# The spinorial $au$-invariant and 0-dimensional surgery

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1 IECN - Institut Élie Cartan de Nancy

Abstract : Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $\chi$. Let $\lambda 1^+g$ be the first non-negative eigenvalue of the Dirac operator on $M,g,\chi$. We set $$\tauM,\chi:= \sup \inf \lambda 1^+g$$ where the infimum runs over all metrics $g$ of volume $1$ in a conformal class $g 0$ on $M$ and where the supremum runs over all conformal classes $g 0$ on $M$. Let $M^\#,\chi^\#$ be obtained from $M,\chi$ by $0$-dimensional surgery. We prove that $$\tauM^\#,\chi^\#\geq \tauM,\chi.$$ As a corollary we can calculate $\tauM,\chi$ for any Riemann surface $M$.

Keywords : Dirac operators conformal geometry gluing formula

Author: Bernd Ammann - Emmanuel Humbert -

Source: https://hal.archives-ouvertes.fr/

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