A generalization of the Monge-Ampère equation on compact Hermitian manifolds
type: Item
instance of: scholarly article
title: A generalization of the Monge-Ampère equation on compact Hermitian manifolds
zbMATH Open document ID: 0858.35036
author: Abdellah Hanani
published in: Bulletin des Sciences Mathématiques
publication date: 1996-08-04T00:00:00Z
review text: Let \((X,g)\) be a compact Hermitian manifold. If \(\varphi\in C^2(X)\), consider the changes of the Hermitian metric defined by \(g_1(\varphi)= -\varphi g+i\partial \overline{\partial}\varphi\) and \(g_2^\pm(\varphi)= g+i\partial \overline{\partial}\varphi\pm \nabla\varphi\otimes \nabla\varphi\), where each \(g_i(\varphi)\) is positive definite. In such cases, \(\varphi\) is said to be admissible. The author proves the existence of an admissible \(C^\infty\) solution to equations of the form \[ [\text{det } g_i(\varphi)] [\text{det}(g)]^{-1}= F(x,\nabla\varphi;\varphi), \] where \(F\in C^\infty(TX\times \mathbb{R})\) is an everywhere strictly positive function satisfying some growth assumptions. For some equations, it is shown that the above solution is also unique. This is an extension to the Hermitian case of results of \textit{P. Delanoë} [J. Funct. Anal. 45, 403-430 (1982; Zbl 0497.58026)].
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 889863
zbMATH Keywords: generalized Monge-Ampère equation
zbMATH Keywords: Hermitian metric
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 889863