## 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

### 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
Full Text:

### References:

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