Yau关于Wiygul堆叠Clifford环面的猜想
Yau's conjecture for the stacked Clifford tori of Wiygul
AI总结：
本文证明Wiygul堆叠Clifford环面的Yau猜想，通过反射引理和谱分析，证明其第一Laplace特征值为2，推广了粘合构造的验证方法。
AI中文摘要：
我们证明了Wiygul堆叠Clifford环面的Yau猜想$\lambda_{1}=2$：对于所有整数$N\ge2$，$k,\ell\ge1$以及每个充分大的$m$，三维球面中亏格为$k\ell m^{2}(N-1)+1$的闭嵌入极小曲面，其形状类似于由小catenoidal隧道连接的$N$个平行Clifford环面副本，其第一Laplace特征值为$2$。对于$N\ge3$，这些曲面是链而非加倍，并且所有先前粘合构造验证所依赖的偶-奇分解不可用。Choe和Soret的反射引理将问题归结为在构造的对称群下不变的函数扇区，我们证明该扇区的最低非零特征值等于$4+O(m^{-1})$。值$4$是精确恒等式的结果：构造的极限腰比形成路径的线图的邻接算子的Perron向量，因此路径的谱隙与平衡条件规定的隧道的总电导相抵消，剩下的就是Clifford环面的Jacobi算子的系数。解析输入包括圆柱上的共形不变通道不等式和带孔环面上的Poincaré不等式。论证所依赖的构造性质被隔离在一个约化定理中，该定理适用于沿有限图的边由薄通道族连接的分块组成的闭曲面，并满足对称性假设以及块上的Poincaré和迹不等式。
英文摘要：
We prove Yau's conjecture $λ_{1}=2$ for the stacked Clifford tori of Wiygul: for all integers $N\ge2$, $k,\ell\ge1$ and every sufficiently large $m$, every closed embedded minimal surface arising from Wiygul's construction, of genus $k\ell m^{2}(N-1)+1$ in the round three-sphere and resembling $N$ parallel copies of the Clifford torus joined by small catenoidal tunnels, has first Laplace eigenvalue $2$. For $N\ge3$ these surfaces are chains rather than doublings, and the even--odd decomposition on which all previous verifications for gluing constructions rest is not available. The reflection lemma of Choe and Soret reduces the problem to the sector of functions invariant under the symmetry group of the construction, and we show that the lowest nonzero eigenvalue of that sector equals $4+O(m^{-1})$. The value $4$ is the outcome of an exact identity in the limiting weighted graph model: the limiting waist ratios of the construction form the Perron vector of the adjacency operator of the line graph of a path, so that the spectral gap of the path cancels against the total conductance of the tunnels prescribed by the balancing conditions, and what survives is the coefficient of the Jacobi operator of the Clifford torus. The analytic input consists of a conformally invariant channel inequality on a cylinder and of a Poincaré inequality on a periodically perforated torus. The properties of the construction on which the argument rests are isolated in a reduction theorem for closed surfaces decomposed into blocks joined by families of thin channels along the edges of a finite graph, subject to a symmetry assumption and to Poincaré and trace inequalities on the blocks.
