First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$
Elham Matinpour
TL;DR
This paper establishes that the first eigenvalue of the drift Laplacian $\mathcal{L}=\Delta-\tfrac{1}{2}\langle x,\nabla\cdot\rangle$ on certain symmetric self-shrinkers in $\mathbb{R}^3$ is universally $\tfrac{1}{2}$. By combining a Courant-type nodal-domain framework for the weighted Laplacian with a two-piece property for self-shrinkers and a detailed analysis of symmetry groups, the authors show that for self-shrinkers invariant under the dihedral group $\mathbb{D}_{g+1}$ or the prismatic group $\mathbb{D}_{g+1}\times \mathbb{Z}_2$, the first eigenvalue satisfies $\lambda_1=\tfrac{1}{2}$. The result leverages the fact that coordinate functions are eigenfunctions with eigenvalue $\tfrac{1}{2}$ and uses symmetry to exclude eigenfunctions with lower eigenvalues via nodal arguments. This provides a unifying spectral property for a broad family of symmetric self-shrinkers, linking Gaussian-weighted spectral geometry to the geometry and stability of self-shrinkers arising in mean curvature flow.
Abstract
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.