Convergence of elastic flows of curves into manifolds (Q2238800)

In this paper the author defines the \(p\)-elastic energy \({\mathcal E}\) of a closed curve \(\gamma\) immersed in a complete Riemannian manifold \((M,g)\) as the sum of the length of the curve and the \(L^p\) norm of its curvature with respect to the length measure. He studies the convergence of the \((L^p,L^{p'})\)-gradient flow of these energies to critical points. In particular the study of the smooth convergence of the flow, that is the existence of the full limit of the evolving flow, is developed assuming the sub-convergence of the flow, which is usually proved by means of parabolic estimates. The author first presents an overview of the general strategy to study the convergence of the flow. The crucial step is the application of a Lojasiewicz-Simon gradient inequality. Then the general strategy is applied to the flow of \({\mathcal E}\) of curves into manifolds, proving the desired sub-convergence to full smooth convergence of the flow. As particular cases the smooth convergence of the flow is proved for \(p=2\) in the Euclidean space \(\mathbb{R}^n\), in the hyperbolic plane \(\mathbb{H}^2\) and in the two-dimensional sphere \(\mathbb{S}^2\). These results imply that the flow remains in a bounded region of \(\mathbb{R}^n\) and \(\mathbb{H}^2\) for any time.