Dynamics of particles on a curve with pairwise hyper-singular repulsion

Figure 1.  The number $ r_0 $ in Lemma 4.1 is the range for which $ {{\bf x}}(z) $ can be approximated by a local Taylor expansion near $ {{\bf x}}(y) $ for any fixed $ y $

Figure 2.  Lemmas 5.1 and 5.2. Left: the summand in the last term of (5.2). The two terms representing the forces from $ z_j $ acting on $ z_{i_M} $ (red) and $ z_{i_M+1} $ (blue), which decreases/increases $ \delta $ respectively. Right: a local uniform distribution like $ \{\tilde{z}_j\} $ makes $ \frac{ \,\mathrm{d}}{ \,\mathrm{d}{t}}\delta \approx 0 $ up to errors from curvature. A possible defect will release the total pushing force on $ \delta $, make $ \frac{ \,\mathrm{d}}{ \,\mathrm{d}{t}}\delta $ positive, and thus violate (5.11)

Figure 3.  Proof of Theorem 2.1. Left: when (5.11) does not hold, $ \delta $ is increasing very fast (i.e., $ \rho_M $ is decreasing very fast). Right: when (5.11) holds, there is almost uniform distribution near $ z_{i_M} $ (red parts) with average density near $ \rho_M $, and the total repulsion at $ z_{i_M} $ is strong (see (5.12)). The rest part has average density at most $ 1+\epsilon $, and Lemma 3.1 applies to give a weak total repulsion cut. The strong/weak total repulsion ((1)-(2) good contribution, $ I_1 $, and (3)-(4) bad contribution, $ I_2 $, see (6.9)) forces the green part to rotate. The parameter $ r_1 $ is to guarantee that (3) or (4) cannot be too short, so that the possible bad contribution from (1)-(4) or (2)-(3) (the term $ I_3 $) can be neglected