Normal form of equivariant maps in infinite dimensions

Tobias Diez, Gerd Rudolph

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Abstract

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

Original languageEnglish
Pages (from-to)159-213
Number of pages55
JournalAnnals of Global Analysis and Geometry
Volume61 (2022)
Issue number1
DOIs
Publication statusPublished - 2021

Keywords

  • Submersion
  • Immersion
  • Group action
  • Equivariant map
  • Kuranishi structure
  • Moduli space
  • Anti-self-dual Yang-Mills
  • Seiberg–Witten
  • Pseudoholomorphic curves

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