An infinite dimensional balanced embedding problem I:existence

TL;DR

This paper proves the existence of an infinite-dimensional balanced embedding for a non-compact complex manifold using a gradient flow approach, addressing challenges in long-term existence and convergence.

Contribution

It introduces a novel method employing a gradient flow in a Hilbert space to establish existence of such embeddings in a model case.

Findings

Existence of a balanced embedding proved for a model case.

Long-time existence of the flow established via perturbation techniques.

Convergence demonstrated using bounds related to Tauber-Hardy-Littlewood theorem.

Abstract

We investigate the problem of balanced embedding of a non-compact complex manifold into an infinite-dimensional projective space. In this paper we prove the existence of such an embedding in a model case. The strategy is by using a gradient flow in a Hilbert space; both the long-time existence and convergence at infinite time are non-trivial. The long time existence is established by choosing a perturbation of the ODE; the convergence depends on a priori bounds that uses techniques in the proof of the Tauber-Hardy-Littlewood theorem.