Mean curvature flow with a forcing term in Minkowski space
The author investigates the mean curvature flow of graphs in Minkowski space \(\mathbb{R}^{n,1}\). More precisely he studies the quasilinear parabolic equation \[ \frac{\partial u}{\partial t}=\sqrt{1-| Du| ^2}\Big[ \text{div} (\frac{Du}{\sqrt{1-| Du| ^2}})-\mathcal{H}\Big] , \] where \(u=u(x,t)\) denotes the height function and \(\mathcal{H}=\mathcal{H}(x)\) the forcing term. For initial data given through a smooth hypersurface \(M_0=\text{ graph}(u_0)\), which is strictly space-like, and for a smooth forcing term \(\mathcal{H}\), he proves longtime existence of smooth solutions. Moreover, requiring the forcing term to be constant and the smooth space-like hypersurface \(M_0\) to be of bounded curvature, the following types of convergences are obtained: (1) If \(M_0\) intersects future null infinity in at least two directions, does not intersect past null infinity at all and if there exists a positive constant mean curvature upper barrier \(M_+\) with the same boundary at future null infinity as \(M_0\), then \(M_t=\text{graph}(u(x,t))\) converges under the flow to a constant mean curvature hypersurface which has the same boundary at future null infinity as \(M_0\). (2) If \(M_0\) does neither intersect future nor past null infinity at all, then \(M_t\) converges under the flow to a convex downward translating soliton.
scientific article; zbMATH DE number 5014202
scientific article