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
paper · pdf · doi:10.48550/arxiv.2105.07504
openalex publication_date 2021/05/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study global aspects of the mean curvature flow of non-separating\nhypersurfaces S in closed manifolds. For instance, if S has non-vanishing\nmean curvature, we show its level set flow converges smoothly towards an\nembedded minimal hypersurface \Γ. We prove a similar result for the flow\nwith surgery in dimension 2. As an application we show the existence of\nmonotone incompressible isotopies in manifolds with negative curvature.\nCombining this result with min-max theory, we show that quasi-Fuchsian and\nhyperbolic 3-manifolds fibered over \S1 admit smooth entire\nfoliations whose leaves are either minimal or have non-vanishing mean\ncurvature. We also conclude the existence of outermost minimal surfaces for\nquasi-Fuchsian ends and study their continuity with respect to variations of\nthe quasi-Fuchsian metric.\n