2004/01/27 by Katrin Wehrheim, Wehrheim, Katrin
Mathematics · #35J65 #53D12 #58B99 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Symplectic Geometry (math.SG) #math.AP #math.SG #msc:35J65 #msc:53D12 #msc:58B99
paper · pdf · doi:10.48550/arxiv.math/0401376
35 pages. This has bubbled off from an earlier preprint (Anti-self-dual instantons with Lagrangian boundary conditions I: Elliptic theory)
arxiv created 2004/01/27 · openalex publication_date 2004/01/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main purpose of this paper is to give a general regularity result for Cauchy-Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic Lp-regularity results hold true under one crucial assumption: The totally real submanifold has to be modelled on an Lp-space or a closed subspace thereof. Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah-Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy-Riemann equation with totally real boundary condition.