Potential Theoretic Hyperbolicity and $\mathbf{L^2}$ extension. Part I: Stein manifolds

TL;DR

This paper generalizes the Ohsawa-Takegoshi $L^2$ extension theorem by weakening curvature assumptions, leveraging potential theoretic positivity properties of the underlying manifold or line bundle, with applications demonstrated through examples.

Contribution

It introduces a new $L^2$ extension theorem that allows weaker curvature conditions under potential theoretic positivity, broadening the theorem's applicability.

Findings

Extension theorem with weakened curvature assumptions

Applicable to manifolds with potential theoretic positivity

Demonstrated through multiple examples

Abstract

We establish a new generalization of an $L^2$ extension theorem of Ohsawa-Takegoshi type. The improvement in the theorem is that it allows the usual curvature assumptions to be significantly weakened in certain favorable settings. The favorable settings come about, for example, when the underlying structure (e.g., the underlying manifold, or the holomorphic Hermitian line bundle whose sections are being extended) has certain potential theoretic positivity. The simplest case occurs when the underlying manifold supports a function with self-bounded gradient in the sense of McNeal, but there are other cases. We demonstrate the improvement over the usual Ohsawa-Takegoshi-type extension theorems through a number of examples.