On the Bergman distance on model domains in \(\mathbb C^n\) (Q2787088)

Let \(D:=\{z=(z_1,z')\in\mathbb C^n: \text{Re}z_1+P(z')<0\}\) be the ``model domain'', where \(P\) is a plurisubharmonic polynomial such that: {\parindent=6mm \begin{itemize}\item[--] there exists \(m_2,\dots,m_n, d\in\mathbb N\) for which \(P(t^{\frac1{2m_2}}z_2,\dots,t^{\frac1{2m_n}}z_n)=t^dP(z')\) for any \(t>0\), \item[--] there exists a \(c_0>0\) such that the function \(P-c_0\sigma\) is plurisubharmonic, where \(\sigma(z'):=\sum_{j=2}^n|z_j|^{2m_j}\). NEWLINENEWLINE\end{itemize}} The author proves the following lower estimate for the Bergman distance \(d_B^B\).NEWLINENEWLINEThere exists a universal constant \(C_\ast>0\) such that \(d_D^B(A,Q)\geq C_\ast\big(\varrho_D(A,Q)+\varrho(Q,A)\big)\), \(A,Q\in D\), where: NEWLINE\[NEWLINE \begin{gathered} \varrho(A,Q):=\frac{\log(1+\frac{\delta(A,Q)}{1+\mathcal L(A)})}{1+\log\log(e^2+\frac{\delta(A,Q)}{1+\mathcal L(A)})+\log(1+\mathcal L(A))},\\ \delta(A,Q):=\frac{|h_A(Q)|}{\mathcal R(A)}+\sum_{\ell=2}^n\frac{|Q_\ell-A_\ell|}{|r(A)|^{\frac1{2m_\ell}}},\\ h_A(Q):=Q_1-A_1+2\sum_{\ell=2}^n\frac{\partial P(A')}{\partial z_\ell}(Q_\ell-A_\ell),\\ r(A):=\text{Re}A_1+P(A'), \quad \widehat\sigma(A):=|A_1|+\sigma(A'),\\ \mathcal R(A):=\sum_{\ell=1}^n\widehat\sigma(A)^{1-\frac1{2m_\ell}}|r(A)|^{\frac1{2m_\ell}},\quad m_1:=1,\quad \mathcal L(A):=\log\Big(1+\frac{\widehat\sigma(A)}{|r(A)|}\Big). \end{gathered} NEWLINE\]NEWLINENEWLINENEWLINEIt is an open problem whether \(\mathcal R(A)\) can be replaced by \(|r(A)|\).NEWLINENEWLINEIn particular, NEWLINE\[NEWLINE d_D^B(A,Q)\gtrsim\frac{\log(1+\frac1{|r(A)|})}{\log\log(e^2+\frac1{|r(A)|})} \text{ when } A\longrightarrow\zeta\in\partial D. NEWLINE\]