论文标题
曲线弹性流到歧管
Convergence of elastic flows of curves into manifolds
论文作者
论文摘要
对于[2，+\ infty）$中给定的$ p \，我们定义了$ p $弹性$ \ mathscr {e} $的封闭曲线$γ：\ mathbb {s}^1 \ to m $ curove us cureve and cureve and curve use cureve and curve us的总和， 措施）。我们对$（l^p，l^{p'}）$的收敛感兴趣 - 这些能量到关键点的梯度流。通过抛物线估计，通常可以证明流动的子连接，即收敛到临界点，直至重新分析，更重要的是，直到环境的等距为等级。假设流量亚汇率，我们有兴趣证明流动的平滑收敛性，即不断发展的流的全部限制。我们首先概述了可以申请证明这种声明的一般策略。至关重要的步骤是应用Lojasiewicz-Simon梯度不等式的应用，我们提出了一种多功能版本。然后，我们将这种策略应用于曲线的$ \ mathscr {e} $的流动，以证明将子连接到完全平滑的流量收敛到关键点的完全平滑收敛。作为推论，我们在欧几里得空间中获得$ p = 2 $的流平滑收敛，在双曲机平面$ \ mathbb {h}^2 $中，以及在二维球体$ \ mather $ \ mathbb {s}^2 $中。特别是，结果意味着$ \ mathbb {r}^n $或$ \ mathbb {h}^2 $中的这种流程在任何时间都保留在该空间的有限区域中。
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.
