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Weak solutions of complex Hessian equations on compact Hermitian manifolds

Part of: Global differential geometry Elliptic equations and systems Pluripotential theory

Published online by Cambridge University Press:  09 September 2016

Sławomir Kołodziej  and
Ngoc Cuong Nguyen
Sławomir Kołodziej
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Łojasiewicza 6, Poland email Slawomir.Kolodziej@im.uj.edu.pl
Ngoc Cuong Nguyen
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Łojasiewicza 6, Poland email Nguyen.Ngoc.Cuong@im.uj.edu.pl
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Abstract

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

Keywords

weak solutionscomplex Hessian equationcomparison principle

MSC classification

Primary: 53C55: Hermitian and Kählerian manifolds 35J96: Elliptic Monge-Ampère equations 32U40: Currents

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2221 - 2248
Copyright
© The Authors 2016 

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