The spinorial τ-invariant and 0-dimensional surgery

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 $$\tau (M,\chi ):=\sup \inf \lambda \_1^{+}(g)$$ 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 $(M^#,χ^#)$ be obtained from $(M,\chi )$ by 0-dimensional surgery. We prove that $$τ(M^#,χ^#)\geq τ(M,χ).$$ As a corollary we can calculate $\tau (M,\chi )$ for any Riemann surface $M$.

References in corpus (6)

• Surgery and Harmonic Spinors
• Surgery and the Spectrum of the Dirac Operator
• The first conformal Dirac eigenvalue on 2-dimensional tori
• The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions
• The first eigenvalue of Dirac and Laplace operators on surfaces
• A spinorial analogue of Aubin's inequality

Cited by in corpus (2)

• Geometry of Spin and Spin^c structures in the M-theory partition function
• Surgery and the spinorial tau-invariant