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The strong halfspace theorem for minimal surfaces. (English) Zbl 0722.53054

This paper contains a proof of the following form of the “maximum principle at infinity” for minimal surfaces: A connected, proper, possibly branched, nonplanar minimal surface M in \({\mathbb{R}}^ 3\) is not contained in a halfspace. The natural generalization of the above result for minimal hypersurfaces \(M^ n\subset {\mathbb{R}}^{n+1},\) \(n\geq 3\), fails as shown by the example of the n-catenoid. The authors show that, nevertheless, the convex hull of such a hypersurface is very special. The paper also contains a discussion of other maximum principles at infinity that appeared recently in the literature.

MSC:

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

[1] Callahan, M., Hoffman, D., Meeks, W.H., III.: Embedded minimal surfaces with an infinite number of ends. Invent. Math.96, 459-505 (1989) · Zbl 0676.53004 · doi:10.1007/BF01393694
[2] Frohman, C., Meeks, W.H., III.: The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in ?3. Preprint · Zbl 0886.57015
[3] Hildebrandt, S.: Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math. Z.128, 253-269 (1972) · Zbl 0253.53005 · doi:10.1007/BF01111709
[4] Hoffman, D., Meeks, W.H., III.: Embedded minimal surfaces of finite topology. Ann. Math.131, 1-34 (1990) · Zbl 0695.53004 · doi:10.2307/1971506
[5] Hoffman, D., Meeks, W.H., III.: Properties of properly embedded minimal surfaces of finite total curvature. Bull. Am. Math. Soc.17, 296-300 (1987) · Zbl 0634.53003 · doi:10.1090/S0273-0979-1987-15566-2
[6] Jorge, L., Xavier, F.: A complete minimal surface in ?3 between two parallel planes. Ann. Math.112, 203-206 (1980) · Zbl 0455.53004 · doi:10.2307/1971325
[7] Langevin, R., Rosenberg, H.: A maximum principle at infinity for minimal surfaces and applications. Duke Math. J.57, 819-828 (1988) · Zbl 0667.49024 · doi:10.1215/S0012-7094-88-05736-5
[8] Meeks, W.H., III., Rosenberg, H.: The geometry of periodic minimal surfaces. Preprint · Zbl 0807.53049
[9] Meeks, W.H., III., Rosenberg, H.: The maximum principle at infinity for minimal surfaces in flat three-manifolds. (to apear) Comment. Math. Helv. · Zbl 0713.53008
[10] Meeks, W.H., III., Rosenberg, H.: The global theory of doubly periodic minimal surfaces. Invent. Math.97, 351-379 (1989) · Zbl 0676.53068 · doi:10.1007/BF01389046
[11] Meeks, W.H., III., Simon, L., Yau, S.T.: The existence of embedded minimal surfaces, exotic spheres and positive Ricci curvature. Ann. Math.116, 221-259 (1982) · Zbl 0521.53007 · doi:10.2307/2007026
[12] Rosenberg, H., Toubiana, E.: A cylindrical type complete minimal surface in a slab of ?3. Bull. Sci. Math. III, 241-245 (1987) · Zbl 0631.53012
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