Première valeur propre du laplacien, volume conforme et chirurgies

Abstract

We define a new differential invariant a compact manifold by $${V_{\mathcal M}(M)={\rm inf}_g V_c(M,[g])}$$ , 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 $${{\rm inf}_g \, {\rm sup}_{{\tilde g}\in[g]} \lambda_1(M,{\tilde g}) {\rm Vol}(M,{\tilde g})^{\frac{2}{n}}}$$ . The proof relies on the study of the behaviour of $${V_{\mathcal M}(M)}$$ when one performs surgeries on M.