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

Abstract

Consider ${\mathbb{R}}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $\Sigma$ be a complete embedded self-shrinker in ${\mathbb{R}}^3$ with the induced metric and weighted measure, and let ${\lambda }_1$ 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 ${\lambda }_1$= 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.