Generic Transversality of Minimal Submanifolds and Generic Regularity of Two-Dimensional Area-Minimizing Integral Currents

Abstract

Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$, and that $\Gamma$ is a smooth submanifold of $N$. This paper proves that for a generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$, if $F$ is any simple $g$-minimal immersion of a closed manifold into N, then $F$ is transverse to $\Gamma$ 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.