Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory

TL;DR

This paper introduces a novel method using Banach manifold maps with corners to construct local models around boundary points in moduli spaces of anti-self-dual connections, with broad applications in geometric analysis.

Contribution

It develops a new approach leveraging differential topology of Banach manifolds with corners for gluing constructions in gauge theory and related nonlinear PDE problems.

Findings

Provides a general framework for gluing in gauge theory

Applicable to a wide class of geometric analysis problems

Enhances understanding of moduli space boundary structures

Abstract

We describe a new approach to the problem of constructing gluing parameterizations for open neighborhoods of boundary points of moduli spaces of anti-self-dual connections over closed four-dimensional manifolds. Our approach employs general results from differential topology for $C^1$ maps of smooth Banach manifolds with corners, providing a method that should apply to other problems in geometric analysis involving the gluing construction of solutions to nonlinear partial differential equations.