Gradients, Preferred Metrics and Asymmetries

Assume the curves xτ flow according to ( ) / x x τ τ τ ∂ ∂ = Φ . Evolutions that only use local data, such as curve shortening, yield partial differential equations. In the present paper global information is permitted and Φ involves integrals of xτ . Specifically, F Φ = −∇ where : F Ω → R is given by ( ) ( ) , , F x f t x x dt = ∫ . The elements of Ω correspond to curves subject to point-wise and isoperimetric constraints. Ideally Ω is an infinite-dimensional smooth ‘surface-like’ submanifold of some suitable Hilbert space X . The choice of inner product in X profoundly affects the formulas for the gradient and also properties of xτ such as smoothness and symmetry. The gradient formulas are derived without the help of any Hilbert basis in three different inner products. The equation ( ) 0 F x ∇ = is shown to be more discriminating than the Euler-Lagrange equation. The main contribution is the identification of two fundamental quantities that radically simplify and streamline all the gradient expressions. One of the three metrics retains both smoothness and symmetry. Interestingly, the gradient in this metric fails to commute with the derivative operator. There is essentially only one type of metric whose gradient commutes with the derivative, but its flow can destroy symmetry and ruin smoothness. There is a non-standard metric of this type whose flow does preserve symmetry. The symbolic form of the gradient is not symmetric with respect to external symmetries and it is possible to exploit this ‘asymmetry’ and choose the metric so that the form of the gradients is greatly simplified.