Mean Convex Hulls and Least Area Disks spanning Extreme Curves

Details Mean Convex Hulls and Least Area Disks spanning Extreme Curves Mean Convex Hulls and Least Area Disks spanning Extreme Curves

2004-12-29

arxiv.org/abs/math/0412529v2

Math Z. 252 (2006) 811--824

Mathematics

Differential Geometry

18 pages

Scientific paper

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes spanning that curve. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C^3-smooth curve in the boundary of C^3-smooth mean convex domain in R^3, there exist a unique least area disk spanning the curve.

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