Normal form of equivariant maps in infinite dimensions

Normal form of equivariant maps in infinite dimensions

Diez, T. (TU Delft Analysis)
Rudolph, Gerd (University of Leipzig)

2021

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.

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

http://resolver.tudelft.nl/uuid:486db0f2-bc96-4db4-9ffe-6bfbec63f4e3

https://doi.org/10.1007/s10455-021-09777-2

0232-704X

Annals of Global Analysis and Geometry, 61 (2022) (1), 159-213

Institutional Repository

journal article

© 2021 T. Diez, Gerd Rudolph