961 research outputs found
The Second Hull of a Knotted Curve
The convex hull of a set K in space consists of points which are, in a
certain sense, "surrounded" by K. When K is a closed curve, we define its
higher hulls, consisting of points which are "multiply surrounded" by the
curve. Our main theorem shows that if a curve is knotted then it has a nonempty
second hull. This provides a new proof of the Fary/Milnor theorem that every
knotted curve has total curvature at least 4pi.Comment: 7 pages, 6 figures; final version (only minor changes) to appear in
Amer.J.Mat
Bending laminations on convex hulls of anti-de Sitter quasicircles
Let and be two bounded measured laminations on the
hyperbolic disk , which "strongly fill" (definition below).
We consider the left earthquakes along and ,
considered as maps from the universal Teichm\"uller space to
itself, and we prove that the composition of those left earthquakes has a fixed
point.
The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism
, the boundary of the convex hull in
of its graph in is the disjoint union of two embedded copies of the hyperbolic plane,
pleated along measured geodesic laminations. Our main result is that any pair
of bounded measured laminations that "strongly fill" can be obtained in this
manner.Comment: 16 page
The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometry
Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature K 2 Ć 1; 0/ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti-de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti-de Sitter spacetimes
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