Low eigenvalues and one-dimensional collapse

TL;DR

This paper demonstrates that when convex domains in Euclidean space collapse to lower dimensions, the fundamental gap between the first two Dirichlet eigenvalues diverges, revealing geometric constraints on spectral properties during collapse.

Contribution

It establishes the divergence of the fundamental gap during one-dimensional collapse for convex domains, extending results to Lipschitz boundaries and polygonal shapes.

Findings

Fundamental gap diverges during domain collapse.

Bounded gap only near rectangles or cylinders.

Eigenvalues cannot be polyhomogeneous on triangle moduli space.

Abstract

Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions.