mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to

integral current In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a ''k''-current in the sense of Georges de Rham is a functional on the space of compactly supported differential ''k''-forms, on a smoot ...

s by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to

Plateau's problem In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is ...

. In 2001 the notion of an integral current was extended to arbitrary metric spaces by Ambrosio and Kirchheim.

Integral currents

A ''k''-dimensional current ''T'' is a linear functional on the space

\Omega^k_c(\mathbb^n)

of smooth, compactly supported ''k''-forms. For example, given a Lipschitz map from a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

into

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

F: N^k \to \mathbb^n

, one has an integral current ''T''(''ω'') defined by integrating the

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...

of the differential ''k''-form, ''ω'', over ''N''. Currents have a notion of boundary

\partial

(which is the usual boundary when ''N'' is a manifold with boundary) and a notion of mass, ''M''(''T''), (which is the volume of the image of ''N''). An integer rectifiable current is defined as a countable sum of currents formed in this respect. An integral current is an integer rectifiable current whose boundary has finite mass. It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.

Flat norm and flat distance

The flat norm , ''T'', of a ''k''-dimensional integral current ''T'' is the infimum of ''M''(''A'') + ''M''(''B''), where the infimum is taken over all integral currents ''A'' and ''B'' such that

T=A + \partial B

. The flat distance between two integral currents is then ''d''_''F''(''T'',''S'') = , ''T'' − ''S'', .

Compactness theorem

Federer-Fleming proved that if one has a sequence of integral currents

T_i

whose supports lie in a compact set ''K'' with a uniform upper bound on

M(T_i) + M(\partial T_i)

, then a subsequence converges in the flat sense to an integral current. This theorem was applied to study sequences of submanifolds of fixed boundary whose volume approached the infimum over all volumes of submanifolds with the given boundary. It produced a candidate weak solution to

References

* * * * {{citation, last=Ambrosio, first=Luigi, author2=Kirchheim, Bernd, title=Currents in Metric Spaces, journal=Acta Mathematica, year=2000, volume=185, pages=1–80, doi=10.1007/bf02392711, doi-access=free Metric geometry Riemannian geometry Convergence (mathematics)