Prescription of finite Dirichlet eigenvalues and area on surface with boundary (Q6123191)

A classical result of \textit{Y. Colin de Verdière} [Ann. Sci. Éc. Norm. Supér. (4) 20, 599--615 (1987; Zbl 0636.58036)] ensures that, for any closed connected differentiable manifold \(M\) of dimension at least \(3\) and any finite set of positive real number \(0=\lambda_1<\lambda_2\leq \dots\leq \lambda_n\), there is a Riemannian metric \(g\) on \(M\) such that the \(k\)-th eigenvalue (counted with multiplicities) of the Laplace-Beltrami operator associated to \((M,g)\) is precisely \(\lambda_k\) for any \(k=1, \dots,n\). The method is based on embedding graphs with a prescribed spectrum into \(M\). This strategy cannot be immediately extended in the case of \(M\) is a surface (i.e.\ \(\dim M=2\)). For instance, one cannot embed a complete graph of order \(n\) into \(M\) unless the genius of \(M\) is large enough. The main theorem of the article under review is the following: given \(M\) a differentiable surface with non-empty boundary and positive real numbers \(\lambda_1<\dots <\lambda_n\) and \(A\), there is a Riemannian metric \(g\) on \(M\) such that \(\operatorname{vol}(M,g)=A\) and the \(k\)-eigenvalue of the Laplacian of \((M,g)\) with Dirichlet condition is \(\lambda_k\) for any \(k=1,\dots,n\). The analogous result for manifolds without boundary and dimension at least \(3\) (i.e.\ the extension of Colin de Verdière's result that prescribes also the volume) seems to be proved in the preprint by \textit{X. He} and \textit{Z. Wang} [``Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum'', Preprint, \url{arXiv:2308.03115}].