  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 $M$.

Differential Geometry;Spectral Theory