For closed \(n\)-dimensional Riemannian manifolds \(M\) with almost positive Ricci curvature, the Laplacian on \(1\)-forms is known to admit at most \(n\) small eigenvalues. If there are \(n\) small eigenvalues, or if \(M\) is orientable and has \(n-1\) small eigenvalues, then \(M\) is diffeomorphic to a nilmanifold, and the metric is almost left invariant. We show that our results are optimal for \(n\geq4\).

\(n\)-Manifolds with \({\rm Ric}\geq n-1\) and \(n\) eigenvalues close to \(n\) are both Gromov-Hausdorff close and diffeomorphic to the canonical sphere. \(n\)-Manifolds with \({\rm Ric}\geq n-1\) and volume close to \({\rm Vol}\;\mathbb{S}^n/\#\pi_1(M)\) are both Gromov-Hausdorff close and diffeomorphic to a smooth space form \(\mathbb{S}^n/\pi_1(M)\). These extend results of T.Colding, P.Petersen and T.Yamaguchi.

Complete \(n\)-manifolds whose part of Ricci curvature less than a positive number is small in \(L^p\) norm (for \(p>n/2\)) have bounded diameter and finite fundamental group. On the contrary, complete metrics with small \(L^{n/2}\)-norm of the same part of the Ricci curvature are dense in the set of metrics of any compact differentiable manifold.

In this paper, we show that the convex domains of \(\mathbb{H}^n\) which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in \(\mathbb{R}^n\) and \(\mathbb{S}^n\).

In this paper we prove a diameter sphere theorem and its corresponding \(\lambda_1\) sphere theorem under \(L^p\) control of the curvature. They are generalizations of some results due to S. Ilias.

Let \((M,g)\) be a compact manifold with Ricci curvature almost bounded from below and \(\pi: \overline{M}\to M\) be a normal, Riemannian cover. We show that, for any non-negative function \(f\) on \(M\), the means of \(f\circ\pi\) on the geodesic balls of \(\overline{M}\) are comparable to the mean of \(f\) on \(M\). Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature \(L^p\)-bounded from below.

For odd dimensional Poincaré-Einstein manifolds \((X^{n+1},g)\), we study the set of harmonic \(k\)-forms (for \(k < n/2\)) which are \(C^m\) (with \(m\in N\)) on the conformal compactification \(\overline{X}\) of \(X\). This is infinite dimensional for small \(m\) but it becomes finite dimensional if \(m\) is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology \(H^k(\overline{X},\partial\overline{X})\) and the kernel of the Branson-Gover differential operators (\(L_k,G_k)\) on the conformal infinity \((\partial\overline{X},[h_0])\). In a second time we relate the set of \(C^{n-2k+1}(\Lambda^k(\overline{X}))\) forms in the kernel of \(d+\delta_g\) to the conformal harmonics on the boundary in the sense of Branson-Gover, providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of \(Q\) curvature for forms.

Hypersurfaces of \(\mathbb{R}^{n+1}\) which are almost extremal for the Reilly inequality on \(\lambda_1\) and have \(L^p\)-bounded mean curvature are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. The same result stands for the Hasanis-Koutroufiotis inequality on extrinsic radius. When a supplementary \(L^q\) bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when \(q > n\), but not necessarily diffeomorphic to a sphere when \(q\leqslant n\).

This paper will not be published since it has been improved and split into the following paper
J.-F. Grosjean, J. Roth, Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces Math. Z. 271 (2012), no. 1-2, 469–488.
and the two papers that follow.

In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or \(\lambda_1\) have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on \(v_M\|B\|_\alpha^n\) and show that their spectral and topological properties depends on the position of \(\alpha\) with respect to the critical value \(\dim M\). The study of the metric shape of these extremal hypersurfaces will be done in \cite{AG1}, using estimates of the present paper.

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large \(\lambda_1\) or small extrinsic radius. The result depends on the \(L^p\) norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for \(p\) equal to the dimension minus \(1\).

We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sectional curvature and on a lower bound on the injectivity radius.

On note (W) l'hypothèse de doublement du volume:
\( \exists C_0,\exists\nu >0\) tels que \(\forall x\in M,\,\forall r'\geq r >0\), on a: \(V_x(r')\leq V_x(r)C_0(\frac{r'}{r})^\nu\) (où \(V_x(r)\) est le volume de la boule géodésique de centre \(x\) et de rayon \(r\)),
 
On note (M) l'inégalité de la moyenne suivante:
(M) \(\exists\lambda >0\) tel que \(\forall x\in M,\, \forall r > 0\), et pour toute fonction sous-harmonique positive sur \(M\), on a: \[\lambda\int_{B_x(r)}f\geq V_x(r)f(x)\] (où \(B_x(r)\) désigne la boule géodésique de centre \(x\) et de rayon \(r\)).
Il existe \(C(C_0,\nu) >0\) tel que si \(M^n\) est une variété Riemannienne complète vérifiant (W) et (M), E est un fibré vectoriel de rang \(m\) sur \(M\) et \(S\) est un sous-espace linéaire de sections de \(E\) vérifiant \(\Delta \|u\|^2\leq 0\) et \(\|u\|\in O(\rho^d)\) \(\forall u\in S\) (où \(p\in M\) et \(\rho(x)=d(x,p)\)), alors \({\rm dim}\, S\leq mC\lambda d^{\nu -1}\).

We give a detailled proof of the T.Colding and P.Petersen sphere theorems.

On s'intéresse à la géométrie des variétés de courbure de Ricci presque supérieure à \(k\) (i.e. telle qu'une norme \(L^p\)-- locale ou globale--de la fonction (\(\underline{\rm Ric}-k)_-\) soit petite, où \(\underline{\rm Ric}\) est la plus petite valeur propre de la courbure de Ricci en \(x\)). On démontre sous cette hypothèse les équivalents des inégalités géométriques classiques de Myers, de Bishop-Gromov, de Lichnerowicz,... puis on caractérise les variétés qui réalisent presque les cas d'égalité (généralisant des travaux de T.Colding et de P.Petersen). Sur une variété compacte \(M^n\) de courbure presque positive, le laplacien sur les \(1\)-formes a au plus \(n\) petites valeurs propres. S'il a exactement \(n\) petites valeurs propres (\(n-1\) suffisent si \(M\) est orientable) alors \(M\) est difféomorphe à une Nilvariété et la métrique est presque invariante à gauche. Ces résultats découlent d'estimées analytiques établies dans la première partie de la thèse.