An essential one sided boundary singularity for a $3$-dimensional area minimizing current in $\mathbb{R}^5$

TL;DR

This paper constructs a 3-dimensional area minimizing current in 5-dimensional space with a boundary singularity, demonstrating the sharpness of a boundary regularity theory and introducing a flexible technique for creating singular examples.

Contribution

It provides a new example of boundary singularity in area minimizing currents, confirming the optimality of existing boundary regularity results and developing a boundary regularity theory for currents with complex boundary topology.

Findings

Constructed a current with a boundary singularity of multiplicity 2.

Showed the boundary regularity theory is dimensionally sharp.

Developed a boundary regularity theory for currents with boundaries meeting along a submanifold.

Abstract

We construct a $3$-dimensional area minimizing current $T$ in $\mathbb{R}^5$ whose boundary contains a real analytic surface of multiplicity $2$ at which $T$ has a density $1$ essential boundary singularity with a flat tangent cone. This example shows that the boundary regularity theory we developed with Reinaldo Resende in another paper, which extends Allard's classical boundary regularity result to higher boundary multiplicity, is dimensionally sharp. The construction of $T$ relies on the prescription of boundary data with non-trivial topology, which makes it a flexible technique and gives rise to a wide family of singular examples. In order to understand the examples, we develop a boundary regularity theory for a class of area minimizing $m$-dimensional currents whose boundary consists of smooth $(m-1)$-dimensional surfaces with multiplicities meeting along an $(m-2)$-dimensional smooth submanifold.