Propriétés génériques des fonctions propres et multiplicité. (Generic properties of eigenfunctions and multiplicity)
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 \textit{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 \textit{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); \textit{B. Colbois} et \textit{Y. Colin de Verdière}, Comment. Math. Helv. 63, No.2, 194-208 (1988; Zbl 0656.53043)] satisfy this condition.
scientific article; zbMATH DE number 4143119