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

Proof. Fix α 0 ∈ S \ Z. Let T = T α0 S be the tangent at α 0 . T is the vectorspace orthogonal to ∇φ i (α 0 ) for i = 1, 2, 3. Let e i = e i (s) ∈ L 2 ∩ Cc∞ be near∇φ i (α 0 ) in L 2 , so that the map (x, y) : T × lR 3 → L 2(x, y) ↦→ α = α 0 + x +3∑e i y i (8.4)is an isomorphism. Let S ′ be S in these coordinates; by the Implicit FunctionTheorem (5.9 in Lang [30]), there exists an open set U ′ ⊂ T , 0 ∈ U ′ , an openV ′ ⊂ lR 3 , 0 ∈ V ′ , and a smooth function F : U → lR 3 such that the local partS ′ ∩ (U ′ × V ′ ) of the manifold S ′ is the graph of y = F (x).We immediately define a smooth projection π : U ′ × V ′ → S ′ by settingπ ′ (x, y) = (x, F (x)); this may be expressed in the original L 2 space; let(x(α), y(α)) be the inverse of (8.4) and U = x −1 (U ′ ); we define the projectionπ : U → S by settingThenπ(α)(s) − α(s) =π(α) = α 0 + x +i=13∑e i F i (x(α))i=13∑e i (s)a i , a i := (F i (x(α)) − y i (α)) ∈ lR (8.5)i=1so if α(s) is smooth, then π(α)(s) is smooth.Let α n be smooth functions such that α n → α in L 2 , then π(α n ) → α 0 ; ifwe choose them to satisfy α n (2π) − α n (0) = 2πh, then, by the formula (8.5),π(α)(2π) − π(α)(0) = 2πh so that π(α n ) ∈ S and it represents a smooth curvewith the assigned rotation index h.8.2 A metric with explicit geodesicsA similar method has been proposed recently in Michor et al. [39], based onan idea originally in [68]. We consider immersed planar curves and again weidentify lR 2 = IC.Proposition 8.9 (Continuous lifting of square root) If ξ : [0, 2π] → IC isan immersed planar curve, then ξ ′ is continuous and ξ ′ ≠ 0, so there exists acontinuous function α :→ IC satisfyingξ ′ (θ) = α(θ) 2 (8.6)and α is uniquely identified up to multiplying by ±1.If the rotation index of ξ is even then α(0) = α(2π), whereas if the rotationindex of ξ is odd then α(0) = −α(2π).56