Slicing of currents

Intuition Beginner

A topographic map shows a mountain by drawing its contour lines: the curves where the land sits at a fixed height. Each contour is a slice of the mountain at one altitude. If you knew every contour, and knew how they stack, you could rebuild the whole mountain. Slicing of currents is this same idea, carried over to the flexible surfaces called currents that you met when surfaces became reporting machines.

Start with a surface and a height function, a rule that assigns a number to every point of space. The level sets of that rule are the places where the height equals a fixed value. Each level set cuts your surface along a curve, one dimension lower than the surface. That cut curve, with its own orientation and counting, is the slice of the surface at that height. A two-dimensional surface sliced at a height gives a one-dimensional curve; a curve sliced at a height gives a few points.

Why bother chopping a surface into slices? Because a slice is simpler than the whole, being one dimension lower, and because hard facts about surfaces can be proved by checking them on every slice and then stacking the answers back up. If you can settle a question for curves, slicing lets you settle the matching question for surfaces, one altitude at a time. This is the engine behind many proofs in the theory of least-area surfaces.

The two facts that make slicing trustworthy are bookkeeping facts. First, the slices do not waste material: adding up the sizes of all the slices, across every height, gives back no more than the size of the original surface, scaled by how steeply the height function climbs. So most heights give a slice of sensible size, and only a few stray heights misbehave. Second, the edge of a slice and the slice of an edge agree up to a sign, so cutting and bounding cooperate rather than fight.

Visual Beginner

Picture a tilted sheet of paper floating in space, and let the height be the up-down coordinate. Sweep a horizontal plane upward through the sheet. At each height the plane meets the sheet along a straight line segment. That segment is the slice of the sheet at that height. As the plane rises from the bottom edge of the sheet to the top edge, the segment slides across the sheet, and the whole family of segments stacked together fills the sheet exactly. The sheet is the integral of its slices.

The centre panel shows the bookkeeping. If you measure the length of each segment and add those lengths over all heights, weighting by how fast the plane climbs through the sheet, you recover the area of the sheet. A steep sheet has short slices spread over a large height range; a shallow sheet has long slices packed into a small height range. The slope of the height function is the conversion factor between slice length summed over height and total area.

The right panel shows how edges behave. The endpoints of a slice segment sit exactly where the cutting plane meets the boundary of the sheet. So if you first take the boundary of the sheet and then slice it, you land on the same endpoints you get by first slicing the sheet and then taking the boundary of the resulting segment. Cutting and bounding commute, up to a sign that records orientation. This is the rule that lets slicing reduce a question about a surface and its rim to the same question about a curve and its endpoints.

Worked example Beginner

We slice a flat sheet by its height and check that the slice lengths add up to the area. Take the unit square sheet sitting in the plane with corners at $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, and let the height be the second coordinate, the value $y$ at the point $(x,y)$. The level set at height $t$ is the horizontal line $y=t$. We track the slices as $t$ runs from $0$ to $1$.

Step 1. Find the slice at height $t$. For a height $t$ between $0$ and $1$, the line $y=t$ crosses the square along the segment from $(0,t)$ to $(1,t)$. This segment has length $1$. For heights below $0$ or above $1$ the line misses the square, so the slice is empty and has length $0$.

Step 2. Add the slice lengths over all heights. Each height from $0$ to $1$ contributes a slice of length $1$, and the range of heights that hit the square has total span $1$. Adding length $1$ across a height range of span $1$ gives $1\times 1=1$.

Step 3. Compare with the area and the slope. The square has area $1$. The height function here is the plain coordinate $y$, which climbs at rate $1$: moving straight up one unit raises the height by one unit. So the slope factor is $1$, and the slice total $1$ equals the area $1$ times the slope factor $1$. The books balance.

Step 4. Tilt the height to see the slope factor work. Replace the height by twice the second coordinate, the value $2y$. Now the level set $2y=t$ is the line $y=t/2$, and as $t$ runs from $0$ to $2$ the slices still each have length $1$, but the height range that hits the square now has span $2$. The slice total is $1\times 2=2$, which equals the area $1$ times the new slope factor $2$. Steepening the height doubled the height range and so doubled the slice total.

What this tells us: summing slice lengths over height recovers the area, once you account for how fast the height climbs. A steeper height spreads the same area over a wider band of heights, so each individual slice stays the same length but there are effectively more of them. This balance is the heart of the slicing rule, and it holds for wild surfaces as well as for flat sheets.

Check your understanding Beginner

Formal definition Intermediate+

Fix integers $1\le k\le n$. Let $T\in {\mathcal{N}}_k({\mathbb{R}}^n)$ be a normal $k$-current, so $\mathbf{M}(T)<\infty$ and $\mathbf{M}(\partial T)<\infty$ in the sense of 02.13.07, and let $f:{\mathbb{R}}^n\to \mathbb{R}$ be Lipschitz. For a Borel set $A\subseteq {\mathbb{R}}^n$ the restriction $T└A$ is the current with mass measure $\Vert T\Vert └A$ and the same orientation, characterised on forms by $(T└A)(\omega )={\int }_A⟨\omega ,\vec{T}⟩d\Vert T\Vert$ ^{[Federer 1969 §4.3]}. The restriction lowers mass, $\mathbf{M}(T└A)\le \mathbf{M}(T)$, and equals $T$ when $A\supseteq \mathrm{spt}T$.

The slice of $T$ by $f$ at level $t\in \mathbb{R}$ is the $(k-1)$-current $$⟨T,f,t⟩=\partial (T└\{f<t\})-(\partial T)└\{f<t\}.$$ The right-hand side is defined for every $t$, since $T└\{f<t\}$ and $(\partial T)└\{f<t\}$ are currents whenever the level set is $\Vert T\Vert$-measurable, which holds for ${\mathcal{L}}^1$-a.e. $t$ because the boundaries $\{f=t\}$ are pairwise disjoint and so charge $\Vert T\Vert +\Vert \partial T\Vert$ at only countably many $t$. The defining combination is exactly the correction that strips the artificial boundary introduced by the cut $\{f<t\}$ and leaves the geometric cut $T\cap \{f=t\}$: one has $\mathrm{spt}⟨T,f,t⟩\subseteq \mathrm{spt}T\cap \{f=t\}$ ^{[Simon 1983 §28]}.

An equivalent description identifies the slice as a derivative in $t$. Writing $T_{<t}=T└\{f<t\}$, the map $t\mapsto T_{<t}$ is nondecreasing in mass, and $$⟨T,f,t⟩=\frac{d}{dt}(\partial T_{<t}-(\partial T)└\{f<t\})$$ in the sense that ${\int }_a^b⟨T,f,t⟩dt=⟨T└\{a<f<b\},f,\cdot ⟩$ assembles back to the restricted current; the precise reconstruction is the coarea formula stated below.

The two structural identities hold for ${\mathcal{L}}^1$-a.e. $t$: $$\partial ⟨T,f,t⟩=-⟨\partial T,f,t⟩,⟨T,f,t⟩=\partial T_{<t}-(\partial T{)}_{<t},$$ where $(\partial T{)}_{<t}=(\partial T)└\{f<t\}$. The first says the boundary operator anticommutes with slicing; it is the current-theoretic counterpart of the fact that the boundary of a level cut and the level cut of a boundary differ only by the geometric slice ^{[Federer 1969 §4.3]}.

A current $T$ is integer-multiplicity rectifiable ($T\in {\mathcal{R}}_k$) as in 02.13.07; the central question of this unit is for which $t$ the slice $⟨T,f,t⟩$ inherits that structure, answered in the next section.

Counterexamples to common slips Intermediate+

• The naive cut $\partial (T└\{f<t\})$ is not the slice. The boundary of the cut current contains both the geometric slice along $\{f=t\}$ and the part of $\partial T$ that already lay below the level. Subtracting $(\partial T)└\{f<t\}$ removes that spurious second piece. Forgetting the correction term gives a current supported partly on $\mathrm{spt}\partial T$, not on $\{f=t\}$.

• Slicing is not defined for every level $t$, only almost every level. At a level $t$ where $\Vert T\Vert$ or $\Vert \partial T\Vert$ charges the hyperplane $\{f=t\}$ with positive mass, the restriction $T└\{f<t\}$ jumps and the slice need not have finite mass. Only countably many such bad levels exist, so the slice is a finite-mass $(k-1)$-current for ${\mathcal{L}}^1$-a.e. $t$.

• The coarea factor uses the Lipschitz constant, not the pointwise gradient norm. The mass bound $\int \mathbf{M}(⟨T,f,t⟩)dt\le \mathrm{Lip}(f)\mathbf{M}(T)$ carries the global Lipschitz constant. The sharper coarea formula with the pointwise factor $|\nabla f|$ along the approximate tangent plane is an equality, but the inequality with $\mathrm{Lip}(f)$ is what is used in the closure-theorem induction.

• Rectifiability of slices is an almost-everywhere statement, never a for-every statement. Even for a smooth $T$, an exceptional level $t$ can produce a slice that is not rectifiable of the expected dimension, for instance when $\{f=t\}$ is tangent to $\mathrm{spt}T$ along a positive-measure set. The theorem controls the slice for ${\mathcal{L}}^1$-a.e. $t$ only.

Key theorem with proof Intermediate+

Theorem (coarea estimate and rectifiability of slices; Federer 1969 §4.3). Let $T\in {\mathcal{R}}_k({\mathbb{R}}^n)$ be integer-multiplicity rectifiable with $\mathbf{M}(T)<\infty$, and let $f:{\mathbb{R}}^n\to \mathbb{R}$ be Lipschitz. Then for ${\mathcal{L}}^1$-a.e. $t\in \mathbb{R}$ the slice $⟨T,f,t⟩$ is an integer-multiplicity rectifiable $(k-1)$-current supported in $\mathrm{spt}T\cap \{f=t\}$, and $${\int }_{\mathbb{R}}\mathbf{M}(⟨T,f,t⟩)dt\le \mathrm{Lip}(f)\mathbf{M}(T).$$ If in addition $\mathbf{M}(\partial T)<\infty$, then for ${\mathcal{L}}^1$-a.e. $t$ one has $\partial ⟨T,f,t⟩=-⟨\partial T,f,t⟩$, and the slice is an integral current.

Proof. The argument represents $T$ by its rectifiable data, applies the classical coarea formula on the rectifiable carrier, and reads off both the mass bound and the rectifiable structure of the slice.

Step 1 (rectifiable representation). By the structure of 02.13.07, $T(\omega )={\int }_M⟨\omega ,\xi ⟩\theta d{\mathcal{H}}^k$ for a countably $k$-rectifiable set $M$, an orienting unit simple $k$-vector field $\xi$ on the approximate tangent planes $T_{x}M$, and an integer multiplicity $\theta \in L^1({\mathcal{H}}^k└M;\mathbb{Z})$. The restriction to $\{f<t\}$ replaces $M$ by $M\cap \{f<t\}$, so $T_{<t}$ is again integer-multiplicity rectifiable, carried by $M\cap \{f<t\}$ with the same $\xi ,\theta$.

Step 2 (the slice as a rectifiable current on the level set). The restriction of the Lipschitz function $f$ to the rectifiable set $M$ is Lipschitz, and by Rademacher's theorem it has an approximate tangential gradient ${\nabla }^{M}f$ defined ${\mathcal{H}}^k$-a.e. on $M$. The coarea formula for the Lipschitz map $f{|}_M$ on the rectifiable set $M$ gives, for every nonnegative Borel $g$, $${\int }_{M}g(x)|{\nabla }^{M}f(x)|d{\mathcal{H}}^k(x)={\int }_{\mathbb{R}}\left({\int }_{M\cap \{f=t\}}gd{\mathcal{H}}^{k-1}\right)dt,$$ so for ${\mathcal{L}}^1$-a.e. $t$ the set $M\cap \{f=t\}$ is countably $(k-1)$-rectifiable with finite ${\mathcal{H}}^{k-1}$-measure ^{[Federer 1969 §4.3]}. Define the candidate slice as integration over $M\cap \{f=t\}$ with multiplicity $\theta$ and the orientation ${\xi }^{\prime }$ obtained by contracting $\xi$ with the unit conormal ${\nabla }^{M}f/|{\nabla }^{M}f|$ along the level set. This candidate is integer-multiplicity rectifiable of dimension $k-1$ by construction, and a direct computation against test forms shows it equals $\partial T_{<t}-(\partial T{)}_{<t}$ for a.e. $t$, hence equals $⟨T,f,t⟩$.

Step 3 (the coarea mass estimate). Applying the coarea formula of Step 2 with $g=\theta$ (nonnegative because $|\theta |$ is the multiplicity of the slice) gives $${\int }_{\mathbb{R}}\mathbf{M}(⟨T,f,t⟩)dt={\int }_{\mathbb{R}}{\int }_{M\cap \{f=t\}}|\theta |d{\mathcal{H}}^{k-1}dt={\int }_M|\theta ||{\nabla }^{M}f|d{\mathcal{H}}^k.$$ The tangential gradient is pointwise bounded by the Lipschitz constant, $|{\nabla }^{M}f|\le \mathrm{Lip}(f)$ ${\mathcal{H}}^k$-a.e., so the last integral is at most $\mathrm{Lip}(f){\int }_M|\theta |d{\mathcal{H}}^k=\mathrm{Lip}(f)\mathbf{M}(T)$. This is the coarea estimate, and it shows in particular that $\mathbf{M}(⟨T,f,t⟩)<\infty$ for a.e. $t$.

Step 4 (boundary identity and integrality). Taking the boundary of the defining identity and using $\partial \partial =0$, $$\partial ⟨T,f,t⟩=\partial \partial T_{<t}-\partial (\partial T{)}_{<t}=-\partial ((\partial T)└\{f<t\}).$$ Applying the slice formula one dimension down to $\partial T$ in place of $T$, the right-hand side is $-(\partial ((\partial T{)}_{<t}))$, which differs from $-⟨\partial T,f,t⟩=-(\partial ((\partial T{)}_{<t})-(\partial \partial T{)}_{<t})$ only by the vanishing term $(\partial \partial T{)}_{<t}=0$. Hence $\partial ⟨T,f,t⟩=-⟨\partial T,f,t⟩$ for a.e. $t$. When $\mathbf{M}(\partial T)<\infty$, the coarea estimate applied to $\partial T$ shows $\mathbf{M}(\partial ⟨T,f,t⟩)=\mathbf{M}(⟨\partial T,f,t⟩)<\infty$ for a.e. $t$, and Step 2 applied to $\partial T$ makes this boundary integer-multiplicity rectifiable. Thus $⟨T,f,t⟩$ has finite mass, finite boundary mass, and integer-multiplicity rectifiable boundary, so it is an integral $(k-1)$-current for a.e. $t$. $\square$

Bridge. The slicing theorem builds toward the inductive architecture of geometric measure theory: it reduces a statement about $k$-currents to a family of statements about $(k-1)$-currents, and the coarea estimate guarantees the reduction loses no more than a Lipschitz factor of mass. The boundary identity $\partial ⟨T,f,t⟩=-⟨\partial T,f,t⟩$ is exactly what makes the induction close, because it lets the boundary of a slice be analysed as the slice of a boundary. This appears again in the structure-theorem proof of 02.13.07, where slicing by coordinate functions is the dimension-reduction step, and it generalises to vector-valued Lipschitz maps $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, where iterated slicing cuts the dimension by $m$. The foundational reason the closure theorem holds is that rectifiability is a slice-by-slice property: a normal current whose slices are a.e. integer-multiplicity rectifiable is itself integer-multiplicity rectifiable, and the bridge is the coarea estimate, which certifies that almost every slice is controlled.

Exercises Intermediate+

Advanced results Master

Slicing extends from scalar to vector-valued Lipschitz maps. For $f=(f_1,\dots ,f_m):{\mathbb{R}}^n\to {\mathbb{R}}^m$ with $m\le k$ and $y\in {\mathbb{R}}^m$, the iterated slice $⟨T,f,y⟩=⟨\cdots ⟨⟨T,f_1,y_1⟩,f_2,y_2⟩,\cdots ,f_m,y_m⟩$ is a $(k-m)$-current, well defined for ${\mathcal{L}}^m$-a.e. $y$, supported in $\mathrm{spt}T\cap f^{-1}\{y\}$, with the coarea bound $${\int }_{{\mathbb{R}}^m}^{\ast }\mathbf{M}(⟨T,f,y⟩)dy\le (\mathrm{Lip}f{)}^m\mathbf{M}(T)$$ and the boundary rule $\partial ⟨T,f,y⟩=(-1{)}^m⟨\partial T,f,y⟩$ ^{[Federer 1969 §4.3]}. The iteration is independent of the order of the $f_j$ for a.e. $y$, a Fubini-type fact that rests on the a.e.-defined approximate Jacobian of $f{|}_M$. When $m=k$ the slices are $0$-currents, finite integer combinations of points whose total weight is the integer-valued degree of $f{|}_M$ over $y$; this is the current-theoretic source of the Brouwer-degree and the multiplicity counts in the area formula.

The reconstruction direction realises a current as the integral of its slices. For $T\in {\mathcal{R}}_k$ and Lipschitz $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $$T└(\varphi \circ f)={\int }_{{\mathbb{R}}^m}\varphi (y)⟨T,f,y⟩\wedge dfdy$$ holds for bounded Borel $\varphi$, exhibiting $T$ as a superposition of its slices weighted by the Jacobian factor $df=d{f}_1\wedge \cdots \wedge d{f}_m$ along the tangent planes; the precise statement is Federer's coarea formula for currents ^{[Federer 1969 §4.3]}. This is the integral-geometric content behind the slicing characterisation of rectifiability: a normal current is integer-multiplicity rectifiable precisely when its slices by a generic family of Lipschitz functions are a.e. integer-multiplicity rectifiable, since rectifiability of the carrier is detected slice by slice through the approximate tangent planes ^{[Simon 1983 §28]}.

Three structural theorems descend from slicing. The boundary-rectifiability theorem of 02.13.07 is proved by inducting on $k$: slicing by a coordinate function reduces a $k$-dimensional integrality question to a family of $(k-1)$-dimensional ones, with the boundary identity $\partial ⟨T,u,t⟩=-⟨\partial T,u,t⟩$ closing the induction. The constancy theorem follows similarly, since a boundaryless top-dimensional current has slices that are boundaryless one dimension down, and the slice multiplicities, being locally constant on connected level sets, force global constancy. The Federer-Fleming closure theorem — the flat limit of integral currents with uniformly bounded mass and boundary mass is integral — uses slicing as its rectifiability criterion: the slices of the flat limit are flat limits of the slices, which are integer-multiplicity rectifiable, and the slicing characterisation transports rectifiability to the limit ^{[Federer-Fleming 1960]}.

Slicing also drives the quantitative regularity theory of mass-minimisers. Monotonicity of the density ratio $r\mapsto r^{-k}\mathbf{M}(T└B_r(x))$ is differentiated against the radial Lipschitz function $f(z)=|z-x|$, and the spherical slices $⟨T,f,r⟩$ on the spheres $\partial B_r(x)$ are exactly the boundary terms in the first-variation computation; the coarea estimate controls them and yields the lower density bound ${\Theta }^k(\Vert T\Vert ,x)\ge 1$ at points of an integral current's support ^{[Simon 1983 §28]}. The same spherical slices construct the tangent cones at singular points, since the blow-ups $T_{x,\lambda }$ have slices converging to the slices of the cone, and the dimension reduction bounding the singular set proceeds by slicing off one dimension at a time.

Synthesis. Slicing is the dimension-reduction operation of geometric measure theory, and the coarea estimate is the conservation law that makes the reduction lossless up to a Lipschitz factor. The slice builds toward the inductive proofs of boundary rectifiability, constancy, and closure, each of which trades a $k$-dimensional question for a measured family of $(k-1)$-dimensional ones; the boundary identity $\partial ⟨T,f,t⟩=-⟨\partial T,f,t⟩$ is exactly what lets the induction commute with the boundary operator, and it is dual to the way the exterior derivative commutes with restriction in the de Rham complex. The coarea estimate identifies the total slice mass with the gradient-weighted mass of the carrier, which generalises the elementary contour-summation of the flat sheet to arbitrary rectifiable currents and Lipschitz functions. Putting these together, the slicing characterisation of rectifiability appears again in the Federer-Fleming compactness theorem, where rectifiability of a flat limit is verified slice by slice, and it connects forward to the regularity theory, where spherical slices furnish the monotonicity formula, the lower density bounds, and the tangent-cone construction that bounds the dimension of the singular set. The whole apparatus is the precise sense in which a current is the integral of its slices, the reconstruction formula being the converse of the cut.

Full proof set Master

Proposition (restriction lowers mass and is additive over disjoint Borel sets). Let $T\in {\mathcal{N}}_k$ and let $A,B$ be disjoint Borel sets. Then $\mathbf{M}(T└A)=\Vert T\Vert (A)$, $\mathbf{M}(T└A)\le \mathbf{M}(T)$, and $T└(A\cup B)=T└A+T└B$.

Proof. By the mass-measure representation of 02.13.07, $T(\omega )=\int ⟨\omega ,\vec{T}⟩d\Vert T\Vert$ with $\Vert T\Vert$ a Radon measure and $\vec{T}$ a unit simple $k$-vector field. The restriction is $(T└A)(\omega )={\int }_A⟨\omega ,\vec{T}⟩d\Vert T\Vert$, which is the current with mass measure $\Vert T\Vert └A$ and the same orientation, so $\mathbf{M}(T└A)=\Vert T└A\Vert ({\mathbb{R}}^n)=\Vert T\Vert (A)$. Monotonicity of the measure gives $\Vert T\Vert (A)\le \Vert T\Vert ({\mathbb{R}}^n)=\mathbf{M}(T)$. For disjoint $A,B$, additivity of $\Vert T\Vert$ on disjoint Borel sets gives, for every $\omega$, ${\int }_{A\cup B}⟨\omega ,\vec{T}⟩d\Vert T\Vert ={\int }_A⟨\omega ,\vec{T}⟩d\Vert T\Vert +{\int }_B⟨\omega ,\vec{T}⟩d\Vert T\Vert$, which is the claimed additivity of the restricted currents. $\square$

Proposition (the slice is supported on the level set). For $T\in {\mathcal{N}}_k$, Lipschitz $f$, and ${\mathcal{L}}^1$-a.e. $t$, $\mathrm{spt}⟨T,f,t⟩\subseteq \mathrm{spt}T\cap \{f=t\}$.

Proof. Fix a point $x_0$ with $f(x_0)\ne t$, say $f(x_0)<t$; choose $r>0$ with $f<t$ on the ball $B_r(x_0)$, possible by continuity of $f$. For any test form $\omega \in {\mathcal{D}}^{k-1}$ supported in $B_r(x_0)$, the set $\{f<t\}$ contains $B_r(x_0)$, so $T_{<t}$ and $T$ agree against forms supported there, and likewise $(\partial T{)}_{<t}$ agrees with $\partial T$ there. Hence $$⟨T,f,t⟩(\omega )=⟨\partial T_{<t},\omega ⟩-⟨(\partial T{)}_{<t},\omega ⟩=⟨\partial T,\omega ⟩-⟨\partial T,\omega ⟩=0,$$ using $⟨\partial T_{<t},\omega ⟩=⟨T_{<t},d\omega ⟩=⟨T,d\omega ⟩=⟨\partial T,\omega ⟩$ since $d\omega$ is also supported in $B_r(x_0)\subseteq \{f<t\}$. The case $f(x_0)>t$ is symmetric, with $B_r(x_0)\subseteq \{f\ge t\}$ making both $T_{<t}$ and $(\partial T{)}_{<t}$ vanish against forms supported in the ball. So the slice annihilates all forms supported away from $\{f=t\}$, placing its support in $\{f=t\}$; and since the slice is built from restrictions of $T$ and $\partial T$, its support also lies in $\mathrm{spt}T$. $\square$

Proposition (lower semicontinuity of total slice mass). Let $T_j\to T$ in the flat norm with ${\sup }_j(\mathbf{M}(T_j)+\mathbf{M}(\partial T_j))<\infty$, and let $f$ be Lipschitz. Then for ${\mathcal{L}}^1$-a.e. $t$, $⟨T_j,f,t⟩\to ⟨T,f,t⟩$ in the flat norm along a subsequence, and $${\int }_{\mathbb{R}}\mathbf{M}(⟨T,f,t⟩)dt\le \underset{j}{\mathrm{liminf}}{\int }_{\mathbb{R}}\mathbf{M}(⟨T_j,f,t⟩)dt.$$

Proof. Flat convergence $T_j\to T$ gives $T_j└\{f<t\}\to T└\{f<t\}$ in the flat norm for a.e. $t$, because the level sets $\{f=t\}$ carry no flat-limit mass except at countably many $t$; the same holds for $\partial T_j$. Taking boundaries (flat-continuous) and subtracting, $⟨T_j,f,t⟩=\partial (T_j└\{f<t\})-(\partial T_j)└\{f<t\}\to ⟨T,f,t⟩$ in the flat norm for a.e. $t$, along a subsequence chosen by a diagonal argument so the exceptional null sets nest. Mass is lower semicontinuous under flat convergence (flat convergence implies weak-$\ast$ convergence, and mass is weak-$\ast$ lower semicontinuous by 02.13.07), so $\mathbf{M}(⟨T,f,t⟩)\le {\mathrm{liminf}}_j\mathbf{M}(⟨T_j,f,t⟩)$ for a.e. $t$. Integrating in $t$ and applying Fatou's lemma to the nonnegative integrands $\mathbf{M}(⟨T_j,f,t⟩)$ yields $\int \mathbf{M}(⟨T,f,t⟩)dt\le \int {\mathrm{liminf}}_j\mathbf{M}(⟨T_j,f,t⟩)dt\le {\mathrm{liminf}}_j\int \mathbf{M}(⟨T_j,f,t⟩)dt$. $\square$

Proposition (slicing characterisation of rectifiability, sufficiency direction). Let $T\in {\mathcal{N}}_k$ have $\mathbf{M}(T)+\mathbf{M}(\partial T)<\infty$, and suppose that for each coordinate $u_i$ and ${\mathcal{L}}^1$-a.e. $t$ the slice $⟨T,u_i,t⟩$ is integer-multiplicity rectifiable. Then $T$ is integer-multiplicity rectifiable.

Proof (reduction). The statement is proved by induction on $k$. For $k=0$ a normal $0$-current of finite mass is a finite signed sum of point masses; the hypothesis is vacuous and integer-multiplicity rectifiability is the integrality of the weights, which holds when $T$ arises as a slice of an integral current of dimension $1$. Assume the claim through dimension $k-1$. Given $T$ at dimension $k$, the hypothesis supplies, for a.e. $t$ and each $i$, an integer-multiplicity rectifiable $(k-1)$-current $⟨T,u_i,t⟩$ carried by a countably $(k-1)$-rectifiable set $M_{i,t}\subseteq \{u_i=t\}$ with integer multiplicity ${\theta }_{i,t}$. The union ${\bigcup }_{i,t}{M}_{i,t}$, fibred over the coordinate hyperplanes, reconstructs a countably $k$-rectifiable carrier for $T$ via the coarea reconstruction formula: integrating the slice carriers against $d{u}_i$ recovers $\Vert T\Vert$ as $\theta {\mathcal{H}}^k└M$ for a measurable integer multiplicity $\theta$ and a countably $k$-rectifiable $M$, because a set whose a.e. coordinate slices are $(k-1)$-rectifiable is itself $k$-rectifiable. The multiplicity $\theta$ is integer-valued ${\mathcal{H}}^k$-a.e. since it agrees with the integer slice multiplicities ${\theta }_{i,t}$ along a.e. level set. Hence $T$ is integer-multiplicity rectifiable, completing the induction. The detailed measurable selection of $M$ and $\theta$ from the slice data is carried out in Federer's structure theory ^{[Federer 1969 §4.3]}. $\square$

Connections Master

The coarea estimate $\int \mathbf{M}(⟨T,f,t⟩)dt\le \mathrm{Lip}(f)\mathbf{M}(T)$ rests on the Carathéodory outer-measure and Hausdorff-measure apparatus of 02.07.02: the rectifiable coarea formula on the carrier $M$ identifies the integrated slice mass with ${\int }_M|{\nabla }^{M}f||\theta |d{\mathcal{H}}^k$, and the disintegration of ${\mathcal{H}}^k└M$ over the level sets into ${\mathcal{H}}^{k-1}$-measures is a Fubini-type theorem for Hausdorff measure built directly on that outer-measure construction.

Slicing is the dimension-reduction step in the boundary-rectifiability and structure theorem of 02.13.07: the proof there slices by coordinate functions to reduce the $k$-dimensional integrality question to the one-dimensional base case, and the boundary identity $\partial ⟨T,u,t⟩=-⟨\partial T,u,t⟩$ proved here is precisely what closes the induction by making the slice of a boundary computable from the boundary of a slice.

The Whitney deformation theorem of 02.13.05 supplies the polyhedral approximation whose slices are the polyhedral slices, and the coarea averaging here is the same Fubini-over-parameter technique that produces the good projection centre there: in both cases a worst-case geometric bound is replaced by an average that is finite because a singular factor is integrable, and the two averaging arguments together drive the Federer-Fleming compactness theorem.

Historical & philosophical context Master

The slicing of geometric chains by level sets appears in Hassler Whitney's 1957 Geometric Integration Theory, where the flat norm and the geometry of flat chains already required cutting a chain along the level sets of a coordinate or a Lipschitz function and controlling the resulting lower-dimensional pieces ^{[Whitney 1957]}. Herbert Federer and Wendell Fleming used slicing as a working tool in their 1960 Annals of Mathematics paper on normal and integral currents, where the rectifiability of the flat limit in the closure theorem is verified through the rectifiability of its slices ^{[Federer-Fleming 1960]}. The systematic theory — the coarea inequality for slices, the a.e. rectifiability of slices, the boundary identity, and the reconstruction formula expressing a current as the integral of its slices — was set down in Federer's 1969 Geometric Measure Theory at §4.3, which remains the encyclopedic source ^{[Federer 1969]}. Leon Simon's 1983 ANU lectures gave the slicing apparatus its standard graduate exposition and connected it to the monotonicity formula and the regularity theory at §28 ^{[Simon 1983]}. Steven Krantz and Harold Parks presented the same material with full proofs and worked examples in their 2008 Geometric Integration Theory ^{[Krantz-Parks 2008]}.

Bibliography Master

@book{whitney1957geometric,
  author    = {Whitney, Hassler},
  title     = {Geometric Integration Theory},
  publisher = {Princeton University Press},
  series    = {Princeton Mathematical Series},
  volume    = {21},
  year      = {1957},
  address   = {Princeton, NJ}
}

@article{federerfleming1960normal,
  author  = {Federer, Herbert and Fleming, Wendell H.},
  title   = {Normal and integral currents},
  journal = {Annals of Mathematics},
  volume  = {72},
  number  = {3},
  pages   = {458--520},
  year    = {1960},
  doi     = {10.2307/1970227}
}

@book{federer1969geometric,
  author    = {Federer, Herbert},
  title     = {Geometric Measure Theory},
  publisher = {Springer-Verlag},
  series    = {Die Grundlehren der mathematischen Wissenschaften},
  volume    = {153},
  year      = {1969},
  address   = {Berlin}
}

@book{simon1983lectures,
  author    = {Simon, Leon},
  title     = {Lectures on Geometric Measure Theory},
  publisher = {Centre for Mathematical Analysis, Australian National University},
  series    = {Proceedings of the Centre for Mathematical Analysis},
  volume    = {3},
  year      = {1983},
  address   = {Canberra}
}

@book{krantzparks2008geometric,
  author    = {Krantz, Steven G. and Parks, Harold R.},
  title     = {Geometric Integration Theory},
  publisher = {Birkh\"auser},
  series    = {Cornerstones},
  year      = {2008},
  address   = {Boston}
}

@book{morgan2016geometric,
  author    = {Morgan, Frank},
  title     = {Geometric Measure Theory: A Beginner's Guide},
  edition   = {5},
  publisher = {Academic Press},
  year      = {2016},
  address   = {Amsterdam}
}