论文标题
在steklov eigenspaces上，单位球中的自由边界最小表面
On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball
论文作者
论文摘要
我们开发了新方法来比较在单位$ n $ n $ -n $ -ball $ \ mathbb {b}^n $中嵌入的自由边界最小submanifold $σ$上的坐标函数的跨度$ \ mathcal {c}（σ）$。 Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{σ_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid.我们还确认$ \ Mathcal {c} = \ Mathcal {e} _ {σ_1} $对于任何免费的边界最小表面嵌入$ \ Mathbb {B}^3 $中，与许多已知或预期的示例的对称性相对，包括：包括：至少在三个discack int 3 disk aint tht三个disk的示例的示例，包括：两个无限的家族$ 0 $ $ 0 $的二面对称示例，以及一个有限的柏拉图对称性的有限家庭；以及任何属属的示例，通过划出几个沿球直径相等的磁盘相交的磁盘。
We develop new methods to compare the span $\mathcal{C}(Σ)$ of the coordinate functions on a free boundary minimal submanifold $Σ$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace $\mathcal{E}_{σ_1}(Σ)$. Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{σ_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\mathcal{C}=\mathcal{E}_{σ_1}$ for any free boundary minimal surface embedded in $\mathbb{B}^3$ with the symmetries of many known or expected examples, including: examples of any positive genus from stacking at least three disks; two infinite families of genus $0$ examples with dihedral symmetry, as well as a finite family with the various Platonic symmetries; and examples of any genus by desingularizing several disks that meet at equal angles along a diameter of the ball.
