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Extremality for the Vafa-Witten bound on the sphere. (English) Zbl 1137.53334

Summary: We prove that the round metric on the sphere has the largest first eigenvalue of the Dirac operator among all metrics that are larger than it. As a corollary, this gives an alternative proof of an extremality result for scalar curvature due to M. Llarull.

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58C40 Spectral theory; eigenvalue problems on manifolds
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