Boundary regularity of rectifiable multiplicity 1 hypercurrents

Question

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of it and the existing older publications. Since I furthermore cannot always bother specific smarter people with me not understanding and the problem is sadly quite important for me, I'm now passing to posting a question here.

Problem setting. My question deals with sufficient conditions for boundary regularity and under which circumstances they can be achieved (hence implying the corresponding regularity near a given regular boundary support). Moreover, this is for me only interesting in codimension 1, ie. we are dealing only with hypercurrents. I am aware of two particular papers about boundary regularity, namely one by Hardt (Comm. PDE) and one by Hardt-Simon (Ann. Math.), and also of some miscellaneous related works (eg. by Brothers about tangent cones). I also tried to consult the book of Francesco Maggi, which unfortunately performs the stunt of relating in its last sentence of the corresponding Part III to the more general works above that I do not get.

Terminology. We consider a rectifiable current $T$ in $ \mathbb{R}^n $ of codimension $1$. We assume moreover that $T$ has multiplicity $1$ almost everywhere. (We may also as well for my problem assume that $$ T=\partial [E] \llcorner \Omega, $$ ie. $T$ is generated as the restriction of an oriented boundary to some differentiable domain $\Omega$.) Let also $\psi$ denote an elliptic parametric integrand of sufficient differentiability on $\mathbb{R}^n$ and assume that $T$ is absolutely minimizing for $\psi$ on $\mathbb{R}^n$. Then let $B$ denote a sufficiently differentiable orientable submanifold of $\mathbb{R}^n$ of codimension 2 without boundary and assume $$ \partial T = [B] $$ with $B$ sufficiently oriented. Assuming that $0\in B$, the above mentioned work by Hardt (Thm. 3.6) assures that $T$ is actually a manifold with boundary near $0$ given $$ \Theta^{n-1}_*(T,a) \leq \frac12, $$ ie. we may bound at least the lower density by what one would expect for a manifold with boundary.

Question. Can one always verify the density-estimate in this case of a multiplicity 1 current? Or can one do this under the assumption of dealing with an oriented boundary? Are there other sufficient conditions on $T$ implying such an estimate in this case (ie. codimension and multiplicity one)?

From what I have read and heard, it should be possible (and intuitively makes sense but that is always very dangerous), but I am kind of blindsided on how to get the tech right (and even what to use to get it right). Thanks to everyone in advance.

Answer 1

Edit: I just realised that I misread your question; let me correct my answer accordingly.

When $\psi$ is the area-functional the paper by Hardt-Simon that you cite seems to answer your question, by giving a complete boundary regularity theorem in codimension one. To paraphrase this result here, it states that if $U$ is an open subset of $\mathbf{R}^{n+1}$, $0 < \alpha < 1$, $T$ is an $n$-dimensional area-minimising current, and $\partial T$ is a connected oriented embedded $C^{1,\alpha}$ submanifold of $U$, then $V \cap \mathrm{spt} \, T$ is a connected embedded $C^{1,\alpha}$ hypersurface with boundary. In particular, near a boundary point $X \in U \cap \mathrm{spt} \partial T$ the Allard-type condition $\Theta(T,X) = 1/2$ is a posteriori satisfied.

Moreover Hardt-Simon remark in the introduction that it when $\psi$ is a general elliptic integrand then the boundary regularity is not known even in dimension $n = 2$, although this might well have changed since then.

I am bit confused about the part of your question relating to the orientability of the boundary, as working with currents imposes this hypothesis. If you are interested in the 'unoriented' setting, you should consult the literature on flat chains mod two.

Although you claim to only be interested in the codimension one case, let me finally point out that Hardt-Simon's boundary regularity result does not extend to higher codimension, even with multiplicity one. White gives the example of $\{ (z^3,z^4) \mid \mathrm{Im} \, z \geq 0 \} \subset \mathbf{C}^2 = \mathbf{R}^4$, which is area-minimising but has a boundary singularity at the origin.