On the Gromov non-hyperbolicity of certain domains in \(\mathbb{C}^n\) (Q2627841)

We say that a Kobayashi hyperbolic domain \(\Omega\subset\mathbb C^n\) is Gromov hyperbolic if there exists a \(\delta\geq0\) such that the Kobayashi metric \(d\) of \(\Omega\) satisfies the relation \((x,y)_w\geq\min\{(x,z)_w, (y,z)_w\}-\delta\), \(x,y,z,w\in\Omega\), where \((x,y)_w:=\frac12\big(d(x,w)+d(w,y)-d(x,y)\big)\). The main results of the paper are the following two theorems: -- Let \(\Omega\subset\mathbb C^n\) be a bounded convex domain and let \(S\) be a complex affine hyperplane with \(S\cap\Omega\neq\emptyset\). Then \(\Omega\setminus S\) is not Gromov hyperbolic. -- Let \(\Omega_\varphi(X):=\{(z,w)\in X\times\mathbb C: |w|<e^{-\varphi(z)}\}\), where \(X\subset\mathbb C^n\) is a bounded convex domain and \(\varphi\) is strictly plurisubharmonic on \(X\) and continuous on \(\overline X\). Then \(\Omega_\varphi(X)\setminus(X\times\{0\})\) is not Gromov hyperbolic.