Tangent-point self-avoidance energies for curves

Details Tangent-point self-avoidance energies for curves Tangent-point self-avoidance energies for curves

2010-06-23

arxiv.org/abs/1006.4566v2

Mathematics

Classical Analysis and ODEs

23 pages, 1 figure

Scientific paper

We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle or to a closed interval. For $q>2$ the energy $E_q$ evaluated on curves in $R^3$ turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in $R^3$ with finite $E_q$-energy that guarantees that these curves are ambient isotopic. This bound depends only on $q$ and the energy values of the curves. Moreover, for all $q$ that are larger than the critical exponent $2$, the arclength parametrization of $\gamma$ is of class $C^{1,1-2/q}$, with H\"{o}lder norm of the unit tangent depending only on $q$, the length of $\gamma$, and the local energy. The exponent $1-2/q$ is optimal.

Affiliated with

Also associated with

No associations

Rating

Tangent-point self-avoidance energies for curves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Tangent-point self-avoidance energies for curves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tangent-point self-avoidance energies for curves will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-278183