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In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator
is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of
and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of
are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of
-forms on the boundary of a
-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.
The normalized eigenvalues Ʌi(M, g) of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics g on a fixed surface M. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in 𝕊3 or 𝕊4. In this paper a family of extremal metrics induced by minimal immersions in 𝕊5 is investigated.
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