Minimal submanifolds with multiple isolated singularities

TL;DR

The work generalizes Smale's singular bridge principle to arbitrary codimension for $n$-dimensional strictly stable minimal cones with $n\ge3$, enabling the gluing of multiple cones via $\varepsilon$-bridges and perturbative analysis of the stability operator. It derives a full Dirichlet/linear theory (Schauder estimates, weighted spaces, asymptotics) and then solves a fixed-point problem to produce strictly stable minimal graphs with finitely many isolated singularities, preserving singularities and prescribing decay to tangent cones. A central novelty is the construction of graphical bridges that yield graphical minimal graphs even in high codimension, culminating in a four-dimensional graphical minimal graph in $\mathbb{R}^7$ with any finite number of isolated singularities, built from copies of the Lawson-Osserman cone. The results provide new explicit singular minimal graphs, sharpen the understanding of stability under gluing, and connect high-codimension bridge techniques with calibrated-geometry examples, significantly expanding the repertoire of known singular minimal submanifolds.

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.

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Corollary 3.2: Global $C^0$ Bound
• proof
• Corollary 3.3: Local $C^0$ Bound