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

Herzlich, M.

Geom. Funct. Anal. 15, No. 6, 1153-1161 (2005).

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