Prescribed Steklov spectra and multiplicities on surfaces

Let $0<a_1\le\cdots\le a_N$ and $H>a_N$. On every compact connected orientable surface with $b\ge1$ boundary components and genus at least an explicit integer $\Gamma(N,b)$, we construct a smooth metric whose first $N$ positive Steklov eigenvalues are $a_1,\ldots,a_N$ and whose next eigenvalue exceeds $H$. Repetitions are allowed. For connected boundary, $\Gamma(N,1)=\lceil N(N-1)/4\rceil$. This is the least genus permitted by the complete-graph band construction used in the proof; no optimality among all metrics is asserted. The metric may also have any sufficiently small prescribed boundary length, any positive prescribed area, and trivial isometry group. All these statements concern the ordinary, unweighted Steklov problem. The proof realises positive graph Laplacians by Dirichlet and boundary forms, estimates the complete finite spectral matrix, and applies degree after smoothing. For a family of invariant perforations of a closed hyperbolic surface, we identify the Steklov spectral subspace converging to the first positive Laplace eigenvalue with its Laplace eigenspace, equivariantly under the finite group used in the construction. The first Steklov multiplicity is at least the smallest dimension of a real irreducible constituent of that Laplace eigenspace; if the latter is irreducible, its multiplicity is preserved exactly. The surfaces of Fortier Bourque, Gruda-Mediavilla, Petri and Pineault yield hyperbolic examples of genera $10$, $17$ and $37$ with first Steklov multiplicities $16$, $21$ and at least $24$, respectively. They disprove Jammes's proposed equality between maximal first multiplicity and the relative chromatic number minus one.