Pincement spectral en courbure positive
Résumé
The subject of my thesis is the study of problems of spectral pinching in positive curvature. In the following, all manifolds are compact, n dimensional and satisfy the curvature assumption : $Ric \geq (n-1)g$ (we let be D the set of such manifolds). Under these assumptions, the first (non zero) eigenvalue of the Laplacian on (M,g) is greater or equal to n (Lichnerowicz's theorem and equality is achieved only for the canonical unit sphere (Obata's theorem). An interesting question is whether or not a Riemannian manifold, belonging to D and satisfying $\lambda_1(M,g) \leq n + \ep $, looks like a sphere, and if so, in which sense ? Unlike the case where the sectional curvature is bounded from below $K \geq 1$, such an assumption does not imply that the manifold is homeomorphic to a sphere. Nevertheless, it is known that for a manifold in D, the following properties are equivalent : the first eigenvalue is close to n and the diameter of the manifold is almost maximal, about $\pi$. More recently, P. Petersen proved, using previous work of T. Colding on volume, that for (M,g) in D, the existence of n+1 eigenvalues close to n is equivalent to the fact that the manifold is Gromov-Hausdorff close to the unit sphere $\mathbb(S)^n$, and such a manifold is diffeomorphic to the sphere thanks to a theorem of J. Cheeger and T. Colding. A natural question is : what happens between these two cases ? What are the properties of a manifold in D with k ($ 1 \leq k \leq n+1$) eigenvalues close to n? One result of my PhD thesis is that a manifold in D has $k$ eigenvalues close to n if and only if it contains a subset close to $\mathbb(S)^(k-1)$ (when k is equal to 1, this is a consequence of results mentioned above). The method used also gives a new proof of Petersen's theorem, the original proof cannot be adapted to this case. In the second part of my thesis, I studied some spectral properties of regular convex domains of a manifold in D with Dirichlet boundary condition. Using a symmetrisation process, P. Berard and D. Meyer proved that the first Dirichlet eigenvalue of a regular domain $\Omega$ is greater or equal to the first Dirichlet eigenvalue of a spherical cap $\Omega^*$ of the canonical unit sphere, with the same relative volume ($ \frac(\vol \Omega)(\vol M )=\frac(\vol \Omega^*)(\vol \mathbb(S)^n)$). Moreover, when there is equality, (\Omega,M,g) is isometric to (\Omega^*,\mathbb(S)^n, can). I proved a stability result corresponding to the case of the hemisphere.
Sur l'ensemble des variétés riemanniennes compactes à courbure de Ricci positive (on normalise par $Ric \geq (n-1)g$), la première valeur propre non nulle du laplacien agissant sur les fonctions atteint son minimum uniquement pour la sphère canonique. Dans cette thèse, nous caractérisons, à l'aide de la distance de Gromov-Hausdorff, les variétés riemanniennes à courbure positive dont les premières valeurs propres du laplacien sont proches de celles de la sphère canonique. Cette propriété de minimimalité du spectre de la sphère s'étend par un procédé de symétrisation, au spectre de Dirichlet des boules géodésiques de la sphère parmi les domaines de variétés à courbure de Ricci positive. Nous étudions les domaines de variétés à courbure de Ricci positive dont la première valeur propre de Dirichlet est presque minimimale. En particulier, nous montrons qu'un domaine convexe dont la première valeur propre de Dirichlet est proche de celle d'un hémisphèere est Gromov-Hausdorff proche d'un hémisphère d'un sinus produit tordu.
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