current

current

Let

Λ

c

m

⁢

(

ℝ

n

)

denote the space of

C

∞

differentiable

m

-forms with compact support in

ℝ

n

. A continuous linear operator

T

:

Λ

c

m

⁢

(

ℝ

n

)

→

ℝ

is called an

m

-

current

. Let

𝒟

m

denote the space of

m

-currents in

ℝ

n

.
We define a

boundary operator

∂

:

𝒟

m

+

1

→

𝒟

m

by

∂

⁡

T

⁢

(

ω

)

:=

T

⁢

(

d

⁢

ω

)

.

We will see that currents

represent

a

generalization

of

m

-surfaces.
In fact if

M

is a

compact

m

-dimensional oriented manifold with

boundary

, we can associate to

M

the current

[

[

M

]

]

defined by

[

[

M

]

]

⁢

(

ω

)

=

∫

M

ω

.

So the definition of boundary

∂

⁡

T

of a current, is justified by
Stokes Theorem:

∫

∂

⁡

M

ω

=

∫

M

𝑑

ω

.

The space

𝒟

m

of

m

-dimensional currents is a real vector space with

operations

defined by

(

T

+

S

)

⁢

(

ω

)

:=

T

⁢

(

ω

)

+

S

⁢

(

ω

)

,

(

λ

⁢

T

)

⁢

(

ω

)

:=

λ

⁢

T

⁢

(

ω

)

.

The sum of two currents represents the

union

of the surfaces they represents.

Multiplication

by a scalar represents a change in the

multiplicity

of the surface. In particular multiplication by

-

1

represents the change of orientation of the surface.

We define the

support

of a current

T

, denoted by

spt

⁢

(

T

)

, the smallest

closed set

C

such that

T

⁢

(

ω

)

=

0

⁢

whenever

ω

=

0

on

C

.

We denote with

ℰ

m

the vector subspace of

𝒟

m

of currents with compact support.

Topology

The space of currents is naturally endowed with the

weak-star

topology

, which will be further simply called

weak convergence

. We say that a

sequence

T

k

of currents, weakly

converges

to a current

T

if

T

k

⁢

(

ω

)

→

T

⁢

(

ω

)

,

∀

ω

.

A

stronger

norm on the space of currents is the

mass norm

. First of all we define the mass norm of a

m

-form

ω

as

||

ω

||

:=

sup

⁡

{

|

⟨

ω

,

ξ

⟩

|

:

ξ

is a unit, simple,

m

-vector

}

.

So if

ω

is a simple

m

-form, then its mass norm is the usual norm of its

coefficient

. We hence define the

mass

of a current

T

as

𝐌

⁢

(

T

)

:=

sup

⁡

{

T

⁢

(

ω

)

:

sup

x

⁡

||

ω

⁢

(

x

)

||

≤

1

}

.

The mass of a currents represents the

area

of the generalized surface.

An intermediate norm, is the

flat norm

defined by

𝐅

⁢

(

T

)

:=

inf

⁡

{

𝐌

⁢

(

A

)

+

𝐌

⁢

(

B

)

:

T

=

A

+

∂

⁡

B

,

A

∈

ℰ

m

,

B

∈

ℰ

m

+

1

}

.

Notice that two currents are close in the mass norm if they coincide apart from
a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation.

Examples

Recall that

Λ

c

0

⁢

(

ℝ

n

)

≡

C

c

∞

⁢

(

ℝ

n

)

so that the following defines a

0

-current:

T

⁢

(

f

)

=

f

⁢

(

0

)

.

In particuar every

signed measure

μ

with finite mass is a

0

-current:

T

⁢

(

f

)

=

∫

f

⁢

(

x

)

⁢

𝑑

μ

⁢

(

x

)

.

Let

(

x

,

y

,

z

)

be the coordinates in

ℝ

3

. Then the following defines a

2

-current:

T

⁢

(

a

⁢

d

⁢

x

∧

d

⁢

y

+

b

⁢

d

⁢

y

∧

d

⁢

z

+

c

⁢

d

⁢

x

∧

d

⁢

z

)

=

∫

0

1

∫

0

1

b

⁢

(

x

,

y

,

0

)

⁢

𝑑

x

⁢

𝑑

y

.

Title

current

Canonical name

Current

Date of creation

2013-03-22 14:27:39

Last modified on

2013-03-22 14:27:39

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Definition

Classification

msc 58A25

Defines

mass

Defines

support

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