We investigate knot-theoretic properties of geometrically defined curvature
energies such as integral Menger curvature. Elementary radii-functions, such as
the circumradius of three points, generate a family of knot energies
guaranteeing self-avoidance and a varying degree of higher regularity of finite
energy curves. All of these energies turn out to be charge, minimizable in
given isotopy classes, tight and strong. Almost all distinguish between knots
and unknots, and some of them can be shown to be uniquely minimized by round
circles. Bounds on the stick number and the average crossing number, some
non-trivial global lower bounds, and unique minimization by circles upon
compaction complete the picture.Comment: 31 pages, 4 figures; version 2 with minor changes and modification
We find conditions under which measures belong to H −1 (R 2 ). Next we show that measures generated by Prandtl, Kaden as well as Pullin spirals, objects considered by physicists as incompressible flows generating vorticity, satisfy assumptions of our theorem, thus they are (locally) elements of H −1 (R 2 ). Moreover, as a by-product, we prove an embedding of the space of Morrey type measures in H −1 .
Abstract. We study two kinds of integral Menger-type curvatures. We find a threshold value of α 0 , a Hölder exponent, such that for all α > α 0 embedded C 1,α manifolds have finite curvature. We also give an example of a C 1,α 0 injective curve and higher dimensional embedded manifolds with unbounded curvature.
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