Let (M, g, σ ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal to g, we denote byλ the first positive eigenvalue of the Dirac operator on (M,g, σ ). We show thatThis inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D = {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ 2 ≤μ, whereμ denotes the first positive eigenvalue of the Laplace operator.

Let (M m , g) be a compact Riemannian manifold isometrically immersed in a simply connected space form (euclidean space, sphere or hyperbolic space). The purpose of this paper is to give optimal upper bounds for the first nonzero eigenvalue of the Laplacian of (M m , g) in terms of r-th mean curvatures and scalar curvature. As consequences, we obtain some rigidity results. In particular, we prove that if (M n , g) is a compact hypersurface of positive scalar curvature immersed in R n+1 and if g is a Yamabe metric, then (M n , g) is a standard sphere.

Abstract. In this paper, we give pinching theorems for the first nonzero eigenvalue λ 1 (M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any ε > 0, there exists a constant C ε depending on the dimension n of M and the L ∞ -norm of the mean curvature H , so that if the, then the Hausdorff-distance between M and a round sphere of radius (n/λ 1 (M)) 1/2 is smaller than ε. Furthermore, we prove that if C is a small enough constant depending on n and the L ∞ -norm of the second fundamental form, then the pinching condition n H 2 2p − C < λ 1 (M) implies that M is diffeomorphic to an n-dimensional sphere. Mathematics Subject Classification (2000). 53A07, 53C21.

Abstract. In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces M of these spaces N . Indeed, we prove that if M is included in a ball of radius small enough then the Hausdorff-distance between M and a geodesic sphere S of N is small. Moreover M is diffeomorphic and quasi-isometric to S. As other application, we obtain rigidity results for almost umbilic hypersurfaces.

Abstract. In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or λ1 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 vM B n α and show that their spectral and topological properties depends on the position of α with respect to the critical value dim M . The study of the metric shape of these extremal hypersurfaces will be done in AG1[3], using estimates of the present paper. IntroductionThroughout the paper, X: M n → R n+1 is a closed, connected, immersed Euclidean hypersurface (with n 2). We set v M its volume, B its second fundamental form, H = 1 n tr B its mean curvature, r M its extrinsic radius (i.e. the least radius of the Euclidean balls containing M ), (λ M i ) i∈N the non-decreasing sequence of its eigenvalues labelled with multiplicities andThe Hasanis-Koutroufiotis inequality asserts that n H 2 2 , once again with equality if and only if M is the sphere S M (we give some short proof of these inequalities in section prel 2). Our aim is to study the spectral properties of the hypersurfaces that are almost extremal for each of this Inequalities. The results and estimates of this paper are used in AG1[3] to study the metric shape of the almost extremal hypersurfaces. We set µ2 the k-th eigenvalue of S M (labelled without multiplicities) and m k its multiplicity. Throughout the paper we shall adopt the notation that τ (ε|n, · · · ) is a positive function which depends on n, · · · and which converges to zero with ε → 0 when n, · · · are fixed.