Comportement semi-classique des systèmes ergodiques
Conformal Geometry and the Composite Membrane Problem
We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.
Conformal gradient vector fields on a compact Riemannian manifold
It is proved that if an n-dimensional compact connected Riemannian manifold (M,g) with Ricci curvature Ric satisfying 0 < Ric ≤ (n-1)(2-nc/λ₁)c for a constant c admits a nonzero conformal gradient vector field, then it is isometric to Sⁿ(c), where λ₁ is the first nonzero eigenvalue of the Laplacian operator on M. Also, it is observed that existence of a nonzero conformal gradient vector field on an n-dimensional compact connected Einstein manifold forces it to...
Construction de laplaciens dont une partie finie (avec multiplicités) du spectre est donnée
Construction de laplaciens dont une partie finie du spectre est donnée
Continuous families of isospectral metrics on simply connected manifolds.
Convergence de variétés et convergence du spectre du laplacien
Convergence of Riemannian manifolds and Laplace operators. I
We study the spectral convergence of compact Riemannian manifolds in relation with the Gromov-Hausdorff distance and discuss the geodesic distances and the energy forms of the limit spaces.
Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below
Convexity and Absolute Continuity of the Laplace-Beltrami Operator.
Correction to : “On isospectral deformations of riemannian metrics. II”
Correction to "On the characterization of flat metrics by the spectrum"
Covering lemmas and BMO estimates for eigenfunctions on Riemannian surfaces.
The principal aim of this note is to prove a covering Lemma in R2. As an application of this covering lemma, we can prove the BMO estimates for eigenfunctions on two-dimensional Riemannian manifolds (M2, g). We will get the upper bound estimate for || log |u| ||BMO, where u is the solution to Δu + λu = 0, for λ > 1 and Δ is the Laplacian on (M2, g). A covering lemma on homogeneous spaces is also obtained in this note.
Cramér's formula for Heisenberg manifolds
Let be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that , where is a specific nonzero constant and is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the -dimensional case.
Curvature Pinching and Eigenvalue Rigidity for Minimal Submanifolds.