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arXiv · 2610.32117

Dirac eigenvalues and the hyper-hemispherical radius

Pengshuai Shi
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Abstract

On a connected compact Riemannian spin manifold with boundary, we prove an upper estimate for the smallest eigenvalue of the Dirac operator under the boundary condition associated with a chirality operator. This upper bound is expressed in terms of the hyper-hemispherical radius. Moreover, the upper bound is realized if and only if the manifold is a hemisphere. As an application, we get a scalar curvature rigidity result for the hemisphere in the case of Lipschitz maps whose differential has a prescribed trace-norm bound. Our estimate extends Bär's result for closed manifolds, where the upper bound is given in terms of hyperspherical radius.

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Pengshuai Shi. 2026-10-23. Dirac eigenvalues and the hyper-hemispherical radius. https://arxiv.org/abs/2610.32117

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