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Poincaré–Lelong equation via the Hodge–Laplace heat equation

Part of: Partial differential equations Complex manifolds Parabolic equations and systems Global differential geometry

Published online by Cambridge University Press:  09 September 2013

Lei Ni  and
Luen-Fai Tam
Lei Ni
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA email lni@math.ucsd.edu
Luen-Fai Tam
Affiliation:
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China email lftam@math.cuhk.edu.hk
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Abstract

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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.

Keywords

Poincaré–Lelong equationHodge-Laplacian heat equationKähler manifoldsConvex exhaustion

MSC classification

Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 32Q15: Kähler manifolds 35B40: Asymptotic behavior of solutions 35K05: Heat equation
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1856 - 1870
Copyright
© The Author(s) 2013 

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