Laplacien de Hodge-de Rham双值结构

Prescription de la multiplicité des valeurs propres du laplacien de Hodge-de Rham

On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\leq p<\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues.

不同形式的效应和小价值

Effondrements et petites valeurs propres des formes différentielles

关于有界曲率坍缩下Hodge Laplacian小特征值的简短综述。

Extrema de valeurs propres dans une classe conforme

Survey about extremum of eigenvalues of geometric operators within a conformal class of a compact riemannian manifold.

Sur la multiplicité des valeurs propres du laplacien de Witten

On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on $p$-form for $0<p<n$. In particular, we prescribe the multiplicity of the first eigenvalues. On 3-dimensional manifolds, we give examples of multiple first eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of embedded surfaces all of whose 1-cohomology is induced by the cohomology of the manifold. In particular, this multiplicity is at least 3.

Première valeur propre du laplacien, volume conforme et chirurgies

We define a new differential invariant a compact manifold by $V_{\mathcal M}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili invariant defined by $\inf_g\sup_{\tilde g\in[g]}λ_1(M,\tilde g)\Vol(M,\tilde g)^{\frac 2n}$. The proof relies on the study of the behaviour of $V_{\mathcal M}(M)$ when one performs surgeries on $M$.

Petite valeurs propres des fibrés principaux en tores

Let M^n be a compact n-dimensional principal T^k-bundle. We consider collapsings of M on N=M/T^k such that the diameter and sectional curvature of M satisfy diam(M)<d and |K(M)|<a, and give examples of collapsings for all k such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of M are bounded above by c(M).inj(M)^2k. Moreover, we prove that the first non-zero eigenvalue of 1-form Laplacian of all T^k-bundle M over N is bounded below by c(n,d,a,N).Vol(M)^2 and c.inj(M)^2k when M collapses on N.