The Fundamental Gap and One-Dimensional Collapse

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. 11 3. Collapsing polygonal domains 15 3.1. Proof of Theorem 4: bounded gap 15 3.2. Proof of Theorem 4: unbounded gap 16 4. The general case 19 References 21

The sound of symmetry

“…The resulting triangle, T k has one angle that is very small and tends to zero as d k → ∞, while the other two angles remain bounded away from zero, and their opposite sides tend toward 1. By the estimates in the proof of Proposition 1 of [23] (c.f. similar estimates in [16])…”