Jean-Pierre Bourguignon

Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS4 with groupG=SU2, SU3 orU2 is either self-dual or anti-self-dual. The analogous statement for SO4-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC0-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.

Spineurs, opérateurs de Dirac et variations de métriques

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.

Upper bound for the first eigenvalue of algebraic submanifolds

1. Statement of results Let M be a compact manifold endowed with a Riemannian metric. The spectrum of the Laplacian, A, acting on functions form a discrete set of the form {0 < ),~ < 22 < �9 �9 �9 < 2k < �9 �9 �9 -~ ~}. In 1970, Joseph Hersch [5] gave a sharp upper bound for the first non-zero eigenvalue 2~ for any Riemannian metric on the 2-sphere in terms of its volume alone. Similar estimates for 21 on any compact oriented surfaces were derived by Yang and Yau [7]. The second and the third authors [6] studied the non-orientable surfaces and pointed out the relationship of 21 and the conformal class of the surface. In fact, their estimates were applied to study the Willmore problem. Another application of these types of upper bounds was found by Choi and Schoen [3] in relation to the set of all minimal surfaces in a compact 3-manifold of positive Ricci curvature. The purpose of this paper is to prove a higher dimensional generalization of the above results. It was pointed out by Marcel Berger [l] that Herschs theorem fails in higher dimensional spheres. In view of the relationship between 21 and the conformal structure of a surface as indicated by Li and Yau [6], we were thus motivated to study the complex category.

The “magic” of Weitzenböck formulas

In recent years, problems of geometric origin have attracted the attention of many analysts. In some instances, Geometry was pointing to the most subtle case of an analytic problem, e.g., the limiting case of Sobolev inequalities in the Yamabe problem or in Yang-Mills theory. Moreover, phenomena of a geometric nature were appearing in a P.D.E. context like the “bubbling off” phenomenon in the harmonic map problem. In many of the geometric situations considered, the problem could be reduced to solving an elliptic scalar equation, most often a non-linear one.

Groupe de Jauge Élargi et Connexions Stables

Dans cet expose, nous nous interessons a certaines formes harmoniques a valeurs dans un fibre, qui apparaissent comme “courbures harmoniques” de connexions.