Minimal submanifolds with multiple isolated singularities

Abstract

We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in ${\mathbb{R}}^{n+1}$ $(n\ge 7$) to arbitrary codimension and each $n\ge 3$. We then apply the procedure to copies of the Lawson-Osserman cone to produce a four dimensional minimal graph in ${\mathbb{R}}^7$ with any finite number of isolated singularities.

55 pages. Revised: This version completes the generalization of Smale's singular bridge principle to high codimension. The bridge constructions and many proofs have been simplified, and the introduction and closing remarks have been expanded. Comments welcome!