Complex manifolds of Sobolev mappings and a Hartogs-type theorem in loop spaces

TL;DR

This paper explores the complex structure of Sobolev loop spaces and establishes a Hartogs-type extension theorem for holomorphic maps in these infinite-dimensional settings.

Contribution

It introduces a complex structure on Sobolev loop spaces and proves a new extension theorem analogous to Hartogs' theorem in this context.

Findings

Complex structure on Sobolev loop spaces established.

Hartogs-type extension theorem proved for holomorphic maps.

Extension results applicable to infinite-dimensional complex manifolds.

Abstract

We recall the complex structure on the generalised loop spaces $W^{k,2}(S,X)$, where $S$ is a compact real manifold with boundary and $X$ is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain domains in generalised loop spaces.