Estimate of the squeezing function for a class of bounded domains (Q1659910)

For a bounded domain \(\Omega\subset\mathbb C^n\) the squeezing function \(s_\Omega\) is defined by the formula: \[ s_{\Omega}(p):=\sup\big\{s_{\Omega,f}(p)\,|\, f:\Omega\longrightarrow\mathbb B(1) \text{ is a holomorphic embedding with } f(p)=0\big\} \] and \[ s_{\Omega,f}(p):=\sup\big\{r>0\,\big|\, \mathbb B(r)\subset f(\Omega)\big\}. \] The main result of the paper is the following theorem. Let \(\Omega\subset\mathbb C^3\) be a bounded domain and let \(q\in\partial\Omega\) be such that: -- \(\Omega\) is smooth and pseudoconvex in a neighborhood of \(q\), -- the Bloom-Graham type of \(\Omega\) at \(q\) is \(d\), -- the regular order of contact at \(q\) is greater than \(2d\) along two smooth complex curves not tangent to each other. Then the squeezing function \(s_\Omega\) has no uniform lower bound near \(q\).