First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$

Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $Σ$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $λ_1$ denote the first eigenvalue of the drift Laplacian in the weighted $L^2$ space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in $\mathbb{S}^3$, we prove that $λ_1$= 1/2 for self-shrinkers invariant under the dihedral group $\mathbb{D}_{g+1}$ or the prismatic group $\mathbb{D}_{g+1}\times \mathbb{Z}_2$. In particular, this holds for known self-shrinkers confirming a universal spectral property tied to their symmetry.