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

2025/09/30 by Matinpour, Elham
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.25617

Abstract

Consider ℝ³ equipped with the Euclidean metric and the Gaussian measure. Let Σ be a complete embedded self-shrinker in ℝ³ with the induced metric and weighted measure, and let λ₁ denote the first eigenvalue of the drift Laplacian in the weighted L² space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in \mathbbS³, we prove that λ₁= 1/2 for self-shrinkers invariant under the dihedral group \mathbbD_g+1 or the prismatic group \mathbbD_g+1× ℤ₂. In particular, this holds for known self-shrinkers confirming a universal spectral property tied to their symmetry.

Citations

• On eigenfunctions and nodal sets of the Witten-Laplacian
• Self-shrinkers whose asymptotic cones fatten
• Fattening in mean curvature flow
• Embedded self-similar shrinkers of genus 0
• An immersed S² self-shrinker
• Generic mean curvature flow I; generic singularities
• Smooth compactness of self-shrinkers

Related

• An eigenvalue estimate for self-shrinkers in a Ricci shirinker
• Eigenvalues of the drifted Laplacian on complete metric measure spaces
• Self-shrinkers whose asymptotic cones fatten
• Index Estimate of Self-Shrinkers in ℝ³ with Asymptotically Conical Ends
• Eigenvalue estimates on shrinkers