Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity
Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity
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低同质性单位球的自由边界极小曲面
DOI：
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发表时间：
2016
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通讯作者：
Peter J. McGrath
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作者：
B. Freidin;Mamikon A. Gulian;Peter J. McGrath
研究了低同调单位球中的自由边界极小曲面。对于满足$m，n>；1$和$m+ngeq8$的每对正整数$(m，n)$，我们构造了一个自由边界极小曲面子集B^{m+n}$(1)在$O(M)与O(N)$下不变.当$m+n<；8$时，所得方程的不稳定性允许我们找到这样的曲面的无限族${sigma_{m，n，k}}_{kin mathbb{N}}$。特别地，${Sigma_{2，2，k}}_{kin Mathbb{N}}$是一族立体环面，当$k$趋于无穷大时，它收敛于Clifford环面上的锥面。这些例子表明，由Fraser和Li得到的自由边界极小曲面的光滑紧性定理一般不能推广到高维。
对于每个$ngeq 3$，我们证明了存在唯一的非平面$so(N)$-不变自由边界极小曲面(“链状”)$Sigma_n子集B^n(1)$。这些曲面推广了Fraser和Schoen研究的$B^3(1)$中的“临界链状”。
We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers $(m,n)$ such that $m, n >1$ and $m+ngeq 8$, we construct a free boundary minimal surface $Sigma_{m, n} subset B^{m+n}$(1) invariant under $O(m) imes O(n)$. When $m+n<8$, an instability of the resulting equation allows us to find an infinite family ${Sigma_{m,n, k}}_{kin mathbb{N}}$ of such surfaces. In particular, ${Sigma_{2, 2, k}}_{kin mathbb{N}}$ is a family of solid tori which converges to the cone over the Clifford Torus as $k$ goes to infinity. These examples indicate that a smooth compactness theorem for Free Boundary Minimal Surfaces due to Fraser and Li does not generally extend to higher dimensions.
For each $ngeq 3$, we prove there is a unique nonplanar $SO(n)$-invariant free boundary minimal surface (a "catenoid") $Sigma_n subset B^n(1)$. These surfaces generalize the "critical catenoid" in $B^3(1)$ studied by Fraser and Schoen.