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

TL;DR

The paper introduces a new conformal volume invariant for compact manifolds, proves it is uniformly bounded, and explores its implications for spectral geometry and manifold surgeries.

Contribution

It defines a novel differential invariant based on conformal volume and establishes its uniform boundedness, linking it to spectral invariants and geometric surgeries.

Findings

The conformal volume invariant is uniformly bounded above.

The invariant provides bounds for the Friedlander-Nadirashvili spectral invariant.

Behavior of the invariant under manifold surgeries is characterized.

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 $M$.