Riemannian metrics with large 𝜆₁

Proceedings of the American Mathematical Society, 2005

We prove that for any p > 1 p>1 , any compact manifold of three or more dimensions carries Riemannian metrics of volume one with the first eigenvalue of the p p -Laplacian arbitrarily large.

Journal of Geometry and Physics, 2008

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.

Duke Mathematical Journal, 2002

On any compact Riemannian manifold (M, g) of dimension n, the L 2normalized eigenfunctions {φ λ } satisfy ||φ λ ||∞ ≤ Cλ n−1 2

Communications on Pure and Applied Mathematics, 1975

Most of the problems in differential geometry can be reduced to problems in differential equations on Riemannian manifolds. Our main purpose here is to study these equations and their applications in geometry.

Arxiv preprint math/0304256, 2003

In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index lemma that will allow us to extend some classical results of finite dimensional Riemannian geometry as Rauch and Berger theorems and the Topogonov theorem in the class of manifolds in which the Hopf-Rinow theorem holds.

Transactions of the American Mathematical Society, 1999

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

International Mathematics Research Notices

We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (M n , g) with C 3 bounded geometry in a suitable sense involving the scalar curvature function. Under C 3 bounds of the geometry, if the supremum of scalar curvature function S g < n(n − 1)k 0 for some k 0 ∈ R, then for small volumes the isoperimetric profile of (M n , g) is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature k 0. This work generalizes Theorem 2 of [12] in which the same result was proved in the case where (M n , g) is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion [29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin-Cartan-Hadamard's conjecture is true in any dimension n in the special case of manifolds with C 3 bounded geometry, and S g < n(n − 1)k 0. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound

1995

This paper is essentially a survey, with however some variations or new points of view on already published results. The main theme is the study of the relationship between various Sobolev type inequalities on manifolds. In the first part, we introduce a scale of dimensions at infinity adapted to manifolds of polynomial growth, in which we recast the results of [C2], [BCLS], and [CL]. In the second one, we show how Poincar@ inequalities allow at the same time to go down in the scale and to refine it into a scale of global inequalities ([CS2], [S],

Indiana University Mathematics Journal, 2000

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

Bulletin of the American Mathematical Society, 1974

Communicated by S. S. Chern, August 1, 1973 1. Introduction. Throughout, M will denote a C 00 compact connected oriented «-manifold, ri^.2. Let p:M-+R be a C 00 function,^ the space of C 00 riemannian metrics on M and where R(g) is the scalar curvature of g. As in Ebin [3], a superscript s will denote objects in the corresponding Sobolev space, £>w/2+l (one can also treat W 8,p spaces in the same way), and we also allow s= oo so *^°° = Jl'. Sign conventions on curvatures are as in Lichnerowicz [10]. Two of our main results follow : THEOREM 1. If p is not identically zero or a positive constant, then *Jl p B is a smooth submanifold of<JK 8 . We can also treat the case p=0. Let 3F 8 denote the set of flat metrics in Jl 8 . Then we have THEOREM 2. Assume J*V0. Writing Ut 8 Q =(je a o \0 r9 )U& r8 9 J(\ is the disjoint union of closed submanifolds. REMARK. If dimM=2, e^J=^" 8 , and if dimM=3, the hypothesis that 1F*J£0 can be dropped.