2023/09/28 by Fujihara, Naotoshi
#53C44 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2309.16449
We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and ℝ. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.