Tubular Excision and Steklov Eigenvalues

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