Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mohammed Anakkar

TL;DR

The paper develops a complex-analytic framework for Sobolev loop spaces $W^{k,2}(S,X)$ with $k> rac{n}{2}$, establishing a natural complex Hilbert-manifold structure on these spaces. It then proves a Hartogs-type extension theorem: holomorphic maps from generalized Hartogs figures in $W^{k,2}(S,H_q^n(r))$ to $q$-Hilbert-Hartogs manifolds extend holomorphically to the envelope $W^{k,2}(S,\Delta^q\times \Delta^n)$. The results generalize classical finite-dimensional Hartogs extension to infinite-dimensional loop spaces and highlight envelope-of-holomorphy phenomena in this setting.

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.

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

TL;DR

The paper develops a complex-analytic framework for Sobolev loop spaces with , establishing a natural complex Hilbert-manifold structure on these spaces. It then proves a Hartogs-type extension theorem: holomorphic maps from generalized Hartogs figures in to -Hilbert-Hartogs manifolds extend holomorphically to the envelope . The results generalize classical finite-dimensional Hartogs extension to infinite-dimensional loop spaces and highlight envelope-of-holomorphy phenomena in this setting.

Abstract

We recall the complex structure on the generalised loop spaces , where is a compact real manifold with boundary and is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain domains in generalised loop spaces.
Paper Structure (3 sections, 14 theorems, 60 equations)

This paper contains 3 sections, 14 theorems, 60 equations.

Table of Contents

  1. Sobolev maps between smooth manifolds
  2. Complex structure on the space of Sobolev maps
  3. A Hartogs-type theorem

Key Result

Theorem 1

Let $k > \frac{n}{2}$, then for every $u \in W^{k,2}_{\text{loc}}(\Omega , \mathbb{R}^m)$ and every $f \in \mathcal{C}^k(\mathbb{R}^m)$ the function $f \circ u$ is in $W^{k,2}_{\text{loc}}(\Omega)$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Proposition 1.1
  • proof
  • Theorem 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • ...and 16 more