Existence and Classification of $$\pmb {\mathbb {S}}^1$$-Invariant Free Boundary Minimal Annuli and Möbius Bands in $$\pmb {\mathbb {B}}^n$$
Existence and Classification of $$\pmb {\mathbb {S}}^1$$-Invariant Free Boundary Minimal Annuli and Möbius Bands in $$\pmb {\mathbb {B}}^n$$
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$$pmb {mathbb {S}}^1$$-$$pmb {mathbb {B}}^n$$ 中不变自由边界最小环带和莫比乌斯带的存在性和分类
DOI：
10.1007/s12220-020-00371-9
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发表时间：
2020
影响因子：
1.1
通讯作者：
Pam Sargent
中科院分区：
文献类型：
--
作者：
A. Fraser;Pam Sargent
We explicitly classify all $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $$\mathbb {S}^1$$-invariant metrics on the annulus and Mobius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all $$\mathbb {S}^1$$-invariant metrics on the Mobius band for each $$k \ge 1$$, and show that it is achieved by the induced metric from a free boundary minimal embedding of the Mobius band into $${\mathbb {B}}^4$$ by kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $$\mathbb {S}^1$$-invariant metrics on the annulus and Mobius band are the induced metrics on explicit free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^3$$ and $${\mathbb {B}}^4$$, including some new families of free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^4$$. Finally, we prove that these are the only $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$.
We explicitly classify all $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $$\mathbb {S}^1$$-invariant metrics on the annulus and Mobius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all $$\mathbb {S}^1$$-invariant metrics on the Mobius band for each $$k \ge 1$$, and show that it is achieved by the induced metric from a free boundary minimal embedding of the Mobius band into $${\mathbb {B}}^4$$ by kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $$\mathbb {S}^1$$-invariant metrics on the annulus and Mobius band are the induced metrics on explicit free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^3$$ and $${\mathbb {B}}^4$$, including some new families of free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^4$$. Finally, we prove that these are the only $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$.
高特征值优化的一些结果
DOI：
10.1007/s00526-020-01802-9
发表时间：
2020
影响因子：
2.1
作者：
Fraser, Ailana;Schoen, Richard
通讯作者：
Schoen, Richard