Free boundary minimal surfaces: a nonlocal approach
Free boundary minimal surfaces: a nonlocal approach
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自由边界最小曲面：一种非局部方法
DOI：
10.2422/2036-2145.201801_008
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
F. Lio
中科院分区：
文献类型：
--
作者：
Alessandro Pigati;F. Lio
给定一个C^k-光滑闭嵌入流形$\mathcal N\子集{\mathbb R}^m$，$k\ge 2$，和一个紧连通光滑黎曼曲面$（S，g）$，$\partial S\neq\emptyset$，我们考虑H^{1/2}（\partial S，\mathcal N）$中的$\frac 1/2 $-调和映射$u\。这些映射是非局部能量\开始{方程}E（f;g）的临界点：|\nabla\widetilde u\big| ^2\，d\text {vol}_g，\end{equation}其中$\widetilde u$是$u$在$S$中的调和延拓。我们将能量表示为$\partial S$的每个边界分量处的$\frac 12$-能量之和（适当地用圆圈$\mathcalS^1 $标识），加上在$H^s（\mathcalS^1）$拓扑中连续的二次项，对于任何$s\in\mathbb R$。我们证明了$\frac 12$-调和映射的$C^{k-1，\delta}$正则性。我们还建立了自由边界极小曲面与E$关于（f，g）$的变分的临界点之间的联系.
Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big|\nabla\widetilde u\big|^2\,d\text{vol}_g,\end{equation} where $\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$-energies at each boundary component of $\partial S$ (suitably identified with the circle $\mathcal S^1$), plus a quadratic term which is continuous in the $H^s(\mathcal S^1)$ topology, for any $s\in\mathbb R$. We show the $C^{k-1,\delta}$ regularity of $\frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichm\"uller space of $S$.