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Propriétés génériques des fonctions propres et multiplicité. (Generic properties of eigenfunctions and multiplicity). (French) Zbl 0697.58056

Let \((X,g_ 0)\) be a compact, smooth \((=C^{\infty})\) Riemannian manifold, \(\Delta\) the Laplacian of \((X,g_ 0)\), \(b_ 0: X\to {\mathbb{R}}^ a \)smooth function and \(C^ k(X)\) the space of real \(C^ k\)- functions on X with large k. Let \(\lambda_ 0\) be an eigenvalue of \(\Delta +b_ 0\) of multiplicity N that verifies the stability condition called the strong Arnold hypothesis [see Y. Colin de Verdière, Comment. Math. Helv. 63, No.2, 184-193 (1988; Zbl 0672.58046)] with respect to the family \(\Delta +b\), where \(b\in C^ k(X)\) is close to \(b_ 0\). Under these assumptions, there exists a neighborhood B of \(b_ 0\) in \(C^ k(X)\) such that the set \(W=\{a\in C^ k(X)\) close to \(b_ 0\) such that \(\Delta +a\) has \(\lambda_ 0\) as eigenvalue with multiplicity \(N\}\) is a submanifold of B. Moreover one can choose a basis \((u_{i,a})_{1\leq i\leq N}\) of the eigenspace \(Ker(\Delta +a-\lambda_ 0)\) which depends differentiably on \(a\in W\). The author proves the existence of an open dense subset \(W_ 1\) of W such that the mapping \(X\to {\mathbb{R}}^ N\), \(x\mapsto (u_{i,a}(x))_{1\leq i\leq N},\) is for \(a\in W_ 1\), (1) an immersion if \(N\geq 2 \dim (X);\) (2) an embedding if \(N\geq 2 \dim (X)+1.\) The same result is true for conformal perturbations of the metric \(g_ 0\). The case of arbitrary variations of the metric is also considered. Some examples of stable eigenvalues of high multiplicity are given. In fact one verifies that the examples of Y. Colin de Verdière [loc. cit.; ibid. 61, 254-270 (1986; Zbl 0607.53028); Ann. Sci. Ec. Norm. Supér., IV. Sér. 20, 599-615 (1987; Zbl 0636.58036); B. Colbois et Y. Colin de Verdière, Comment. Math. Helv. 63, No.2, 194-208 (1988; Zbl 0656.53043)] satisfy this condition.
Reviewer: M.Craioveanu

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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