Annales Academić Scientiarum Fennicć
Mathematica
Volumen 34, 2009, 195-214

A GEOMETRIC CURVATURE DOUBLE INTEGRAL OF MENGER TYPE FOR SPACE CURVES

Pawel Strzelecki, Marta Szumanska and Heiko von der Mosel

Uniwersytet Warszawski, Instytut Matematyki
ul. Banacha 2, PL-02-097 Warsaw, Poland; pawelst 'at' mimuw.edu.pl

Politechnika Warszawska, Wydzial Matematyki i Nauk Informacyjnych
Plac Politechniki 1, PL-00-661 Warsaw, Poland; M.Szumanska 'at' mini.pw.edu.pl

RWTH Aachen, Institut für Mathematik
Templergraben 55, D-52062 Aachen, Germany; heiko 'at' instmath.rwth-aachen.de

Abstract. We consider rectifiable closed space curves for which the energy

I_p(\gamma) := \int_\gamma \int_\gamma 1 / inf_z R^p(x,y,z) dH¹(x) dH¹(y), p\ge 2,

is finite. Here, R(x,y,z) denotes the radius of the smallest circle passing through x, y, and z. It turns out that I_p is a self-avoidance energy (curves of finite energy have no self-intersections). For p > 2, we study regularizing effects of I_p: we prove that the arclength parametrization \Gamma of a curve \gamma with I_p(\gamma) < \infty is everywhere differentiable, and its derivative, \Gamma', is Hölder continuous with exponent 1 - 2/p. Moreover, we obtain compactness results for classes of curves with uniformly bounded I_p energy, and briefly discuss their variational applications.

2000 Mathematics Subject Classification: Primary 28A75, 49J45, 49Q10, 53A04, 57M25.

Key words: Menger curvature, global curvature, self-avoidance energies, minimizing knots, imbedding theorems.

Reference to this article: P. Strzelecki, M. Szumanska and H. von der Mosel: A geometric curvature double integral of Menger type for space curves. Ann. Acad. Sci. Fenn. Math. 34 (2009), 195-214.

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