Convergence of elastic flows of curves into manifolds

TL;DR

This paper proves that the gradient flow of a generalized elastic energy for curves in manifolds converges smoothly to critical points, extending known results to various geometries using a Lojasiewicz-Simon inequality.

Contribution

It introduces a versatile Lojasiewicz-Simon gradient inequality approach to establish full smooth convergence of elastic flows in manifolds, improving upon sub-convergence results.

Findings

Flow converges smoothly to critical points in Euclidean space, hyperbolic plane, and sphere.

Flow remains in a bounded region of the ambient space for all time.

Method applies to a broad class of elastic energies with different p-values.

Abstract

For a given $p\in[2,+\infty)$, we define the $p$-elastic energy $\mathscr{E}$ of a closed curve $\gamma:\mathbb{S}^1\to M$ 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). We are interested in the convergence of the $(L^p,L^{p'})$--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 Lojasiewicz-Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of $\mathscr{E}$ 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 $\mathbb{R}^n$, in the hyperbolic plane $\mathbb{H}^2$, and in the two-dimensional sphere $\mathbb{S}^2$. In particular, the result implies that such flow in $\mathbb{R}^n$ or $\mathbb{H}^2$ remains in a bounded region of the space for any time.