# De Giorgi's structure theorem and Federer's characterization theorem in higher codimension

We say that a $$\mathscr{H}^{m}$$-measurable set $$M\subset\mathbb{R}^n$$ is $$m$$-rectifiable if there are countably many Lipschitz maps $$f_j:\mathbb{R}^m\to \mathbb{R}^n$$ and a $$\mathscr{H}^m$$-null set $$M_0$$ such that $$M\subset M_0\bigcup f_j(\mathbb{R}^m)$$.

Let $$M\subset\mathbb{R}^n$$ be a $$\mathscr{H}^{m}$$-locally finite $$m$$-rectifiable set. Define the $$m$$-dimensional measure-theoretic interior, exterior and boundary of $$M$$ as follows. \begin{aligned} \text{Int}( M):=\{x\in\mathbb{R}^n \,|\,\lim_{r\to 0}\frac{\mathscr{H}^m(M\cap B_r(x))}{\omega_m r^m}=1\},\\ \text{Ext} (M):=\{x\in\mathbb{R}^n \,|\,\lim_{r\to 0}\frac{\mathscr{H}^m(M\cap B_r(x))}{\omega_m r^m}=0\},\\ \partial^* M:=\mathbb{R}^n\setminus (\text{Int}(M)\cup \text{Ext}(M)).\\ \end{aligned}

It can be proven that $$\mathscr{H}^m(M\Delta \text{Int}(M))=0$$ and that $$\mathscr{H}^m((\mathbb{R}^n\setminus M)\Delta \text{Ext}(M))=0$$ ($$\Delta$$ denotes the simmetric difference).

Now endow $$M$$ with a measurable orientation, so that $$M$$ becomes an oriented, $$\mathscr{H}^{m}$$-locally finite $$m$$-rectifiable set in $$\mathbb{R}^n$$. Denote with $$[[M]]$$ the current canonically associated to $$M$$. Note that then finite perimeter sets are just as the (obviously $$\mathscr{H}^n$$-locally finite and trivially oriented) $$n$$-rectifiable sets $$E\subset\mathbb{R}^n$$ such that $$\mathbb{M}(\partial[[E]])<\infty$$, where $$\partial$$ and $$\mathbb{M}$$ are respectively the boundary operator and the mass operator for currents. The following two results are well known.

(1) De Giorgi's Structure Theorem: If a measurable set $$E\subset\mathbb{R}^n$$ has finite perimeter, then $$\partial^*E$$ is an oriented $$(n-1)$$-rectifiable set such that $$\mathscr{H}^{n-1}(\partial^*E)<\infty$$ and $$\partial [[E]]=[[\partial^* E]]$$.

(2) Federer's Characterization Theorem: If a measurable set $$E\subset \mathbb{R}^n$$ is such that $$\mathscr{H}^{n-1}(\partial^*E)<\infty$$, then $$E$$ has finite perimeter.

It is natural to ask if $$(1)$$ and $$(2)$$ can be generalized to hold for oriented, $$\mathscr{H}^{m}$$-locally finite $$m$$-rectifiable sets $$M\subset\mathbb{R}^n$$. In fact, Federer and Fleming proved that De Giorgi's theorem generalizes to the following.

(1)' Boundary Rectifiability Theorem: If an oriented, $$\mathscr{H}^{m}$$-locally finite $$m$$-rectifiable set $$M\subset\mathbb{R}^n$$ satisfies $$\mathbb{M}(\partial[[M]])<\infty$$, then there exists an $$(m-1)$$-dimensional integer-multiplicity current $$S=\langle N,\theta,\xi\rangle$$ such that $$\partial [[M]]=S$$.

The notation $$S=\langle N,\theta,\xi\rangle$$ means that $$S$$ acts on test forms as the measure $$\mathscr{H}^{m-1}_{|N}(dx)\theta(x)\xi(x)$$, where $$N$$ is a $$(m-1)$$-rectifiable set, $$\theta$$ a measurable integer-value function positive and $$\mathscr{H}^{m-1}$$-locally integrable on $$N$$, and for $$\mathscr{H}^{m-1}$$-a.e. $$x$$ $$\xi(x)$$ is a $$(m-1)$$-multivector of norm $$1$$ orienting $$T_xN$$. An important observation is that the multiplicity $$\theta$$ must be integer-valued but might be different from $$1$$, so in general $$S$$ cannot be identified with a set.

I have two questions.

First question: Is it true that $$\mathscr{H}^{m-1}(N\Delta \partial^*M))=0$$ (i.e. that we can take $$N=\partial^*M$$ in the theorem)?

Second question: Is something known about the generalization of Federer's theorem? I would conjecture the following.

(2)' Conjecture. If an oriented, $$\mathscr{H}^{m}$$-locally finite $$m$$-rectifiable set $$M\subset\mathbb{R}^n$$ is such that $$\mathscr{H}^{m-1}(\partial^*M)<\infty$$, then $$\mathbb{M}(\partial [[M]])<\infty$$. More generally, if $$T=\langle M,\theta,\xi\rangle$$ is an $$m$$-dimensional integer-multiplicity current such that $$\mathscr{H}^{m-1}(\partial^* M)<\infty$$, then $$\mathbb{M}(\partial T)<\infty$$.

• What is a boundary in the sense of currents? (The linked paper by Harrison does not refer to currents.) And what would qualify a notion of boundary as measure-theoretic — what can it depend on or what transformations should preserve it? Jan 20, 2022 at 16:43
• If $T$ is a current that acts on compactly supported $m$-forms of $\mathbb{R}^n$, its boundary $\partial T$ is defined to be the current that acts on $k-1$-forms as $<\partial T, \omega >=<T,d\omega>$. If $M^m$ is a smooth submanifold of $\mathbb{R}^n$, can think of $M$ as a current $T_M$ via $<T_M,\omega>=\int_M \omega$. Stokes theorem says that $\partial T_M=T_{\partial M}$ (ie boundary in the sense of currents and topological boundary coincide). As a corollary of the result of Harrison, we know that this continues to hold when $M$ is assumed only Lipschitz. Jan 21, 2022 at 13:36
• I think notation needed in the body of the question should be in the question, not the comments. Jan 31, 2022 at 21:10
• Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already more than 20 times, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts. -- Usually, the vast majority of minor edits can be avoided by writing and proofreading a question or an answer (or any update to such) carefully before posting it. Sep 16, 2022 at 17:03