Density theorems for complete minimal surfaces in R^3

Abstract

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C^k convergence on compact sets, for any k. As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in R^3 with uncountably many ends.

32 pages, 8 figures. To appear in G.A.F.A

References in corpus (1)

• On the existence of a proper minimal surface in $R^3$ with the conformal type of a disk

Cited by in corpus (10)

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• Complete minimal surfaces in R3 with a prescribed coordinate function
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• Compact complete minimal immersions in R^3
• Properly embedded minimal planar domains with infinite topology are Riemann minimal examples
• Limit sets for complete minimal immersions
• On the existence of a proper conformal maximal disk in L^3
• Compact complete proper minimal immersions in strictly convex bounded regular domains of R^3
• Non-proper helicoid-like limits of closed minimal surfaces in 3-manifolds
• Complete minimal surfaces and harmonic functions