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arXiv 2007.00582math.APmath.DGmath.FA
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曲线到流形的弹性流的收敛性

Convergence of elastic flows of curves into manifolds

Marco Pozzetta

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摘要：

对于一个给定的$p\in[2,+\infty)$，我们定义的$p$ -弹性能量$\mathscr{E}$的封闭曲线$γ:\mathbb{S}^1\to M$沉浸在一个完整的黎曼流形$(M,g)$作为曲线的长度和$L^p$ -范数的曲率（相对于长度测量）。我们感兴趣的是$(L^p,L^{p'})$的收敛--这些能量向临界点的梯度流。通过抛物线估计，通常可以证明流的次收敛，即收敛到重新参数化的临界点，更重要的是，收敛到环境的等距。假设流是次收敛的，我们感兴趣的是证明流的光滑收敛性，即演化流的全极限的存在性。我们首先给出一个概述的一般策略，可以申请证明这样一个声明。关键的一步是Lojasiewicz-Simon梯度不等式的应用，我们提出了一个通用的版本。然后，我们将这种策略应用到流的$\mathscr{E}$的曲线到流形，证明所需的改善的次收敛到全光滑收敛的流的临界点。作为推论，我们得到了$p=2$流在欧氏空间$\mathbb{R}^n$、双曲平面$\mathbb{H}^2$和二维球面$\mathbb{S}^2$中的光滑收敛性.特别地，该结果意味着在任何时间，这种流在$\mathbb{R}^n$或$\mathbb{H}^2$保持在空间的有界区域中。

英文摘要：

For a given $p\in[2,+\infty)$, we define the $p$-elastic energy $\mathscr{E}$ of a closed curve $γ:\mathbb{S}^1\to M$ 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.

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