Pincement spectral en courbure positive

Jerome Bertrand
Abstract : 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.
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Jerome Bertrand. Pincement spectral en courbure positive. Mathématiques [math]. Université Paris Sud - Paris XI, 2003. Français. ⟨tel-00008705⟩

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