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The maximum principle at infinity for minimal surfaces in flat three manifolds. (English) Zbl 0713.53008

Maximum principles have been extensively used in the study of minimal surfaces and of surfaces of constant mean curvature. It it often used for interior points or points on the boundary of the surfaces in consideration. Recently in the literature various forms have appeared of what could be called a maximum principle at infinity.
In this paper, the authors prove a strong maximum principle at infinity for immersed minimal surfaces, which reads as follows: Let N be a complete, flat, 3-dimensional manifold, and let \(M_ 1\) and \(M_ 2\) be disjoint, connected, properly immersed minimal surfaces in N with compact boundaries \(\partial M_ 1\), \(\partial M_ 2\) (possibly empty). Then: 1) If both \(\partial M_ 1\) and \(\partial M_ 2\) are empty, \(M_ 1\) and \(M_ 2\) are flat. 2) If \(\partial M_ 1\) or \(\partial M_ 2\) is nonempty, there exist a point \(x\in \partial M_ 1\) and a point \(y\in M_ 2\) (after reindexing, if necessary) such that \(dist(x,y)=dist(M_ 1,M_ 2)\).
Reviewer: M.P.do Carmo

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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