A maximum principle at infinity for minimal surfaces and applications. (English) Zbl 0667.49024

The main results:
(1) Let \(f_ 1\), \(f_ 2\) be solution of the exterior Plateau problem on \(\Omega =\{X=(x_ 1,x_ 2)|\) \(x^ 2_ 1+x^ 2_ 2\geq 1\}\). Suppose \(f_ 1(X)\geq f_ 2(X)\) for \(X\in \Omega\) and there exists a sequence \(X_ i\in \Omega\) such that \(| X_ i| \to \infty\) and \(| f_ 1(X_ i)-f_ 2(X_ i)| \to 0\). Then \(f_ 1=f_ 2.\)
(2) Minimal herissons with catenoid type ends are rigid.
(3) Let f solve the exterior Plateau problem on \(\Omega\). Then \(0\leq a(f)\leq 1\), where a(f) is the coefficient of log R in the asymptotic expansion of f near infinity. If \(a(f)=1\), then \(f(R)=\cosh^{-1}(R)\), i.e., f defines a catenoid.
Reviewer: C.Udrişte


49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
Full Text: DOI


[1] A. D. Alexandrov, Sur la théorie des volumes mixtes des corps convexes, I, II, III, IV , Math. Sbornik. 4-45 (1937-1938).
[2] R. Langevin, G. Levitt, and H. Rosenberg, Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss) , Singularities (Warsaw, 1985), Banach Center Publication, vol. 20, PWN, Warsaw, 1988, pp. 245-253. · Zbl 0658.53004
[3] W. Meeks, A survey of the geometric results in the classical theory of minimal surfaces , Bol. Soc. Brasil. Mat. 12 (1981), no. 1, 29-86. · Zbl 0577.53007 · doi:10.1007/BF02588319
[4] R. Osserman, A Survey of Minimal Surfaces , Van Nostrand Reinhold, New York, 1969. · Zbl 0209.52901
[5] H. Rosenberg and E. Toubiana, Complete minimal surfaces and minimal herissons , to appear in J. Differential Geom. · Zbl 0657.53004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.