Un probl\`eme de type Yamabe sur les vari\'et\'es compactes spinorielles compactes

TL;DR

This paper investigates a conformal eigenvalue problem for the Dirac operator on compact spin manifolds, establishing bounds and conditions for strict inequalities, with implications for conformal spin geometry.

Contribution

It introduces a conformal invariant related to the Dirac operator's eigenvalues and provides conditions for strict inequalities compared to the sphere case.

Findings

Established bounds for the conformal eigenvalue invariant.

Identified conditions for strict inequality with the sphere case.

Applied results to problems in conformal spin geometry.

Abstract

Let $(M,g,\si)$ be a compact spin manifold of dimension $n \geq 2$. Let $\lambda_1^+(\tilde{g})$ be the smallest positive eigenvalue of the Dirac operator in the metric $\tilde{g} \in [g]$ conformal to $g$. We then define $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. We show that $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %=\frac{n}{2} \om_n^{{1 \over n}}$ . We find sufficient conditions for which we obtain strict inequality $\lamin(M,[g],\si) < \lamin(\mS^n)$. This strict inequality has applications to conformal spin geometry. ----- Soit $(M,g,\si)$ une vari\'et\'e spinorielle compacte de dimension $n \geq 2$. %Si $\tilde{g} \in [g]$ est une m\'etrique conforme \`a $g$, On note $\lambda_1^+(\tilde{g})$ la plus petite valeur propre $>0$ de l'op\'erateur de Dirac dans la m\'etrique $\tilde{g} \in [g]$ conforme \`a $g$. On d\'efinit $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. On montre que $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %= \frac{n}{2} \om_n^{{1 \over n}}$ On trouve des conditions suffisantes pour lesquelles on obtient l'in\'egalit\'e stricte $\lamin(M,[g],\si) < \lamin(\mS^n)$. Cette in\'egalit\'e stricte a des applications en g\'eom\'etrie spinorielle conforme.