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

2. Now• at t = 1/2 it achieves the maximum α(k, 1/2) = arctan(1/(2k)), andalso• |∂ θ C| = 1/ sin(α).|π N ∂ t C| 2 |∂ θ C| = |∂ t C| 2 |∂ θ C| sin 2 (α) ∼ sin(α)so if k is large, then the angle α is small and then E ⊥ (C) < ε ∼ 1/k.3. We now smooth C just a bit to obtain Ĉ, so that E⊥(Ĉ) < 2ε.4. We then apply Prop. 11.3 to Ĉ to finally obtain ˜C, that moves only in thenormal direction, soas we wanted to show.E ⊥ (Ĉ) = E ⊥( ˜C) = E( ˜C) < 2ε11.4 Existence of critical geodesics for H jIn contrast, it is possible to show that a Sobolev-type metric does admit criticalgeodesics.Theorem 11.6 (4.3 in Michor and Mumford [36]) Consider the Sobolevtype metric∫〈h, k〉 G n c= 〈h, k〉 + 〈Ds n h, Ds n k〉 dscwith n ≥ 1. Let k ≥ 2n + 1; suppose c, h are in the (usual and parametric) H kSobolev space (that is, c, h admit k-th (weak)derivative, that is square integrable).Then it is possible to shoot the geodesic, starting from c and in direction h, forshort time.In §4.8 in [36] it is suggested that the theorem may similarly hold for Sobolevmetrics with length-dependent scale factors.11.5 Parameterization invarianceLet M be the manifold of (freely)immersed curves. Let 〈h 1 , h 2 〉 c be a Riemannianmetric, for h 1 , h 2 ∈ T c M; let ‖h‖ c = √ 〈h, h〉 c be its associated norm. LetE(γ) := ∫ 10 ‖ ˙γ(v)‖2 γ(v)dv be the action.Let C : [0, 1] × S 1 → S 1 be a smooth homotopy. We recall the definition thatwe saw in Section 4.6, and add a second stronger version.Definition 11.7 A Riemannian metric is• curve-wise parameterization invariant when the metric does not dependon the parameterization of the curve, that is ‖˜h‖˜c = ‖h‖ c when˜c(t) = c(ϕ(t)) and ˜h(t) = h(ϕ(t));92