On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem

2021/12/07 by Yannick Sire, Sire, Yannick, Tian Xu +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Crystallography and Radiation Phenomena #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2112.03640

openalex publication_date 2021/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a closed spin manifold of dimension m≥6 equipped with a Riemannian metric \ig and a spin structure \sa. Let \lm₁⁺(\ig) be the smallest positive eigenvalue of the Dirac operator D_\ig on M with respect to a metric \ig conformal to \ig. The Bär-Hijazi-Lott invariant is defined by \lm_min⁺(M,\ig,\sa)=inf_\ig∈[\ig]\lm₁⁺(\ig)\Vol(M,\ig)^(1)/(m). In this paper, we show that \lm_min⁺(M,\ig,\sa)lt;\lm_min⁺(S^m,\ig_S^m,\sa_S^m)=\frac m2\Vol(S^m,\ig_S^m)^\frac1m provided that \ig is not locally conformally flat. This estimate is a spinorial analogue to an estimate by T. Aubin, solving the Yamabe problem in this setting.

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