Tubular excision and Steklov eigenvalues

Abstract

Given a compact manifold M and a closed connected submanifold N ⊂ M of positive codimension, we study the Steklov spectrum of the domain Ωε ⊂ M obtained by removing the tubular neighbourhood of size ε around N . All non-zero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of N in M . Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.

Metric upper bounds for Steklov and Laplace eigenvalues

“…The interplay of these eigenvalues with the geometry of M has been an active area of investigation in recent years. See [20] for a survey and [11,12,15,23,4,19,18] for recent relevant results.…”

“…Then b = 1 while N Σ , Λ, Γ and |M| are uniformy bounded. It is proved in[4] that σ −− → +∞. It follows that the presence of |Σ| in the denominator of inequality (18) is required.…”