Modified mean curvature flow of graphs in Riemannian manifolds

We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for this extrinsic flow with smooth initial data.

💡 Research Summary

The paper studies a modified mean curvature flow (M‑MCF) of graphs in a class of warped product manifolds endowed with a non‑vanishing Killing vector field. Let (\bar M) be a complete, non‑compact Riemannian manifold that carries a Killing field (X) whose orthogonal distribution is integrable. Choosing an integral leaf (M) of this distribution and using the flow (\Phi) generated by (X), one obtains a global isometry (\Phi: M\times\mathbb R\to\bar M). In the induced coordinates ((x,s)) the ambient metric takes the warped product form \