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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.05148
- arXiv:
- arXiv:1901.05148
- Bibcode:
- 2019arXiv190105148W
- Keywords:
-
- Mathematics - Differential Geometry;
- 53A10 (primary);
- and 49Q05;
- 53C42 (secondary)
- E-Print:
- 21 pages. The revised version (Dec, 2019) has a new section about generic regularity of $2$-dimensional area minimizers