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Sur les fonctions propres positives des variétés de Cartan-Hadamard. (On positive eigenfunctions of Cartan-Hadamard manifolds.). (French) Zbl 0689.53027

The author completes the study begun in a previous paper [Ann. Math., II. Ser. 125, 495-536 (1987; Zbl 0652.31008)]. Let M be a Cartan-Hadamard manifold of dimension \(n\geq 2\) with sectional curvature pinched between the constants \(-a^ 2\) and \(-b^ 2\) \((0<a\leq b<+\infty)\). The author proves that if \({\mathcal L}\) is an elliptic operator of second order on M of a certain type (the Laplace-Beltrami operator \(\Delta\) on M is of this type) and \(\lambda_ 1\) the first eigenvalue of \({\mathcal L}\), then for each \(t\in (0,\lambda_ 1)\) there exist \({\mathcal L}_+tI\)-harmonic functions on M tending to zero at infinity.
One should mention that there is an important difference as against the case of the constant curvature concerning the speed of the convergence to zero at infinity. An estimate of quotients of Green functions on M relative to the same pole and different levels t, \(t'\) for the case \({\mathcal L}=\Delta\) allows to specify a little the shaping at infinity of the eigenfunctions constructed in the first section. Some examples of manifolds without Green functions relative to \(\Delta +\lambda_ 1I\) or with the \(\lambda_ 1\)-eigenfunctions which do not vanish at infinty are given. Finally, a partial generalization in the setting of manifolds with negative curvature of S. J. Patterson’s results [Acta Math. 136, 241-273 (1976; Zbl 0336.30005)] concerning the connections between eigenfunctions and limit sets of Fuchsian groups on the hyperbolic space is pointed out.
Reviewer: M.Craioveanu

MSC:

53C20 Global Riemannian geometry, including pinching
31C12 Potential theory on Riemannian manifolds and other spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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