Limiting Case of the Spin Hypersurface Dirac Operator arising in the positive mass theorem for black holes

Serhan Eker

doi:10.2478/amns.2020.2.00070

Abstract

AbstractIn this paper, we give an explicit form of the scalar curvaure for the limiting case of the eigenvalue of the hypersurface Dirac operator which arises in the positive mass theorem for black holes. Then, we show that the hypersurface is an Einstein.

Highlights

On a compact Riemannian Spin−manifold, mathematicians and physicists have been investigated the spectrum of the Dirac operator to obtain subtle information about the topology and geometry of the manifold and its hypersurface [2,3,4, 6, 10, 13, 14, 17]
We introduce some basic facts concerning hypersurface Dirac operator
On a compact Riemannian Spin−manifold N, one can construct a spinor bundle denoted by S and globally defined along M called hypersurface spinor bundle of M [18]

Summary

Introduction

On a compact Riemannian Spin−manifold, mathematicians and physicists have been investigated the spectrum of the Dirac operator to obtain subtle information about the topology and geometry of the manifold and its hypersurface [2,3,4, 6, 10, 13, 14, 17]. The author improved inequality (2) by using the conformal covariance of the Dirac operator on a compact Riemannian Spin−manifold (M, g) of dimension n ≥ 3, λ n 4(n − 1) μ1,. Where R is the scalar curvature of M associated to a conformal deformation of metric g and for some real−valued function on N [12] They investigated the limiting case of the above inequality and they show that the hypersurface is an Einstein. They obtain the following estimates for the eigenvalue of the hypersurface Dirac operator Dp defined in [12],. We show that the hypersurface manifold is Einstein manifold with constant Ricci curvature by considering the limiting case of the above inequality

Hypersurface Dirac Operator

Limiting Case of the Hypersurface manifold endowed with Spin Structure