R²×S¹(m)中Karcher鞍塔的Morse指数
Morse index of Karcher saddle towers in $\mathbb{R}^2 \times \mathbb{S}^1(m)$
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中文总结 AI 辅助
本文研究R²×S¹(m)中Karcher构造的单周期极小曲面子族的Morse指数与零空间维数，利用对称性和Dirichlet-to-Neumann映射完成相关指数与维数的公式推导及证明。
中文摘要 AI 辅助
对于每个整数k≥3，赫尔曼·卡歇尔（Hermann Karcher）构造了一个完全单周期极小曲面Ξₖ（在相似变换下唯一），它具有2k个端点，渐近于k个沿同一直线等角相交的平面的并集，且在基本平移商下亏格为0。记Ξₖ,ₘ为Ξₖ经过m≥1个基本周期平移得到的商曲面，本文研究子族Ξₖ,₂和Ξ₃,ₘ的Morse指数与零空间维数。在m=2的情形，我们证明对所有k≥3，Ξₖ,₂的Morse指数为4k-3，零空间维数为3；在k=3的情形，我们证明存在实数α*∈(1/3,1/2)，使得对所有m≥1，Ξ₃,ₘ的Morse指数为6m−4⌊mα*⌋−3，零空间维数为3，仅当mα*为整数时零空间维数为7。两个证明均利用各曲面的对称性及半周期上的Dirichlet-to-Neumann映射，经共形变换后将问题简化为单位圆上的傅里叶分析计算。
英文摘要
For each integer $k \geq 3$ Hermann Karcher identified a complete singly periodic minimal surface $Ξ_k$ (unique up to similarity) with $2k$ ends asymptotic to the union of $k$ planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing $Ξ_{k,m}$ for the quotient of $Ξ_k$ by translation through $m \geq 1$ fundamental periods, we study the Morse index and nullity of the subfamilies $Ξ_{k,2}$ and $Ξ_{3,m}$. In the $m=2$ case we prove for all $k \geq 3$ that $Ξ_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number $α^* \in (1/3,1/2)$ such that for all $m \geq 1$ the Morse index of $Ξ_{3,m}$ is $6m- 4 \lfloor mα^* \rfloor - 3$ and its nullity is $3$ unless $mα^*$ is an integer, in which case its nullity is $7$. Both proofs exploit the symmetries of each surface and the Dirichlet-to-Neumann map on a half period to reduce the problem to Fourier-analytic computations on the unit circle (after a conformal change).
