Convergence of elastic flows of curves into manifolds

Abstract

For a given p∈[2,+∞), we define the p-elastic energy E of a closed curve γ:S1→M immersed in a complete Riemannian manifold (M,g) as the sum of the length of the curve and the Lp-norm of its curvature (with respect to the length measure). We are interested in the convergence of the (Lp,Lp′)-gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow.We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p=2 in the Euclidean space Rn, in the hyperbolic plane H2, and in the two-dimensional sphere S2. In particular, the result implies that such flow in Rn or H2 remains in a bounded region of the space for any time.