THE ENERGY-MOMENTUM TENSOR ON LOW DIMENSIONAL Spin^c MANIFOLDS

Abstract

On a compact surface endowed with any Spin c structure, we give a formula involving the Energy-Momentum tensor in terms of geometric quantities. A new proof of a Bär-type inequality for the eigenvalues of the Dirac operator is given. The round sphere S 2 with its canonical Spin c structure satisfies the limiting case. Finally, we give a spinorial characterization of immersed surfaces in S 2 × R by solutions of the generalized Killing spinor equation associated with the induced Spin c structure on S 2 × R.

A note on twisted Dirac operators on closed surfaces

“…For this reason the eigenvalue estimate (1.2) can also be interpreted as an eigenvalue estimate of the Dirac operator associated to any Spin c structure and (1.4) as an estimate on the nodal set of a spinor that is in the kernel of the Spin c Dirac operator. Theorem 1.1 is similar to Theorem 3.4 in [13] taking also into account the zero set of ψ. However, note that the constant in front of the second term in (1.2) is different since we are estimating the Chern-form instead of the curvature two-form of the line bundle associated to the Spin c structure.…”

“…However, note that the constant in front of the second term in (1.2) is different since we are estimating the Chern-form instead of the curvature two-form of the line bundle associated to the Spin c structure. In addition, Theorem 3.4 in [13] also discusses the equality case. It turns out that one possible case of equality in (1.2) is achieved under the same assumptions.…”

“…We can think of the Dirac operator associated to a Spin c structure as the twisted Dirac operator acting on spinors which are twisted by a line bundle. For more details on Spin c structures see [19, Appendix D] and [13,23] for eigenvalue estimates of the Spin c Dirac operator. For this reason the eigenvalue estimate (1.2) can also be interpreted as an eigenvalue estimate of the Dirac operator associated to any Spin c structure and (1.4) as an estimate on the nodal set of a spinor that is in the kernel of the Spin c Dirac operator.…”

“…In addition, Theorem 3.4 in [13] also discusses the equality case. It turns out that one possible case of equality in (1.2) is achieved under the same assumptions.…”

Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

“…In the basis {e 1 , e 2 = Xe 1 , e 3 = ξ}, the Ricci identity (6) gives that The same computation holds for the unit vector fields e 2 and e 3 and we get R 2331 − (a 12 a 33 − a 13 a 23 ) = −g(d ∇ E(e 2 , e 3 ), e 2 ) R 2332 + R 2112 − (a 22 a 33 + a 22 a 11 − a 2 13 − a 2 12 + 5c) = g(d ∇ E(e 2 , e 3 ), e 1 ) + g(d ∇ E(e 1 , e 2 ), e 3 ) + c R 2113 − (a 23 a 11 − a 12 a 13 ) = −g(d ∇ E(e 1 , e 2 ), e 2 ) g(d ∇ E(e 1 , e 2 ), e 1 ) = g(d ∇ E(e 2 , e 3 ), e 3 ) R 3221 − (a 13 a 22 − a 23 a 21 ) = −g(d ∇ E(e 2 , e 3 ), e 3 ) R 3112 − (a 32 a 11 − a 31 a 12 ) = g(d ∇ E(e 1 , e 3 ), e 3 ) R 3113 + R 3223 − (a 22 a 33 − a 11 a 33 + a 2 13 + a 2 23 ) = g(d ∇ E(e 2 , e 3 ), e 1 ) − g(d ∇ E(e 1 , e 3 ), e 2 ) g(d ∇ E(e 2 , e 3 ), e 2 ) = −g(d ∇ E(e 1 , e 3 ), e 1 )…”