Symmetries and the First Laplace Eigenvalue of Lawson Surfaces

In this paper, we study the first eigenvalue of the Laplace--Beltrami operator on the Lawson minimal surfaces $\xi_{m,k}$ embedded in the unit three-sphere $\mathbb{S}^3$. Motivated by Yau's conjecture on the first eigenvalue of closed embedded minimal hypersurfaces in the sphere, we develop a symmetry-based approach to the equality $\lambda_1(\xi_{m,k})=2$ for the family of Lawson surfaces with $m$ and $k$ even. Our method exploits the discrete reflection symmetries intrinsic to Lawson's construction, together with the algebraic structure of the associated reflection group, Courant's nodal domain theorem, and the coordinate eigenfunctions arising from Takahashi's theorem. More precisely, we show that the equality $\lambda_1(\xi_{m,k})=2$ follows once a natural topological obstruction for invariant nodal sets in the fundamental patch is verified.