Rotationally symmetric translating solitons in \(\mathbb{C}^2\) (Q6634427)

Let \((M,g)\) be a Kähler-Einstein surface. If \(\Sigma\) is a smooth real surface in \(M\) and \(\omega\) is the Kähler form on \(M\) then the Kähler angle of \(\Sigma\) in \(M\) is defined by \(\omega_{\Sigma}=\cos\alpha d\mu_{\Sigma}\) where \(d\mu_{\Sigma}\) is an area element induced by the Kähler metric \(g\). \(\Sigma\) is called symplectic if \(\cos\alpha>0\) and holomorphic if \(\cos\alpha=1\).\N\NA surface \(\Sigma\) in \(\mathbb C^2\) is called a translating soliton if there exists a constant nonzero vector \(T\in\mathbb C^2\), such that \(T^{\perp}= H\) (where \(T^{\perp}\) is the projection of \(T\) onto normal bundle of \(\Sigma\)) holds on \(\Sigma\), where H is the mean curvature vector of \(\Sigma\) in \(\mathbb C^2\).\N\NThe author studies complete, rotationally symmetric solitons in \(\mathbb C^2\) which are represented by \[F(r,\theta)=(h(r)\cos\theta, h(r)\sin\theta, f(r),g(r)), (r,\theta)\in [0,R_0)\times [0,2\pi]\] where \(h,f,g\in C^2(0,R_0)\) for some \(R_0>0\) and \(h(r)>0\) for \(r>0\) and \(h(0)=0\).\N\NThe main theorem the author proves is\N\NTheorem. For any given \(\delta>0\), if a complete, rotationally symmetric translating soliton in \(\mathbb C^2\) satisfies \(\cos\alpha\ge\delta\), then it must be a plane.\N\NAs a consequence the author proves also that any complete, rotationally symmetric soliton in \(\mathbb C^2\) cannot arise as a finite time blow up limit of symplectic mean curvature flow.