Geometric plurisubharmonicity and convexity: an introduction
type: Item
instance of: scholarly article
title: Geometric plurisubharmonicity and convexity: an introduction
zbMATH Open document ID: 1251.31003
author: F. Reese Harvey
author: H. Blaine jun. Lawson
published in: Advances in Mathematics
publication date: 2012-07-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.3875
reviewed by: Guy Roos
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 6060975
zbMATH Keywords: nonlinear partial differential equations
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
OpenAlex ID: W2963911347
arXiv ID: 1111.3875
cites work: Q4357414
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: $C^\infty$ approximations of convex, subharmonic, and plurisubharmonic functions
cites work: An introduction to potential theory in calibrated geometry
cites work: Duality of positive currents and plurisubharmonic functions in calibrated geometry
cites work: Dirichlet duality and the nonlinear Dirichlet problem
cites work: Plurisubharmonicity in a General Geometric Context
cites work: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds
cites work: Q4604158
cites work: On the General Notion of Fully Nonlinear Second-Order Elliptic Equations
cites work: Stetige streng pseudokonvexe Funktionen
cites work: Parallel calibrations and minimal submanifolds
DOI: 10.1016/J.AIM.2012.03.033
review text: The paper is an essay on potential theory for geometric plurisubharmonic functions on a Riemannian manifold \(X\), endowed with a closed subset \(\mathbb{G}\) of the Grassmann bundle \(G(p,TX)\) of tangent \(p\)-planes.NEWLINENEWLINEA smooth function \(u\) on \(X\) is called \(\mathbb{G}\)-plurisubharmonic (resp. strictly \(\mathbb{G}\)-plurisubharmonic) if \(\text{tr}_{W} \text{Hess}_{x}u\geq0\) (resp. \(\text{tr}_{W} \text{Hess}_{x}u>0\)) for all \(W\in\mathbb{G}_{x}\) and all \(x\in X\). (Here \(\text{Hess}_{x}u\) denotes the Riemannian Hessian of \(u\) at \(x\in X\); if \(X\) is the Euclidean space \(\mathbb{R}^{n}\), the Riemannian Hessian \(\text{Hess}_{x}u\) may be identified with the second derivative \(D_{x}^{2}u\)). If \(X\) is an open subset of \(\mathbb{C}^{n}\) and \(\mathbb{G}\) is the complex tangent line bundle, then the \(\mathbb{G} \)-plurisubharmonic functions are the usual plurisubharmonic functions on \(X\).NEWLINENEWLINEThe paper presents the definitions and properties of \(\mathbb{G}\)-convexity, boundary \(\mathbb{G}\)-convexity, upper semicontinuous \(\mathbb{G} \)-plurisubharmonic functions, \(\mathbb{G}\)-harmonic functions and the Dirichlet problem, distributions which are distributionally \(\mathbb{G}\)-pluri\-subharmonic.
description: scientific article; zbMATH DE number 6060975