Mean curvature type flows of graphs with nonzero Neumann boundary data in product manifold \(M^n \times \mathbb{R} \) (Q2235926)

Let \((M^n,\sigma)\), \(n\ge 2\), be an \(n\)-dimensional complete Riemannian manifold with metric \(\sigma\) and let \(\Omega\subset M^n\) be a bounded domain with \(C^3\) boundary. The authors study the evolution of a graphic hypersurface of the form \((x,u(x,t))\), \(x\in\Omega\), \(t>0\), in the product space \(M^n\times\mathbb{R}\) with a nonparametric mean curvature type flow. They prove a gradient estimate for the flow under suitable conditions. As a consequence long time existence for the flow is obtained.