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Given a compact manifold M, we prove that every critical Riemannian metric g for the functional " first eigenvalue of the Laplacian " is λ 1-minimal (i.e., (M, g) can be immersed isometrically in a sphere by its first eigenfunctions) and give a sufficient condition for a λ 1-minimal metric to be critical. In the second part, we consider the case where M is(More)
Let M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k ≥ 0, we consider the conformal invariant λ c k (C) defined as the supremum of the k-th eigenvalue λ k (g) of the Laplace-Beltrami operator ∆ g , where g runs over C. First, we give a sharp universal lower bound for λ c(More)
We establish inequalities for the eigenvalues of Schrö-dinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend(More)
We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the isoperimetric ratio of the domain. Consequently , the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with(More)
In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time T if and only if its energy is negative at some(More)
For any bounded regular domain Ω of a real analytic Rie-mannian manifold M , we denote by λ k (Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λ k and as a functional upon the set of domains of fixed volume in M. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain(More)
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution g0 = 9 + (1 + 8 cos 2 v) 2 1 + 8 cos 2 v " du 2 + dv 2 1 + 8 cos 2 v « , 0 ≤ u < π 2 , 0 ≤ v < π, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of(More)