Metrics of curves in shape optimization and analysis - Andrea Carlo ...

11.3 The H 0 distance is degenerateIn §C in [67], or §3.10 in [37], the following theorem is proved.Theorem 11.5 The H 0 -induced distance is degenerate: the distance betweenany two curves is 0.This results is generalized in [38] to L 2 -type metrics of submanifolds of anycodimension.Here we will sketch the main idea of the proof for a very simple case: it ispossible to connect two segmentsc 0 (u) = (u, 0) , c 1 (u) = (u, 1)with a family of “zigzag” homotopies C k (t, θ) so that the H 0 action is infinitesimalwhen k → ∞. A snapshot of the curves along the homotopy, for t = 0, 1/8 . . . 1and k = 5, is in Fig. 18.t = 1/2t = 1/4t = 1/8t = 0t = 1t = 7/8t = 3/4t = 1/2case t ∈ [0, 1/2] 1/k∂C∂θα∂C∂vcase t ∈ [1/2, 1]Figure 18: Zig-zag homotopyIn the following, let C = C k for k large. We recall that the H 0 action isE(C) =and we also defineE ⊥ (C) =∫ 10∫∫ 10C∫C|∂ t C| 2 ds dt =|π N ∂ t C| 2 ds dt =∫ 1 ∫ 100∫ 1 ∫ 100|∂ t C| 2 |∂ θ C| dθ dt ;|π N ∂ t C| 2 |∂ θ C| dθ dt (11.1)(this “action” will be explained in Sec. 11.10). The argument goes as follows.1. The angle α is (the absolute value of) the angle between ∂ θ C and thevertical direction (that is also the direction of ∂ t C). Note that• α = α(k, t), and91