Half-space type theorems in warped product spaces with one-dimensional factor. (English) Zbl 1165.53362
Summary: This work states some half-space type theorems in a warped product space of the form \(I \times_{\rho } M\), where \({I \subseteq {\mathbb R}}\) is an open interval and \(M\) is either a compact \(n\)-manifold, or a complete simply connected surface with constant curvature \(c \leq 0\). Such theorems generalize the classical half-space theorem for minimal surfaces in \({\mathbb R}^{3}\), obtained by D. Hoffman and W. H. Meeks III [Invent. Math. 101, No. 2, 373–377 (1990; Zbl 0722.53054)], and recent results for surfaces contained in a slab of \({\mathbb R}\times _{\rho } M\), obtained by M. Dajczer and L. J. Alías [Comment. Math. Helv. 81, No. 3, 653–663 (2006; Zbl 1110.53039)].
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
References:
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