Abstract
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric \(\tilde g\) conformal to g, we denote by \(\tilde\lambda\) the first positive eigenvalue of the Dirac operator on \((M, \tilde g, \sigma)\) . We show that
This 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 \(2\tilde\lambda^2 \leq \tilde\mu\) , where \(\tilde\mu\) denotes the first positive eigenvalue of the Laplace operator.
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Ammann, B., Grosjean, J.F., Humbert, E. et al. A spinorial analogue of Aubin’s inequality. Math. Z. 260, 127–151 (2008). https://doi.org/10.1007/s00209-007-0266-5
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DOI: https://doi.org/10.1007/s00209-007-0266-5