Federer-Fleming deformation theorem

The Federer–Fleming deformation theorem can be considered as one of the central results in the theory of integral currents, created by H. Federer and W.H. Fleming at the end of 1950s, [a1]. The notion of an integral current was introduced to meet all the requirements of the concept of "k-dimensional domain of integration in Euclidean n-space" and combines the smoothness properties of differentiable manifolds and the combinatorial structure of polyhedral chains with integer coefficients. The notion of integral current provides a powerful instrument for solving geometrical variational problems like the Plateau problem.

The deformation theorem shows how one can deform normal and integral currents into similar currents with supports in the appropriate skeletons of a standard cubical cell complex with given cube edge, and gives estimates for the masses of the currents.

Let $ C $ denote the standard cubical complex in the Euclidean space $ \mathbf R ^ {n} $ generated by the cubes with edge $ 2 $ and with centres at the integer points. Let $ C _ \varepsilon $ denote the complex obtained from $ C $ by $ \varepsilon $- homothety (cf. Homothety). As always, let $ {\mathcal N} _ {k} ( \mathbf R ^ {n} ) $ denote the space of normal currents in $ \mathbf R ^ {n} $, and let $ { \mathop{\rm mass} } ( T ) $ be the mass of the current $ T $( cf. also Differential form; Mass and co-mass).

The Federer–Fleming deformation theorem reads: Let $ \varepsilon > 0 $ be an arbitrary positive number. Then any $ k $- dimensional normal current $ T $ in $ \mathbf R ^ {n} $ can be represented as a sum

$$ T = P + Q + \partial S, $$

where

$$ P \in {\mathcal N} _ {k} ( \mathbf R ^ {n} ) , Q \in {\mathcal N} _ {k} ( \mathbf R ^ {n} ) , S \in {\mathcal N} _ {k + 1 } ( \mathbf R ^ {n} ) , $$

with the following properties:

1) The current $ P $ is a polyhedral chain of $ C _ \varepsilon $ with real coefficients. If $ T $ is an integral current, then the coefficients of $ P $ are integers.

2) The supports of $ P $ and $ S $ lie in the $ 2n \varepsilon $- neighbourhood of the support of $ T $, and the supports of $ \partial P $ and $ Q $ lie in the $ 2n \varepsilon $- neighbourhood of $ \partial T $.

3) For the masses of $ T $, $ P $, $ Q $, and $ S $ the following estimates hold:

$$ { \frac{ { \mathop{\rm mass} } ( P ) }{\varepsilon ^ {k} } } \leq 2n ^ {k} \left [ \left ( \begin{array}{c} n \\ k \end{array} \right ) { \frac{ { \mathop{\rm mass} } ( T ) }{\varepsilon ^ {k} } } + \left ( \begin{array}{c} n \\ {k - 1 } \end{array} \right ) { \frac{ { \mathop{\rm mass} } ( \partial T ) }{\varepsilon ^ {k - 1 } } } \right ] , $$

$$ { \frac{ { \mathop{\rm mass} } ( \partial P ) }{\varepsilon ^ {k - 1 } } } \leq 2n ^ {k - 1 } \left ( \begin{array}{c} n \\ {k - 1 } \end{array} \right ) { \frac{ { \mathop{\rm mass} } ( \partial T ) }{\varepsilon ^ {k - 1 } } } , $$

$$ { \frac{ { \mathop{\rm mass} } ( Q ) }{\varepsilon ^ {k} } } \leq 6n ^ {k} \left ( \begin{array}{c} n \\ {k - 1 } \end{array} \right ) { \frac{ { \mathop{\rm mass} } ( \partial T ) }{\varepsilon ^ {k - 1 } } } , $$

$$ { \frac{ { \mathop{\rm mass} } ( S ) }{\varepsilon ^ {k + 1 } } } \leq 4n ^ {k + 1 } \left ( \begin{array}{c} n \\ k \end{array} \right ) { \frac{ { \mathop{\rm mass} } ( T ) }{\varepsilon ^ {k} } } . $$

4) If $ T $ is an integral current, then so are $ P $, $ Q $ and $ S $.

5) If $ T $ is an (integral) Lipschitz chain, so are $ P $, $ Q $ and $ S $.

6) If $ \partial T $ is an (integral) Lipschitz chain, so is $ Q $.

As a consequence, one can obtain the following result: For each integral current $ T $ there exists a sequence of integral Lipschitz chains $ T _ {i} $ such that

$$ {\lim\limits } _ {i \rightarrow \infty } T _ {i} = T \textrm{ and } {\lim\limits } _ {i \rightarrow \infty } N ( T _ {i} - T ) = 0, $$

where $ N ( T ) = { \mathop{\rm mass} } ( T ) + { \mathop{\rm mass} } ( \partial T ) $.

As other applications of the theorem, one can obtain the theorem about isomorphism between the integer singular homology groups of $ ( A,B ) $ and the homology groups of the integral current chain complex $ {\mathcal I} _ {*} ( A ) / {\mathcal I} _ {*} ( B ) $, where $ A $ and $ B $ are local Lipschitz neighbourhood retracts in $ \mathbf R ^ {n} $; the isoperimetric inequalities for integral currents; etc. Some modern applications of the theorem can be found in [a2], [a3].

References

[a1]	H. Federer, W.H. Fleming, "Normal and integral currents" Ann. of Math. , 72 : 3 (1960) pp. 458–520 MR0123260 Zbl 0187.31301
[a2]	Dao Trong Thi, A.T. Fomenko, "Minimal surfaces, stratified multivarifolds and the Plateau problem" , Amer. Math. Soc. (1991)
[a3]	A.T. Fomenko, "Variational principles in topology: multidimensional minimal surface theory" , Kluwer Acad. Publ. (1990)