Relate the Minkowski boundary measure to the singular first variation

Determine how the total mass of the boundary measure associated with an integral varifold through the modified Minkowski contents and the Hahn–Banach-selected Radon measure \(\gamma^V\) relates, in general, to the mass of the singular part \(\|\delta V\|_{\mathrm{sing}}\) of the varifold’s first variation.

Background

For an integral varifold with locally bounded first variation, the paper identifies the generalized boundary set with the support of the singular part of the first variation, $\operatorname{spt}(\|\delta V\|_{\mathrm{sing}})$. It defines upper and lower modified Minkowski contents of this set and uses the Hahn–Banach theorem to obtain a Radon measure $\gamma^V$ lying between those contents. The measure $\gamma^V$ serves as the boundary measure in a boundary Michael–Simon–Sobolev inequality and in the resulting isoperimetric inequality.

Theorem 3 resolves the relationship in a special case where the generalized boundary is a compact smooth $(n-1)$-dimensional submanifold and an appropriate modified normal field exists: the upper and lower Minkowski contents coincide, $\gamma^V$ is unique, and it is bounded above by a multiple of $\|\delta V\|_{\mathrm{sing}}$. The general relationship between these notions of boundary mass remains unresolved.