2021/05/16 by Marco A. M. Guaraco, Guaraco, Marco A. M., Vanderson Lima +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #math.GT
paper · pdf · doi:10.48550/arxiv.2105.07504
31 pages, 7 figures
arxiv created 2021/05/16 · openalex publication_date 2021/05/16 · arxiv updated 2021/05/18 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study global aspects of the mean curvature flow of non-separating hypersurfaces S in closed manifolds. For instance, if S has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface Γ. We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic 3-manifolds fibered over S1 admit smooth entire foliations whose leaves are either minimal or have non-vanishing mean curvature. We also conclude the existence of outermost minimal surfaces for quasi-Fuchsian ends and study their continuity with respect to variations of the quasi-Fuchsian metric.