Restrictions on scalar curvature and first Dirac eigenvalue of closed manifolds

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Under suitable restrictions on the Riemannian metric, the scalar curvature K of a Riemannian manifold cannot become arbitrarily large everywhere. The results of Lichnerowicz and Gromov-Lawson show that min к ≤ 0 for large families of manifolds. Results of Gromov, Llarull and others show that other manifolds admit positive upper bounds for min к, which are optimal in certain special cases. Similarly, on a closed Riemannian spin manifold, the Dirac eigenvalue λ1 of smallest absolute value is 0 if the Â-genus is nonzero. In other cases, there exist universal upper bounds for |λ1| by Vafa-Witten and Gromov, again under suitable restrictions on the Riemannian metric. Herzlich and others gave optimal upper bounds for |λ1| for certain manifolds.Note that both problems above are directly related by the Friedrich inequality, which bounds min к from above by |λ1|. Also, some of the methods employed for both problems are similar. In this project, we want to find good upper bounds for both the scalar curvature and the first Dirac eigenvalue on larger classes of Riemannian manifolds.