Poincaré–Lelong equation via the Hodge–Laplace heat equation

2011/09/30 by Lei Ni, Luen-Fai Tam
Mathematics · #Algebraic Geometry and Number Theory #Applied mathematics #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Heat equation #Laplace transform #Laplace's equation #Mathematical analysis #Mathematics #Partial differential equation #math.DG

paper · pdf · doi:10.1112/s0010437x12000322

published as Compositio Math. 149 (2013) 1856-1870

arxiv created 2012/10/26 · openalex publication_date 2013/09/09 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• Multiple Integrals in the Calculus of Variations
• On the parabolic kernel of the Schrödinger operator
• R-Torsion and the Laplacian on Riemannian manifolds
• Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature
• A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature