Poincar\'e-Lelong equation via the Hodge Laplace heat equation

TL;DR

This paper introduces a novel approach to solving the Poincaré-Lelong equation by analyzing the large-time behavior of solutions to the Hodge-Laplace heat equation, yielding optimal results under specific curvature conditions.

Contribution

It develops a new method linking the Poincaré-Lelong equation with the Hodge-Laplace heat equation and offers an alternative proof for a recent gap theorem.

Findings

Effective solution method for Poincaré-Lelong equation

Optimal results for manifolds with nonnegative bisectional curvature

Alternative proof of a recent gap theorem

Abstract

In this paper, we develop a method of solving the Poincar\'e-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on $(1, 1)$-forms. The method is effective in proving an optimal result when $M$ has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author.