Two-dimensional graphs moving by mean curvature flow
The authors consider the motion of an immersed surface in a 4-dimensional manifold \(M\), \({\mathbf F}_0 : \Sigma\to M\), moving by its mean curvature in \(M\), i.e., a one-parameter family \({\mathbf F}_t ={\mathbf F}(\cdot,t)\) of surfaces with corresponding images \(\Sigma_t = {\mathbf F}_t(\Sigma)\) is considered such that \[ \frac{d}{dt}{\mathbf F}(x,t)={\mathbf H}(x,t),\quad {\mathbf F}(x,0)={\mathbf F}_0(x),\tag{1} \] where \({\mathbf H}(x,t)\) is the mean curvature vector of \(\Sigma_t\) at \({\mathbf F}(x,t)\). The initial surface \(\Sigma_0\) is a graph in \(\mathbb R^4\), if there exists a unit constant 2-form \(\omega\) in \(\mathbb R^4\) such that \(v=(e_1\wedge e_2,\omega) \geq v_0 > 0\) for some constant \(v_0\), where \(\{e_1,e_2\}\) is an orthonormal frame on \(\Sigma_0\). The authors prove that if \(v_0\geq\frac{1}{\sqrt{2}}\) on \(\Sigma_0\), then the initial problem (1) has a global solution \({\mathbf F}\) and the scaled surfaces converge to a self-similar solution. A surface \(\Sigma_0\) is a graph in \(M_1\times M_2\), where \(M_1\) and \(M_2\) are Riemann surfaces, if \((e_1\wedge e_2,\omega_1) \geq v_0 > 0\), where \(\omega_1\) is a Kähler form on \(M_1\). It is proved that, if \(M\) is a Kähler-Einstein surface with scalar curvature \(R\), \(v_0\geq\frac{1}{\sqrt{2}}\) on \(\Sigma_0\), then the mean curvature flow has a global solution and it sub-converges to a minimal surface. If, in addition, \(R\geq 0\), it converges to a totally geodesic surface which is holomorphic.
estimation of the second fundamental form
scientific article; zbMATH DE number 1885821