Higher Dimensional Surgery and Steklov Eigenvalues

Abstract

We show that for compact Riemannian manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions 3 and higher. Our result generalizes the 1-dimensional surgery in Fraser and Schoen (Adv Math 348:146–162, 2019) to higher dimensional surgeries and to higher eigenvalues. It is proved in Fraser and Schoen (Adv Math 348:146–162, 2019) that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in $\mathbb {R}^n$, for $n \ge 3$. We show that this is also true for higher Steklov eigenvalues. Using similar ideas, we show that in $\mathbb {R}^n$, for $n\ge 3$, the j-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of j unit balls, in contrast to the case in dimension 2 (Girouard and Polterovich in Funct Anal Appl 44:106–117, 2010).