Convergence of elastic flows of curves into manifolds

Abstract

For a given $p\in [2,+\infty )$, we define the $p$-elastic energy $\mathcal{E}$ 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 $(L^p,L^{p^{\prime }})$--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 $\mathcal{E}$ 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.

References in corpus (3)

• A survey of the elastic flow of curves and networks
• The Lojasiewicz-Simon inequality for the elastic flow
• The elastic flow of curves in the hyperbolic plane

Cited by in corpus (12)

• A survey of the elastic flow of curves and networks
• The Lojasiewicz-Simon inequality for the elastic flow
• Existence and convergence of the length-preserving elastic flow of clamped curves
• General rigidity principles for stable and minimal elastic curves
• The free elastic flow for closed planar curves
• On the convergence of the Willmore flow with Dirichlet boundary conditions
• Asymptotic convergence of evolving hypersurfaces
• Complete classification of planar p-elasticae
• Singularities of the hyperbolic elastic flow: Convergence, quantization and blow-ups
• On p-biharmonic curves
• Variational stabilization of degenerate p-elasticae
• Łojasiewicz-Simon inequalities for minimal networks: stability and convergence