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Minimal surfaces in \(\mathbb{R}^3\) properly projecting into \(\mathbb{R}^2\). (English) Zbl 1252.53005

The authors give a construction, applicable to any open Riemann surface \(\mathcal N\) and any real number \(\theta\) with \(0< \theta < \pi/2\), of a minimal immersion \[ X=(X_1,X_2,X_3):{\mathcal N}\to{\mathbb R}^3 \] for which the real-valued function \(X_3+ \tan(\theta)|X_1|\) is positive and proper. Additionally, the flux map of \(X\) may be arbitrarily prescribed. A corollary of this result is that any open Riemann surface \(\mathcal N\) admits a conformal minimal immersion \(X\) into \({\mathbb R}^3\) such that \((X_1,X_3)\) is a proper harmonic map. The main tools used by the authors come from the theory of approximation by meromorphic functions H. L. Royden, [J. Anal. Math. 18, 295–327 (1967; Zbl 0153.39801)], [S. Scheinberg, Ann. Math. (2) 108, 257–298 (1978; Zbl 0423.30035), Math. Ann. 243, 83–93 (1979; Zbl 0393.30031)].

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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