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Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant

Jurgen Julio-Batalla

Abstract

We consider on a closed Riemannian spin manifold $(M^n,g,σ)$ the spinorial Yamabe type equation $D_g\varphi=λ|\varphi|^{\frac{2}{n-1}}\varphi$, where $\varphi$ is a spinor field and $λ$ is a positive constant. For a normalized solution $\varphi$ of this equation we find a positive lower bound for $λ^2$. As an application we obtain an explicit lower bound of the Bär-Hijazi-Lott invariant for some spin manifolds with positive scalar curvature.

Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant

Abstract

We consider on a closed Riemannian spin manifold the spinorial Yamabe type equation , where is a spinor field and is a positive constant. For a normalized solution of this equation we find a positive lower bound for . As an application we obtain an explicit lower bound of the Bär-Hijazi-Lott invariant for some spin manifolds with positive scalar curvature.
Paper Structure (4 sections, 6 theorems, 39 equations)

This paper contains 4 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimation of Parameters
  4. Explicit lower bound

Key Result

Theorem 1.1

Let $(M^n,g,\sigma)$ be a closed Riemannian spin manifold of positive scalar curvature. Assume $n\geq 3$ and there is an open bounded domain $\Omega\subset M$ such that $(\Omega,g)$ is conformally flat. If $\varphi$ is a solution of the equation Dirac with $\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_g=1$. where $\omega_n$ is the volume of the round sphere $(\mathbb{S}^n,g_0)$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 2 more