Fully non-linear elliptic equations on compact Hermitian manifolds
type: Item
instance of: scholarly article
title: Fully non-linear elliptic equations on compact Hermitian manifolds
zbMATH Open document ID: 1409.53062
author: Gabor Szekelyhidi
published in: Journal of Differential Geometry
publication date: 2018-05-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1501.02762
full work available at URL: https://projecteuclid.org/euclid.jdg/1527040875
review text: The author the author considers fully nonlinear elliptic equations on compact Hermitian manifolds \((M,\alpha)\). Let \(\chi\) be a fixed \((1,1)\)-form on \(M\). For a \(C^2\) function \(u:M\rightarrow M\) and \(g=\chi+i\partial\bar{{\partial}}u\) let \(A=\alpha^{i\bar{p}}g_{j\bar{p}}\) be the endomorphism of \(T^{(1,0}M\) which is a Hermitian endomorphism with respect to the metric \(\alpha\). The author considers the equation for u of the form \[F(A)=h,\tag{\(*\)}\] where \(h\) is a given function on \(M\) and \(F(A)=f(\lambda_1,\lambda_2,\dots,\lambda_n)\) is a smooth, symmetric function of the eigenvalues of \(A\). The author assumes that \(f\) is defined on an open, symmetric cone \(\Gamma\subset \mathbb R^n\) with vertex at the origin, containing the positive orthant \(\Gamma_n\). A function \(\underline{u}\) is called a \( C\)-subsolution of \((*)\) if for each \(x\in M\) the set \(\{\lambda'\in\Gamma:f(\lambda')=h(x) \text{ and } \lambda'-\lambda(B(x))\in \Gamma_n\}\) is bounded, where \(\lambda(B(x))\) denotes the \(n\)-tuple of eigenvalues of \(B(x), B=B^i_j=\alpha^{i\bar{p}}(\chi_{j\bar{p}}+\partial_j\partial_{\bar{p}})\). The main result is the following: Theorem. Suppose that \(u\) is a solution and \(\underline{u}\) is a \( C\)-subsolution of \((*)\). If we normalize \(u\) so that \(\sup_Mu=0\), then we have the estimate \(||u||_{C^{2,\alpha}}<C\), where \(C\) depends on the given data \(M,\alpha,\chi,h\) and the subsolution \(\underline{u}\). The author applies the result to the class of Hessian quotient equations on Kähler manifolds.
reviewed by: Włodzimierz Jelonek
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6877022
zbMATH Keywords: Hermitian manifold
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: Hessian equation
MaRDI profile type: Publication
OpenAlex ID: W2963898678
Wikidata QID: Q115165671
arXiv ID: 1501.02762
DOI: 10.4310/JDG/1527040875
Recommended article: Fully non-linear elliptic equations on compact almost Hermitian manifolds
Recommended article: On a class of fully nonlinear elliptic equations on Hermitian manifolds
Recommended article: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds
Recommended article: Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds
Recommended article: Fully non-linear parabolic equations on compact Hermitian manifolds
Recommended article: On a Class of Fully Nonlinear Elliptic Equations Containing Gradient Terms on Compact Hermitian Manifolds
Recommended article: Fully nonlinear elliptic equations on Hermitian manifolds for symmetric functions of partial Laplacians
Recommended article: On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II:L∞Estimate
Recommended article: Fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metric
Recommended article: A Monge-Ampère type fully nonlinear equation on Hermitian manifolds
description: scientific article