We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator. (C) 2011 Elsevier Inc. All rights reserved

We consider a compact, orientable minimal hypersurfaces of the unit sphere and prove a comparison theorem between the spectrum of the stability operator and that of the Laplacian on 1-forms. As a corollary, we show that the index is bounded below by a linear function of the first Betti number; in particular, if the first Betti number is large, then the immersion is highly unstable

Let M be a Riemannian manifold and ΩΩ a compact domain of M with smooth boundary. We study the solution of the heat equation on ΩΩ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? This problem is the analogous (for the heat equation) of the classical Serrin’s problem for harmonic domains. In this paper we give an affirmative answer to the above question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.

Abstract. -We control the gap between the mean value of a function on a submanifold (or a point), and its mean value on any tube around the submanifold (in fact, we give the exact value of the second derivative of the gap). We apply this formula to obtain comparison theorems between eigenvalues of the Laplace-Beltrami operator, and then to compute the first three terms of the asymptotic time-expansion of a heat diffusion process on convex polyhedrons in euclidean spaces of arbitrary dimension. We also write explicit bounds for the remainder term of the above expansion, which hold for all values of time. The results of this paper have been announced, without proof, in [16]. Résumé (Un lemme de valeur moyenne et quelques applications)On contrôle l'écart entre la valeur moyenne d'une fonction sur une sous-variété d'une variété riemannienne, et sa valeur moyenne sur un voisinage tubulaire autour de la sous-variété (on donne, en effet, la valeur exacte de la dérivée seconde de cet ecart). On applique ensuite cette formule afin d'obtenir des théorèmes de comparaison pour les valeurs propres et les fonctions propres de l'opérateur de Laplace-Beltrami, et pour calculer les trois premiers termes du développement asymptotique relatifà un problème de diffusion de la chaleur sur les polyèdres convexes dans un espace euclidien de dimension quelconque. On donne enfin des bornes explicites des restes du développement susdit, qui sont valable pour toute valeur du temps. Les résultats de cet article ontété annoncés (sans démonstrations) dans [16].