Let ( , ) be a smooth and complete surface, Ω ⊂ be a domain in , and Δ be the Laplace operator on . The spectrum of the Dirichlet-Laplace operator on Ω is a sequence 0 < 1 (Ω) ≤ 2 (Ω) ≤ ⋅ ⋅ ⋅ ↗ ∞. A classical question is to ask what is the domain Ω * which minimizes (Ω) among all domains of a given area, and what is the value of the corresponding (Ω * ). The aim of this article is to present a numerical algorithm using shape optimization and based on the nite element method to nd an

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... n of a candidate for Ω * . Some veri cations with existing numerical results are carried out for the rst eigenvalues of domains in ℝ 2 . Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.