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Spinor inequality for magnetic fields on spin manifolds

Jurgen Julio-Batalla

Abstract

This paper is concerned with the zero mode equation $D_g\varphi=iA\cdot\varphi$ on closed spin manifold $(M^n,g,σ)$ 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 $(\mathbb{S}^n,g_{can},σ_{can})$ this result confirms that the inequality obtained in \cite{Frank} is not sharp.

Spinor inequality for magnetic fields on spin manifolds

Abstract

This paper is concerned with the zero mode equation on closed spin manifold of positive scalar curvature. Here is a real one form on . We proved that if is a non trivial solution of the zero mode equation then where is the Yamabe constant of and . In the case of the round sphere this result confirms that the inequality obtained in \cite{Frank} is not sharp.
Paper Structure (5 sections, 4 theorems, 56 equations)

This paper contains 5 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spin manifolds
  4. Invariance properties
  5. Proof of the inequality

Key Result

Theorem 1.1

Let $(M^n,g,\sigma)$ be a closed spin manifold of positive scalar curvature $s_g$. If $(\varphi,A)$ is a nontrivial zero mode then where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$ and, $v_n=(n-1)/2$ if $n$ is odd and $v_n=n/2$ if $n$ is even.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Claim 1
  • proof
  • ...and 1 more