A spinorial analogue of Aubin’s inequality

Abstract

Let (M, g, σ ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal to g, we denote byλ the first positive eigenvalue of the Dirac operator on (M,g, σ ). We show thatThis inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D = {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ 2 ≤μ, whereμ denotes the first positive eigenvalue of the Laplace operator.

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“…Assertions (i) and (ii) follow directly by simple calculations, and (iii) follows directly from (5). For (iv) we have by definition of…”

On a spin conformal invariant on manifolds with boundary

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