Spinor inequality for magnetic fields on spin manifolds

TL;DR

This paper establishes a lower bound on the norm of the differential of a real one-form for solutions to the zero mode equation on closed spin manifolds with positive scalar curvature, refining previous inequalities especially on the sphere.

Contribution

It proves a new inequality relating the differential of the one-form to the Yamabe constant, improving understanding of zero modes on spin manifolds with positive scalar curvature.

Findings

The inequality holds for solutions on closed spin manifolds.

On the sphere, the inequality is shown to be not sharp.

The result connects zero modes with geometric invariants like the Yamabe constant.

Abstract

This paper is concerned with the zero mode equation $D_g\varphi=iA\cdot\varphi$ on closed spin manifold $(M^n,g,\sigma)$ of positive scalar curvature. Here $A$ is a real one form on $M$. We proved that if $(\varphi, A)$ is a non trivial solution of the zero mode equation then $$\parallel dA\parallel_{n/2}>Y(M^n,[g])/(4v_n^{1/2}),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$ and $v_n=\left[\frac{n}{2}\right]$. In the case of the round sphere $(\mathbb{S}^n,g_{can},\sigma_{can})$ this result confirms that the inequality obtained in \cite{Frank} is not sharp.