First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in ℝ³

Consider R 3 \mathbb {R}^3 equipped with the Euclidean metric and the Gaussian measure d μ = e − | x | 2 4 d x d\mu =e^{\frac {-\vert x\vert ^2}{4}}dx . Let Σ ⊂ R 3 \Sigma \subset \mathbb {R}^3 be a complete embedded self-shrinker with the induced metric and weighted measure, and let λ 1 \lambda _1 denote the first eigenvalue of the drift Laplacian Δ μ = Δ − 1 2 ⟨ x , ∇ . ⟩ \Delta _{\mu }=\Delta -\frac {1}{2}\langle x,\nabla .\rangle in the weighted space L 2 ( Σ , e − | x | 2 4 d x ) L^2(\Sigma , e^{\frac {-\vert x\vert ^2}{4}}dx) . Inspired by Choe and Soret’s estimate, of the first eigenvalue of the Laplacian on symmetric minimal surfaces in