Premi\`ere valeur propre du laplacien, volume conforme et chirurgies

Abstract

We define a new differential invariant a compact manifold by $V_{\mathcal M}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili invariant defined by $\inf_g\sup_{\tilde g\in[g]}\lambda_1(M,\tilde g)\Vol(M,\tilde g)^{\frac 2n}$. The proof relies on the study of the behaviour of $V_{\mathcal M}(M)$ when one performs surgeries on $M$.