A spinorial analogue of Aubin’s inequality
By Bernd Ammann, J. Grosjean, E. Humbert, B. Morel. 2003
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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$${\rm inf}_{\tilde{g} \in [g]} \tilde\lambda {\rm Vol}(M,\tilde g)^{1/n} \leq (n/2) {\rm Vol}(S^n)^{1/n}.$$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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/60fc20b0db9972c7445fb05541c194e9f8c449bf
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Smooth Yamabe invariant and surgery (2008, methodology): “The proof is straightforward; we will just give a sketch using notation from [8] which coincides with that of [26].”
- The Dirac Spectrum (2009, unknown): “3] and [22] respectively for the details.”