Higher Dimensional Surgery and Steklov Eigenvalues

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