Geometric plurisubharmonicity and convexity: an introduction
nonlinear partial differential equations
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.
scientific article; zbMATH DE number 6060975
scientific article