Mean curvature flow of graphs in warped products (Q2841362)

Let \(M\) be an \(n\)-dimensional manifold and \((\bar{M},\bar{g})\) an \((n+1)\)-dimensional Riemannian manifold.NEWLINENEWLINENEWLINEA map \(F: M \times [0,T[\to \bar{M}\) such that every \(F_t=F(\cdot ,t): M\to \bar{M}\) is an immersion is called the mean curvature flow of \(F_0\) if it is a solution of the equation NEWLINE\[NEWLINE\frac{\partial F}{\partial t}= \vec{H},NEWLINE\]NEWLINE where \(\vec{H}(\cdot ,t)\) is the mean curvature vector of the immersion \(F_t.\)NEWLINENEWLINENEWLINEIn this article, the authors consider a complete Riemannian manifold which is either compact or has a pole, a positive smooth function \(\varphi\) on \(M\) and the warped product \(M\times_\varphi\mathbb R\) to study the flow by the mean curvature of a locally Lipschitz continuous graph on \(M\). They prove that the flow in this case exists for all time, and that the evolving hypersurface is \(C^\infty\) for \(t>0\) and is a graph for all \(t\). By imposing on the graph on \(M\) the condition that its distance to \(M\) is bounded on it, and on \(M\) a condition on its sectional curvature related to the Hessian of \(\varphi\), they prove that these flow has a well-defined limit.