Embedded minimal disks: Proper versus nonproper—global versus local

By T. Colding, W. Minicozzi. 2002

53 citations recorded by the source corpus.

We construct a sequence of (compact) embedded minimal disks in a ball in R 3 with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8b09c9384b69aae1f95e481a31d987c2f4636b34

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In Applied Mathematics

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Citations in the corpus

Minimal submanifolds (2005, unknown): “However, local examples constructed in [24] show the difficulty with such an argument.”
The classical theory of minimal surfaces (2011, background): “In their paper [29], Colding and Minicozzi constructed a sequence of compact, embedded minimal disks Dn ⊂ B(1) with boundary in S(2), that converges to a singular minimal lamination L of the closed ball B(~0, 1) which has an isolated singularity at ~0.”
The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus (2005, methodology): “To summarize, [CM04a] shows that (N-P) can occur for a sequence of disks Σi ⊂ BRi with ∂Σi ⊂ ∂BRi ; however, [CM04e] shows that (N-P) cannot occur for disks if the radii Ri go to infinity.”
Stable constant mean curvature surfaces (2008, background): “However, the Colding-Minicozzi example in [9] (see also Example II in [36]) describes a singular minimal lamination L1 of the open unit ball A = B(1) ⊂ R3 with singular set S1 being the origin {~0}; the related”
The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected (2002, background): “However, we showed in [ CM15 ] that smoothness and properness of the limit can fail in the local case; cf. fig. 12.”