Multiple eigenvalues and the width

We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k Neumann eigenvalues. The number k depends on the ratio of the corresponding width over the diameter of the domain. For convex thin domain in $R^n$, we obtain the eigenvalue comparison with collapsing segment.

💡 Research Summary

The paper investigates the multiplicity of Neumann eigenvalues for convex “thin’’ domains in Euclidean space and for compact thin Riemannian manifolds with non‑negative Ricci curvature. The central geometric quantity is the width W(Ω), defined as the minimal cross‑sectional diameter over all directions, and the diameter D(Ω). The authors prove that when the ratio W·D⁻¹ is sufficiently small, the first Neumann eigenvalue (and, in the planar case, the first k eigenvalues) is simple, i.e., has multiplicity one.

In two dimensions, for any positive integer k they define an explicit constant
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