Local plurisubharmonic defining functions on the boundary (Q785770)

Conditions concerning the existence of local plurisubharmonic defining functions on the boundary of a smooth domain in \(\mathbb{C}^2\) are studied. Existence of (gobal) plurisubharmonic defining functions on the boundary have consequences for the regularity of the Bergman projection and the \(\bar{\partial}\)-Neumann operator [\textit{H. P. Boas} and \textit{E. J. Straube}, Math. Z. 206, No. 1, 81--88 (1991; Zbl 0696.32008)]. While the current article deals with local results, it should be noted that there has been some success in using such local results to construct a global plurisubharmonic defining function on the boundary, under certain conditions [\textit{A. Noell}, Pac. J. Math. 176, No. 2, 421--426 (1996; Zbl 0865.32011)]. The article identifies a necessary and sufficient condition for a smooth domain in \(\mathbb{C}^2\) to locally admit a plurisubharmonic defining function on its boundary, near a boundary point. Let \(r\) be a defining function of a smooth domain in \(\mathbb{C}^2\). Then the plurisubharmonic defining function is locally given by \(\rho = r(1+Kr+T)\) for a constant \(K>0\) and a real function, \(T\). The condition relates the complex Hessian of \((1+T)r\) to \(\mathcal{L}_r\), the Levi form of \(r\), in a neighborhood of boundary point. Furthermore, (in Theorems 1.4, 4.5) it is shown that under a lower bound hypothesis, \(|r_z|^2 \le C \mathcal{L}_r\), of the Levi form of \(r\), the local existence of a plurisubharmonic defining function is equivalent to \(T\) satisfying a certain differential condition. The results presented here hold up to the known cases (for instance in the case of strongly pseudoconvex domains [\textit{J. J. Kohn}, Trans. Am. Math. Soc. 181, 273--292 (1973; Zbl 0276.35071)], in which case \(T=0\)) which admit plurisubharmonic defining functions on the boundary. While the results are mainly restricted to complex dimension 2, Section 6 of the article deals with the case of dimension \(n\), where many simultaneous conditions, which are necessary for the existence of a local plurisubharmonic defining function on the boundary of the form \(\rho = r\cdot h\), are presented.