A GEOMETRIC CURVATURE DOUBLE INTEGRAL OF MENGER TYPE FOR SPACE CURVES

作者：Strzelecki Pawel^*; Szumanska Marta; Von der Mosel Heiko

来源：Annales Academiae Scientiarum Fennicae-Mathematica, 2009, 34(1): 195-214.

• Menger curvature
• global curvature
• self-avoidance energies
• minimizing knots
• imbedding theorems

We consider rectifiable closed space curves for which the energy
I(p)(gamma) (sic) integral(gamma)integral(gamma)1/inf(z)R(p)(x,y,z)dH(1)(x)dH(1)(y), p >= 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) < infinity is everywhere differentiable, and its derivative, Gamma', is Holder 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.

• 出版日期2009