly immersed holomorphic curves in B2 , and embedded ones in B3 , with any given topology [8], and also those with the complex structure of any given bordered Riemann surface [2,3]. A related result in higher dimension was obtained by Drinovec Drnovšek [11]. Parallel development
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