Global analysis of the generalised Helfrich flow of closed curves immersed in Rn

In this paper we consider the evolution of regular closed elastic curves γ immersed in Rn. Equipping the ambient Euclidean space with a vector field : Rn → Rn and a function f : Rn → R, we assume the energy of γ is smallest when the curvature κ of γ is parallel to c0 = ( ◦ γ)+(f ◦ γ)τ, where τ is the unit vector field spanning the tangent bundle of γ. This leads us to consider a generalisation of the Helfrich functional Hc0 λ , defined as the sum of the integral of |κ − c0| 2 and λ-weighted length. We primarily consider the case where f : Rn → R is uniformly bounded in C∞(Rn) and : Rn → Rn is an affine transformation. Our first theorem is that the steepest descent L2-gradient flow of Hc0 λ with smooth initial data exists for all time and subconverges to a smooth solution of the Euler-Lagrange equation for a limiting functional Hc∞ λ . We additionally perform some asymptotic analysis. In the broad class of gradient flows for which we obtain global existence and subconvergence, there exist many examples for which full convergence of the flow does not hold. This may manifest in its simplest form as solutions translating or spiralling off to infinity. We prove that if either and f are constant, the derivative of is invertible and non-vanishing, or (f, γ0) satisfy a 'properness' condition, then one obtains full convergence of the flow and uniqueness of the limit. This last result strengthens a well-known theorem of Kuwert, Sch¨atzle and Dziuk on the elastic flow of closed curves in Rn where f is constant and vanishes.