The Energy-Momentum tensor on low dimensional $\Spinc$ manifolds

TL;DR

This paper derives a formula for the Energy-Momentum tensor on low-dimensional $ ext{Spin}^c$ manifolds, proves a Bär-type eigenvalue inequality, and characterizes immersed surfaces in $ ext{S}^2 imes ext{R}$ via spinorial methods.

Contribution

It provides a new geometric formula involving the Energy-Momentum tensor, offers a novel proof of an eigenvalue inequality, and characterizes surfaces in $ ext{S}^2 imes ext{R}$ using spinors.

Findings

Formula for Energy-Momentum tensor on $ ext{Spin}^c$ surfaces.

Proof of Bär-type eigenvalue inequality.

Characterization of immersed surfaces via spinors.

Abstract

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