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

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.