Prescription of the Robin spectrum

Let $M$ be a compact connected smooth manifold with smooth boundary, and let $\rho$ be a positive continuous function on the boundary which is served as the Robin parameter. In this paper, we study three problems concerning the prescription of finite Robin spectrum: (1) Prescribing finitely many Robin eigenvalues and the volume. (2) Within a fixed conformal class, prescribing the multiplicities of finitely many Robin eigenvalues. (3) Within a fixed conformal class, prescribing finitely many distinct Robin eigenvalues and the volume. A key step in our method is to solve the corresponding problems for the Dirichlet spectrum. As a consequence, we also obtain analogous results for the Dirichlet case.