Scholarlink

Let \((M,\mathit {g})\) be an m-dimensional compact manifold. We assume that M is spin, and we fix a spin structure \(\sigma \) on M. We denote by \(\mathbb {S}(M)=Spin(TM)\times _\rho \mathbb {S}_m\) the spinor bundle on M with hermitian metric \((\cdot ,\cdot )\) and compatible spin connection \(\nabla ^{\mathbb {S}}\) . The Clifford multiplication \(\displaystyle TM\otimes \mathbb {S}(M)\to \mathbb {S}(M) \) is denoted by \(X\otimes \psi \mapsto X\cdot \psi \) . Let \(D=D_{\mathit {g}}\) be the (Atiyah-Singer) Dirac operator defined on \(\Gamma (\mathbb {S}(M))\) , i.e. \(D=\sum _{k=1}^m e_k\cdot \nabla ^{\mathbb {S}}_{e_k}\) for a local orthonormal frame \(\{e_1,\dots ,e_m\}\) of TM.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

The Spinorial Brezis-Nirenberg Problem

  • 章节

  • 发表日期：

    2025年

查看全文
  • Yanheng Ding,
  • Tian Xu

摘要

Let \((M,\mathit {g})\) be an m-dimensional compact manifold. We assume that M is spin, and we fix a spin structure \(\sigma \) on M. We denote by \(\mathbb {S}(M)=Spin(TM)\times _\rho \mathbb {S}_m\) the spinor bundle on M with hermitian metric \((\cdot ,\cdot )\) and compatible spin connection \(\nabla ^{\mathbb {S}}\) . The Clifford multiplication \(\displaystyle TM\otimes \mathbb {S}(M)\to \mathbb {S}(M) \) is denoted by \(X\otimes \psi \mapsto X\cdot \psi \) . Let \(D=D_{\mathit {g}}\) be the (Atiyah-Singer) Dirac operator defined on \(\Gamma (\mathbb {S}(M))\) , i.e. \(D=\sum _{k=1}^m e_k\cdot \nabla ^{\mathbb {S}}_{e_k}\) for a local orthonormal frame \(\{e_1,\dots ,e_m\}\) of TM.