Positive metric currents and holomorphic chains in Hilbert spaces

TL;DR

This paper explores the theory of positive metric currents in Hilbert spaces, characterizing certain currents as positive closed rectifiable currents and solving the boundary problem for holomorphic chains.

Contribution

It introduces a new framework for understanding currents in infinite-dimensional Hilbert spaces and solves the boundary problem for holomorphic chains within this setting.

Findings

Characterization of positive closed rectifiable currents in Hilbert spaces

Solution to the boundary problem for holomorphic chains

Extension of metric current theory to infinite-dimensional spaces

Abstract

We present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed $(k,k)-$rectifiable currents and solve the boundary problem for holomorphic chains.