Current (mathematics)

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 smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M.

Definition[]

Let ${\displaystyle \Omega _{c}^{m}(M)}$ denote the space of smooth m-forms with compact support on a smooth manifold ${\displaystyle M}$. A current is a linear functional on ${\displaystyle \Omega _{c}^{m}(M)}$ which is continuous in the sense of distributions. Thus a linear functional

is an m-dimensional current if it is continuous in the following sense: If a sequence ${\displaystyle \omega _{k}}$ of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when ${\displaystyle k}$ tends to infinity, then ${\displaystyle T(\omega _{k})}$ tends to 0.

The space ${\displaystyle {\mathcal {D}}_{m}(M)}$ of m-dimensional currents on ${\displaystyle M}$ is a real vector space with operations defined by

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current ${\displaystyle T\in {\mathcal {D}}_{m}(M)}$ as the complement of the biggest open set ${\displaystyle U\subset M}$ such that

whenever ${\displaystyle \omega \in \Omega _{c}^{m}(U)}$

The linear subspace of ${\displaystyle {\mathcal {D}}_{m}(M)}$ consisting of currents with support (in the sense above) that is a compact subset of ${\displaystyle M}$ is denoted ${\displaystyle {\mathcal {E}}_{m}(M)}$.

Principal Current[]

To define a Principal current, we must first define a principal function.

Principal Function[]

Let ${\displaystyle T_{z}=T-zI}$ and construct self-adjoint operators ${\displaystyle T_{z}^{*}T_{z}}$, ${\displaystyle T_{z}T_{z}}$ with spectral resolutions ${\displaystyle E_{\lambda }^{z}}$ and ${\displaystyle F_{\lambda }^{z}}$ respectively.

If ${\displaystyle z}$ is not in the essential spectrum of ${\displaystyle T}$, then both spectral projections are constant finite-rank operators for sufficiently small ${\displaystyle \lambda }$. Thus if ${\displaystyle p(T)}$ is a polynomial in ${\displaystyle T}$

and when ${\displaystyle z}$ is in the spectrum of ${\displaystyle T}$, let where ${\displaystyle J}$ is presumably the Jacobian and ${\displaystyle \psi }$ is a real, smooth compactly-supported function normalised so that ${\textstyle \pi \int _{0}^{\infty }\psi ^{2}(s)\,ds=1}$, and ${\displaystyle J_{\varepsilon }(p)(z)=J(p)(z)}$ for small ${\displaystyle \varepsilon }$ when ${\displaystyle z}$ is not in the essential spectrum of ${\displaystyle T}$.

Since it is hypothesised ${\displaystyle T^{*}T-TT^{*}}$ is trace class, then there must exist an ${\displaystyle L^{1}}$, real-valued, compactly supported function ${\displaystyle g_{T}}$ called the principal function of ${\displaystyle T}$ so that

Since it has been shown for almost everywhere ${\displaystyle \lambda }$,

if ${\displaystyle \sigma _{z}(\lambda )}$ is the phase shift for the perturbation problem ${\displaystyle T_{z}^{*}T_{z}\to T_{z}T_{z}^{*}}$

If ${\displaystyle g_{T}}$ has a Lebesgue point at ${\displaystyle z}$, then

Thus, If ${\displaystyle T^{*}T-TT^{*}}$ has finite dimensional range, then ${\displaystyle \mathrm {ind} (T_{z}^{*}H,T_{z}H)=g_{T}(z)}$ for ${\displaystyle L^{2}}$ almost all ${\displaystyle z}$.^[1]

Principal Current[]

The principal function defined above is for a two-dimensional Lebesgue measure.

In cases where the essential spectrum is curve-like, the principal function can be defined on the curve as an average of the values on both of its sides, even when ${\displaystyle g_{T}}$ is weakly differentiable i.e, when ${\textstyle {\frac {\delta g}{\delta x}}}$ and ${\textstyle {\frac {\delta g}{\delta y}}}$ are measures.

For ${\displaystyle H^{1}}$ almost all ${\displaystyle b}$ in ${\displaystyle R^{2}}$ and a weakly-differentiable function ${\displaystyle f}$^[2]

whenever ${\displaystyle \psi }$ satisfies the conditions for a principal function.

It follows that if subspaces ${\displaystyle T_{z}H}$ and ${\displaystyle T_{z}^{*}H}$ are ${\displaystyle F}$-comparable, then ${\displaystyle \sigma _{z}^{*}(0)}$ is equal to the average of the approximated upper and lower limits of principal function ${\displaystyle g_{T}}$ at ${\displaystyle z}$.

This expression can be interpreted as a redefinition of the principal function on the set of points ${\displaystyle z}$ where the rotationally-symmetric regularisation of the principal function converge Hausdorff one-measure almost everywhere, which implies ${\displaystyle \mathrm {ind} (T_{z}^{*}H,T_{z}H)=g_{T}(z)}$ ${\displaystyle H^{1}}$ almost everywhere.

The principal function of a single operator ${\displaystyle T}$ can be used to define a principal current of the unital C*-algebra corresponding to ${\displaystyle T}$.

By definition, the principal current for the C* algebra which corresponds to a ${\displaystyle T}$ such that ${\displaystyle [T,T^{*}]}$ is in trace class is^[3]

The two-current defined by this relation has certain basic properties: with a suitable functional calculus

1. ${\displaystyle \delta T\left({\frac {df}{f}}\right)=\operatorname {index} f(T)}$ when ${\displaystyle f\neq 0}$ on ${\displaystyle \sigma _{ess}(T)}$
2. ${\displaystyle \delta T(fdh)=\mathrm {trace} [h(T^{*},T),f(T^{*},T)]}$
3. ${\displaystyle \mathrm {Mass} (T)<{\frac {1}{2}}||[T^{*},T]||_{tr}}$
4. ${\displaystyle T_{\Omega }=\Omega _{\#}(T)}$ where ${\displaystyle \Omega (T,T^{*})}$ is a smooth function of ${\displaystyle T}$ and ${\displaystyle T^{*}}$ and ${\displaystyle T_{\Omega }}$ is the current formed from the operator ${\displaystyle \Omega (T,T^{*})}$
5. ${\displaystyle T_{T}=T_{T+k}}$ for ${\displaystyle k}$ in trace class.

Homological theory[]

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by ${\displaystyle [[M]]}$:

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:

This relates the exterior derivative d with the boundary operator ∂ on the homology of M.

In view of this formula we can define a boundary operator on arbitrary currents

via duality with the exterior derivative by for all compactly supported m-forms ω.

Certain subclasses of currents which are closed under ${\displaystyle \partial }$ can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.

Topology and norms[]

The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence T_k of currents, converges to a current T if

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ω is an m-form, then define its comass by

So if ω is a simple m-form, then its mass norm is the usual L^∞-norm of its coefficient. The mass of a current T is then defined as

The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.

An intermediate norm is Whitney's flat norm, defined by

Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation.

Examples[]

Recall that

so that the following defines a 0-current:

In particular every signed regular measure ${\displaystyle \mu }$ is a 0-current:

Let (x, y, z) be the coordinates in R³. Then the following defines a 2-current (one of many):

See also[]

• Georges de Rham
• Herbert Federer
• Differential geometry
• Varifold

Notes[]

References[]

• de Rham, G. (1973), Variétés Différentiables, Actualites Scientifiques et Industrielles (in French), 1222 (3rd ed.), Paris: Hermann, pp. X+198, Zbl 0284.58001.
• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801.
• Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series, 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.
• Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), 1, Beijing/Boston: Science Press/International Press, pp. x+237, ISBN 978-1-57146-125-4, MR 2030862, Zbl 1074.49011
• Carey, Richard W.; Pincus, Joel D. (7 January 1985), "Principal Currents", Integral Equations and Operator Theory, 8 (5): 614–640, doi:10.1007/BF01201706, S2CID 189878392
• Carey, Richard W.; Pincus, Joel D. (1986), "Index Theory for Operator Ranges and Geometric Measure Theory", Proceedings of Symposia in Pure Mathematics, Geometric Measure Theory and the Calculus of Variations, American Mathematical Society, 44: 149–161, doi:10.1090/pspum/044/840271

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