Metric shape of hypersurfaces with small extrinsic radius or large λ1 \lambda_1

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

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results 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 give rigidity results for almost umbilic hypersurfaces

On the cohomology algebra of some classes of geometrically formal manifolds

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal K\"ahler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat. Finally we prove that a six-dimensional manifold with $b_1 \neq 1, b_2 \geqslant 2$ and not having the cohomology algebra of $\mathbb{T}^3 \times S^3$ carries a symplectic structure as soon as it admits a formal metric.Comment: Final version. Accepted in Proc.London Math.So

Hypersurfaces with small extrinsic radius or large λ1\lambda_1 in Euclidean spaces

We prove that hypersurfaces of $\R^{n+1}$ which are almost extremal for the Reilly inequality on $\lambda_1$ and have $L^p$-bounded mean curvature ($p>n$) are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that 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 $q\leqslant n$.Comment: 24 page

EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS

We consider operators defined on a Riemannian manifold $M^m$ by $\lt(u)=-div(T\nabla u)$ where $T$ is a positive definite $(1,1)$-tensor such that $div(T)=0$. We give an upper bound for the first nonzero eigenvalue \lat of $\lt$ in terms of the second fundamental form of an immersion $\phi$ of $M^m$ into a Riemannian manifold of bounded sectional curvature. We apply these results to a particular family of operators defined on hypersurfaces of space forms and we prove a stability result

Minimal submanifolds with a parallel or a harmonic pp-form.

International audienceThe purpose of this paper is to study the relations between the existence of minimal immersions of a Riemannian manifold $M$ into another and some structural or topological properties of $M$. The properties on $M$ which we consider are the existence of a parallel or a harmonic $p$-form

pp-Laplace operator and diameter of manifolds.

International audienceLet \var be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the $p$-Laplacian \lap and we prove that the limit of \sqrt[p]{\lap} when $p\rightarrow\infty$ is $2/d(M)$ where $d(M)$ is the diameter of $M$. Moreover if \var is an oriented compact hypersurface of the Euclidean space $\R$ or \Snpi, we prove an upper bound of \lap in term of the largest principal curvature $\kappa$ over $M$. As applications of these results we obtain optimal lower bounds of $d(M)$ in term of the curvature. In particular we prove that if $M$ is a hypersurface of $\R$ then : $d(M)\geq\pi/\kappa$