LIOUVILLE AND CALABI-YAU TYPE THEOREMS FOR COMPLEX HESSIAN EQUATIONS

We prove a Liouville type theorem for entire maximal m-subharmonic functions in ℂn with bounded gradient. This result, coupled with a standard blow-up argument, yields a (nonexplicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the progra...

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Bibliographic Details
Published inAmerican journal of mathematics Vol. 139; no. 2; pp. 403 - 415
Main Authors Dinew, Slawomir, Kołodziej, Slawomir
Format Journal Article
LanguageEnglish
Published Baltimore Johns Hopkins University Press 01.04.2017
Subjects
Geometry
Partial differential equations
Theorems
Topological manifolds
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Summary:We prove a Liouville type theorem for entire maximal m-subharmonic functions in ℂn with bounded gradient. This result, coupled with a standard blow-up argument, yields a (nonexplicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the program, initiated by Hou, Ma, and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right-hand side in some Lp, with a sharp bound on p.
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2017.0009