Zusammenfassung
Let (M, g, sigma) be a compact Riemannian spin manifold of dimension >= 2. For any metric (g) over tilde conformal to g, we denote by (lambda) over tilde the first positive eigenvalue of the Dirac operator on (M, (g) over tilde, sigma). We show that inf((g) over tilde is an element of[g]) (lambda) over tilde Vol(M, (g) over tilde)(1/n) <= (n/2) Vol (S-n)(1/n). This inequality is a spinorial ...
Zusammenfassung
Let (M, g, sigma) be a compact Riemannian spin manifold of dimension >= 2. For any metric (g) over tilde conformal to g, we denote by (lambda) over tilde the first positive eigenvalue of the Dirac operator on (M, (g) over tilde, sigma). We show that inf((g) over tilde is an element of[g]) (lambda) over tilde Vol(M, (g) over tilde)(1/n) <= (n/2) 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 not equal {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2 (lambda) over tilde (2) <= (mu) over tilde where <(mu)over tilde denotes the first positive eigenvalue of the Laplace operator.
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