2019/01/16 by Brian White, White, Brian · 2 citations
Mathematics · #53A10 (primary) #53C42 (secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #and 49Q05
paper · pdf · doi:10.48550/arxiv.1901.05148
openalex publication_date 2019/01/16 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28
Suppose that N is a smooth manifold with a smooth Riemannian metric g0, and that Γ is a smooth submanifold of N. This paper proves that for a generic (in the sense of Baire category) smooth metric g conformal to g0, if F is any simple g-minimal immersion of a closed manifold into N, then F is transverse to Γ and F is self-transverse. The theorem remains true with "transverse" and "self-transverse" replaced by "strongly transverse" and "strongly self-transverse". The theorem also holds for hypersurfaces of constant mean curvature or, more generally, of prescribed mean curvature. The paper also proves that for a generic ambient metric, every 2-dimensional surface (integral current or flat chain mod 2) without boundary that minimizes area in its homology class has support equal to a smoothly embedded minimal surface.