On a class of fully nonlinear elliptic equations on Hermitian manifolds
type: Item
instance of: scholarly article
title: On a class of fully nonlinear elliptic equations on Hermitian manifolds
zbMATH Open document ID: 1326.32054
author: Bo Guan
author: Wei Sun
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-09-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.5863
review text: On a compact \(n\)-dimensional Hermitian manifold \((M, \omega )\) with smooth boundary, consider a smooth real \((1,1)\)-form \(\chi \) and the equation for the unknown \(u\) (such that \(\chi +dd^c u >0\)) \[ (\chi +dd^c u)^n =f (\chi +dd^c u)^{n-k} \wedge \omega ^k, \] for \(1\leq k\leq n\). For \(k=1\), if \(M\) is a compact Kähler manifold, \(\chi\) is Kähler and \(f\) constant, the equation is satisfied by a stationary point of the J-flow first studied in [\textit{S. K. Donaldson}, Asian J. Math. 3, No. 1, 1--15 (1999; Zbl 0999.53053)] and [\textit{X. Chen}, Int. Math. Res. Not. 2000, No. 12, 607--623 (2000; Zbl 0980.58007)]. For general \(k\), the equation was introduced in [\textit{H. Fang} et al., J. Reine Angew. Math. 653, 189--220 (2011; Zbl 1222.53070)]. The main result of the present paper says that, for smooth \(f\) and smooth boundary condition \(\phi\), if there exists a \(C^2\) (subsolution) \(v\) such that \[ (\chi +dd^c v)^n \geq f (\chi +dd^c v)^{n-k} \wedge \omega ^k \] and \(v=\phi\) on \(\partial M\), then a solution exists as well. The main ingredient of the proof are \(C^2\) estimates generalizing those from the last cited paper to the Hermitian setting.
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 6481192
zbMATH Keywords: fully nonlinear elliptic equations
zbMATH Keywords: Hermitian manifold
zbMATH Keywords: J-flow
reviewed by: Slawomir Kołodziej
MaRDI profile type: Publication
OpenAlex ID: W2056768051
arXiv ID: 1301.5863
cites work: Q4151333
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: The dirichlet problem for nonlinear second-order elliptic equations. II. Complex monge-ampère, and uniformaly elliptic, equations
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: A new parabolic flow in Kähler manifolds
cites work: Q5601113
cites work: Q3766544
cites work: The Dirichlet problem for the Monge-Ampère equations in Hermitian metric
cites work: Liouville and Calabi-Yau type theorems for complex Hessian equations
cites work: Q4946338
cites work: Moment maps and diffeomorphisms
cites work: On a class of fully nonlinear flows in Kähler geometry
cites work: The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
cites work: The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function
cites work: Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds
cites work: Complex Monge-Ampère equations and totally real submanifolds
cites work: A Monge-Ampère type fully nonlinear equation on Hermitian manifolds
cites work: The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds
cites work: Boundary-value problems on \(S^ n\) for surfaces of constant Gauss curvature
cites work: The extremal function associated to intrinsic norms.
cites work: Q3575854
cites work: A uniqueness theorem in Kähler geometry
cites work: Equations of Monge-Ampère type on compact Hermitian manifolds.
cites work: A second order estimate for complex Hess equations on a compact Kähler manifold
cites work: On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian
cites work: Complex Monge Ampere Equations
cites work: The problem of dirichlet for quasilinear elliptic differential equations with many independent variables
cites work: On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy
cites work: Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds
cites work: The complex Monge-Ampère equation on compact Hermitian manifolds
cites work: On the Dirichlet problem for Hessian equations
cites work: Convergence of the \(J\)-flow on Kähler surfaces
cites work: On the \(J\)-flow in higher dimensions and the lower boundedness of the Mabuchi energy
cites work: On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I
cites work: A Priori Estimates for Complex Monge-Ampere Equation on Hermitian Manifolds
DOI: 10.1007/S00526-014-0810-1
description: scientific article