The intrinsic geometry on bounded pseudoconvex domains (Q1651388)

Bounded pseudoconvex domains have bounded plurisubharmonic exhaustion functions. In the smooth case (\(C^{2}\)) such exhaustion functions can be manufactured from defining functions: a classical result of Diederich and Fornæss says that there exist \(\tau\in (0,1]\) and a defining function \(\rho=\rho_{\tau}\) such that \(-(-\rho_{\tau})^{\tau}\) is (strictly) plurisubharmonic in the domain. The Diederich-Fornæss index \(\eta\) of the domain is defined to be the supremum over these \(\tau\). Domains that admit a plurisubharmonic defining function (so that \(1\) is one of the \(\tau\) values, whence \(\eta=1\), but not necessarily vice versa!) enjoy desirable properties such as global regularity in the \(\overline{\partial}\)-Neumann problem and existence of a Stein neighborhood basis for the closure. It is thus of interest to understand to what extent quantified versions of such properties hold when \(\eta<1\). In turn, it is of interest to analyze how the boundary, specifically the interplay of the set of weakly pseudoconvex boundary points with the rest of the boundary, affects \(\eta\), or, conversely, what restrictions a specific value of \(\eta\) places on that interplay. The present paper studies this latter question for domains in \(\mathbb{C}^{2}\). The restriction to \(\mathbb{C}^{2}\) has the effect that the situation is at least somewhat manageable. Denote by \(L\) and \(N\) the normalized tangential and normal \((1,0)\) fields respectively. There are two main necessary conditions for the Diederich-Fornæss index to have a particular value. {Theorem 1.2.} Let \(\Omega\) be a bounded pseudoconvex domain with smooth boundary in \(\mathbb{C}^{2}\) and \(\Sigma\) be the Levi-flat set of \(\partial\Omega\). If the Diederich-Fornæss index is \(\eta_{0}\), then for any \(\eta<\eta_{0}\), there exists a smooth defining function \(\rho\) such that \[ 3+4C\frac{\|\nabla\rho\|}{\mathrm{Hess}_{\rho}(N,L)}+0.5\left|L\left(\frac{\|\nabla\rho\|}{\mathrm{Hess}_{\rho}(N,L)}\right)\right| \geq \frac{1}{1-\eta} \] on \(\Sigma\). Here \(\mathrm{Hess}_{\rho}\) denotes the complex Hessian of \(\rho\). The constant \(C\) is explicit in terms of quantities that characterize the interplay between \(\Sigma\) and the boundary. For the second condition, note that any defining function can be written as \(e^{\psi}\delta\), where \(\delta\) denotes the signed boundary distance. The second condition is given in terms of the function \(\psi\), and for index $1$. {Theorem 1.5.} Let \(\Omega\) be a bounded pseudoconvex domain with smooth boundary in \(\mathbb{C}^{2}\) and \(\Sigma\) be the Levi-flat set of \(\partial\Omega\). If the Diederich-Fornæss index is $1$, then there is a family of real smooth functions \(\psi_{n}\) defined on a neighborhood of \(\Sigma\) such that on all points of \(\Sigma\) \[ 0 \leq \left|\frac{1}{2}\overline{L}\psi_{n} + g(\nabla_{N}\nabla\delta, L)\right| \leq \mu_{n} + \nu_{n}|\mathrm{Hess}_{\psi_{n}}(L, L)|\;, \] for all $n\in\mathbb{N}$ where $\mu_{n}\;,\;\nu_{n}\rightarrow 0$ as $n\rightarrow\infty$. Here $g$ is the Euclidean metric, so that $g(\nabla_{N}\nabla\delta, L) = \mathrm{Hess}_{\delta}(N, L)$. Note that the quantities \(\mathrm{Hess}_{\rho}(N,L)\) and \((1/2)\overline{L}\psi + \mathrm{Hess}_{\delta}(N,L)\) are precisely the ones that are relevant for studying global regularity of the \(\overline{\partial}\)-Neumann problem via `good vector fields' or via (approximate) exactness of a certain \(1\)-form. This is not surprising, since these properties and plurisubharmonicity of a defining function, in an appropriate weak sense, are equivalent (see [the reviewer and \textit{M. K. Sucheston}, Monatsh. Math. 136, No. 3, 249--258 (2002; Zbl 1008.32023)]).