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

Langevin, Rémi; Rosenberg, Harold

Duke Math. J. 57, No. 3, 819-826 (1988).

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

MSC:

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)

Keywords:

minimal surfaces; exterior Plateau problem; catenoid

References:

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