Global symplectic coordinates on gradient Kähler-Ricci solitons (Q368523)

The authors construct explicitly a global symplectomorphism between a gradient Kähler-Ricci soliton \((\mathbb{C}^n, \omega_{RS})\) and \((\mathbb{R}^{2n}, \omega_{0})\). They classify the totally geodesic submanifolds of \((\mathbb{C}^n, \omega_{C,n})\) through the origin and show that, up to unitary transformation of \(\mathbb{C}^n\), they must be \((\mathbb{C}^k, \omega_{C,k})\). As an example of gradient Kähler-Ricci solitons they consider \((\mathbb{C}^n, \omega_{C,n})\) -- the product of \(n\)-copies of the Cigar soliton -- where Ciriza's property holds and prove that there exists a symplectomorphism \(\psi_{C,n}\) between \((\mathbb{C}^n, \omega_{C,n})\) and \((\mathbb{R}^{2n}, \omega_{0})\) with the property that \(\psi_{C,n}(0)=0\) and \(\psi_{C,n}\) takes the complete complex totally geodesic submanifolds through the origin to complex linear subspaces of \(\mathbb{C}^n\).