Poincaré-Lelong equation via the Hodge Laplace heat equation

Abstract

In this paper, we develop a method of solving the Poincaré-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.

References in corpus (3)

• An optimal gap theorem
• Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature
• A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

Cited by in corpus (15)

• An optimal gap theorem
• Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds
• Chern-Ricci flows on noncompact complex manifolds
• Kähler manifolds with almost non-negative curvature
• Comparison and vanishing theorems for Kähler manifolds
• Manifolds with positive orthogonal Ricci curvature
• On the sharp dimension estimate of CR holomorphic functions in Sasakian Manifolds
• The fundamental group, rational connectedness and the positivity of Kaehler manifolds
• Longtime existence of the Kähler-Ricci flow on ${\mathbb{C}}^n$
• Gap theorem on Kähler manifold with nonnegative orthogonal bisectional curvature
• Kähler manifolds and cross quadratic bisectional curvature
• On the three-circle theorem and its applications in Sasakian manifolds
• Gromov-Hausdorff limits of Kähler manifolds with bisectional curvature lower bound I
• A new positivity condition for the curvature of Hermitian manifolds
• Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian Manifolds