First boundary Dirac eigenvalue and boundary capacity potential

DOI10.1007/S00023-022-01233-6arXiv2206.13286MaRDI QIDQ6403257

Simon Raulot

Publication date: 27 June 2022

Abstract: We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface $Sigma$ bounding a noncompact domain in a spin asymptotically flat manifold (M n , g) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of $Sigma$ in (M n , g) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Spin and Spin({}^c) geometry (53C27) Global Riemannian geometry, including pinching (53C20)

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