A Viscosity Method for the Min-Max Construction of Closed Geodesics

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension $1$ and $2$ to the $\varepsilon$-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index of our sequence of critical points converging towards a closed non-trivial geodesic

The Classification of Branched Willmore Spheres in the 33-Sphere and the 44-Sphere

We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in $S^4$, generalising a theorem of Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4\pi$.Comment: 74 pages, 1 figur

Télévision, moyens d’information et comportement électoral

La campagne électorale qui a précédé les consultations de l'automne 1962 et l'analyse des résultats qui les ont suivies ont fourni à la presse l'occasion d'un grand nombre d'articles sur l'effet des moyens d'information, et en particulier de la télévision (...)