The introduction of the concepts of energy machinery and energy structure on a manifold makes it possible a large class of energy representations of gauge groups including, as a very particular case, the ones known up to now. By using an adaptation of methods initiated by I. M. Gelfand, we provide a sufficient condition for the irreducibility of these representations.
We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...
We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on . This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on .
Equivalence and zero sets of certain maps on infinite dimensional spaces are studied using an approach similar to the deformation lemma from the singularity theory.
We show under some assumptions that a differentiable function can be transformed globally to a polynomial or a rational function by some diffeomorphism. One of the assumptions is that the function is proper, the number of critical points is finite, and the Milnor number of the germ at each critical point is finite.
The -convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations , where is the elementary symmetric function of order , , of the eigenvalues of the Hessian matrix . For example, is the Laplacian and is the real Monge-Ampère operator det , while -convex functions and -convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative -convex functions, and give several...
The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to...
À l’aide de la théorie des itinéraires et des suites de tricotage, nous étudions la conjugaison topologique des fonctions unimodales. Nous introduisons la notion de conjugaison macroscopique, caractérisée par l’égalité des suites de tricotage. Puis nous présentons un théorème de classification des fonctions unimodales. Pour illustrer ces résultats, nous montrons que l’ensemble des solutions de l’équation de Feigenbaum contient une infinité de classes topologiques.
Nous construisons une famille de surfaces de Riemann hyperelliptiques, de genre variable, munies de fonctions méromorphes de degré deux et d’indice un, ce qui apporte une réponse positive à une conjecture de S. Montiel et A. Ros.