Yau's conjecture for the stacked Clifford tori of Wiygul

Abstract

We prove Yau's conjecture $\lambda_{1}=2$ for the stacked Clifford tori of Wiygul: for all integers $N\ge2$, $k,\ell\ge1$ and every sufficiently large $m$, the closed embedded minimal surface of genus $k\ell m^{2}(N-1)+1$ in the round three-sphere which resembles $N$ 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: 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 Poincar\'e inequality on a 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 Poincar\'e and trace inequalities on the blocks.