Minimal submanifolds with multiple isolated singularities

TL;DR

This paper extends the singular bridge principle for minimal cones to higher codimensions and constructs minimal graphs in 7 with multiple isolated singularities, advancing understanding of minimal submanifolds.

Contribution

It generalizes Smale's principle to arbitrary codimension and applies it to produce minimal graphs with multiple singularities in higher dimensions.

Findings

Extended the singular bridge principle to arbitrary codimension

Constructed minimal graphs in 7 with any finite number of isolated singularities

Demonstrated the existence of complex minimal submanifolds with prescribed singularities

Abstract

We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. 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.