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).

我们表明，对于维数至少为 3 且边界为非空的紧凑黎曼流形，我们可以通过执行 2 维或更高维的手术来修改流形，同时保持 Steklov 谱几乎不变。这表明，在考虑维度 3 及更高维度的 Steklov 特征值的形状优化问题时，域拓扑的某些变化不会产生影响。我们的结果将 Fraser 和 Schoen (Adv Math 348:146–162, 2019) 中的一维手术推广到更高维度的手术和更高的特征值。在 Fraser 和 Schoen (Adv Math 348:146–162, 2019) 中证明，单位球不会最大化\(\mathbb {R}^n\)中可收缩域中的第一个非零归一化 Steklov 特征值，对于\( n \ge 3\). 我们表明，对于更高的 Steklov 特征值也是如此。使用类似的想法，我们表明在\(\mathbb {R}^n\) 中，对于\(n\ge 3\)，第j个归一化的 Steklov 特征值不会通过一系列可收缩域退化而在极限范围内最大化与维度 2 中的情况相反（Girouard 和 Polterovich 在 Funct Anal Appl 44:106–117, 2010 中）的j 个单位球的不相交联合。