On a spin conformal invariant on manifolds with boundary
Raulot, Simon
In: Mathematische Zeitschrift, 2009, vol. 261, no. 2, p. 321-349
Ajouter à la liste personnelle- Summary
- Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by: $$\lambda_{\rm min}(M, \partial M) := \underset{\overline{g}\in[g]}{\rm inf}|\lambda_1^\pm(\overline{g})|{\rm Vol}(M, \overline{g})^{\frac{1}{n}},$$ where $$\lambda_1^\pm(\overline{g})$$ is the smallest eigenvalue of the Dirac operator under the chiral bag boundary condition $${\mathbb{B}}^\pm_{\overline{g}}$$ . More precisely, we show that if n ≥2 then: $$\lambda_{\rm min}(M, \partial M) \leq \lambda_{\rm min}({\mathbb{S}}_+^n, \partial{\mathbb{S}}_+^n).$$