Flat convergence of integral currents and the size of rectifiable sets in metric spaces

概要

Flat convergence of integral currents and the
size of rectifiable sets in metric spaces
著者
学位授与機関
学位授与番号
URL

Takeuchi Shu
Tohoku University
11301甲第19614号
http://hdl.handle.net/10097/00132985

博士論文

Flat convergence of integral
currents and the size of rectifiable
sets in metric spaces
(距離空間における整カレントの
♭収束と修正可能集合のサイズ)

竹内 秀
令和２年

In this thesis, there are two purposes.
The first purpose is to prove Theorem 8, which states the lower semicontinuity of the size of integral currents in separable Hilbert spaces. Currents
are originally defined in Euclidean spaces by de Rham in [4], which is a nice
tool in geometric measure theory. In fact, Theorem 8 is already proved by
Almgren in [1]. Note that to state Theorem 8, we must define the notion
of currents in metric spaces, which was introduced by Ambrosio and Kirchheim in [2]. In the sequel we briefly review the theory of currents in metric
spaces:
Let X be a complete metric space, then for each k ∈ Z≥0 we define
Dk (X) by
Dk (X) := Lipb (X) × (Lip(X))k ,

(1)

where Lip(X) is the collection of Lipschitz functions on X and Lipb (X) is
the collection of bounded Lipschitz functions on X. Now currents in metric
spaces are defined as in the following Definition 1:
Definition 1 (Currents, [2, Definition 3.1]). Let X be a complete metric
space, k ∈ Z≥0 and T : Dk (X) → R. We say that T is a k-dimensional
current in X if and only if T satisfies the following:
(1) (linearity) T is multilinear, i.e. T (f, π1 , π2 , . . . , πk ) is linear with respect to for each component f, π1 , π2 , . . . , πk .
(2) (continuity) Let (f, π1 , π2 , . . . , πk ) ∈ Dk (X) and (f, π1,j , π2,j , . . . , πk,j )
∈ Dk (X) for j ∈ Z≥1 , and assume that for all i ∈ {1, 2, . . . , k}
sup Lip(πi,j ) < ∞

(2)

j

and πi,j pointwisely converges to πi as j → ∞. Then we have
lim T (f, π1,j , π2,j , . . . , πk,j ) = T (f, π1 , π2 , . . . , πk ).

j→∞

(3)

(3) (locality) Let (f, π1 , π2 , . . . , πk ) ∈ Dk (X) and assume that there exist
some i ∈ {1, 2, . . . , k} and a neighborhood N of {x ∈ X | f (x) ̸= 0}
such that πi |N is constant. Then we have T (f, π1 , π2 , . . . , πk ) = 0.
(4) (finite mass) There exist a finite Borel measure µ on X such that for
any (f, π1 , π2 , . . . , πk ) ∈ Dk (X) we have
|T (f, π1 , π2 , . . . , πk )| ≤

k
∏
i=1

∫
|f | dµ,

Lip(πi )

(4)

X

where Lip(πi ) is the least Lipschitz constant of πi .
The collection of k-dimensional currents in X is denoted by Mk (X).
Proposition 2 (The mass of currents). Let X be a complete metric space,
k ∈ Z≥0 and T ∈ Mk (X). Then there exists a unique finite Borel measure
∥T ∥ in X such that the following holds:

(1) Let µ be a finite Borel measure on X which satisfies (4). Then for any
Borel set B ⊂ X we have ∥T ∥(B) ≤ µ(B).
(2) ∥T ∥ also satisfies (4).
We say that ∥T ∥ is the mass of T .
Let T ∈ Mk (X) and it is possible to define the value T (f, π1 , π2 , . . . , πk )
for a bounded Borel measurable function f and πi ∈ Lip(X). Now for any
Borel set B ⊂ X we define T ⌞ B ∈ Mk (X) by
T ⌞ B(f, π1 , π2 , . . . , πk ) := T (f χB , π1 , π2 , . . . , πk ),

(5)

where χB is the characteristic function of B. If k ∈ Z≥1 , we define ∂T :
Dk−1 (X) → R by
∂T (f, π1 , π2 , . . . , πk−1 ) := T (1, f, π1 , π2 , . . . , πk−1 ).

(6)

Now we are in the place to define integer-rectifiable currents and integral
currents in metric spaces. We say that a Borel set S ⊂ X is a countably
Hk -rectifiable set if and only if there exist a countable family of compact
sets {Ai }∞
i=1 and Lipschitz maps fi : Ai → X such that we have
(
)
∞
∪
k
H S\
fi (Ai ) = 0,
(7)
i=1

where Hk is the Hausdorff measure on X.
Definition 3 (Integer-rectifiable currents, [2, Definition 4.2]). Let X be a
complete metric space and k ∈ Z≥1 . We say that T ∈ Mk (X) is a kdimensional integer-rectifiable current in X if and only if T satisfies
the following:
(1) There exists a countably Hk -rectifiable set S ⊂ X such that ∥T ∥(X \
S) = 0.
(2) For any Borel set N ⊂ X with Hk (N ) = 0 we have ∥T ∥(N ) = 0.
(3) For any φ ∈ Lip(X, Rk ) and an open set O ⊂ X, there exists θ ∈
L1 (Rk , Z) such that we have
)
(
∫
∂πi
dLk ,
(T ⌞ O)(f ◦ φ, π1 ◦ φ, π2 ◦ φ, . . . , πk ◦ φ) =
f θ det
∂x
k
j
R
(8)
for any (f, π1 , π2 , . . . , πk ) ∈ Dk (Rk ), where Lk is the k-dimensional
Lebesgue measure on Rk .
The collection of k-dimensional integer-rectifiable currents in X is denoted
by Ik (X).

Definition 4 (0-dimensional integer-rectifiable currents). Let X be a complete metric space. We say that T ∈ M0 (X) is a 0-dimensional integerrectifiable current in X if and only if T satisfies the following: There
exist countable points x1 , x2 , · · · ∈ X and θ1 , θ2 , · · · ∈ Z such that for any
f ∈ Lipb (X) we have
T (f ) =

∞
∑

θi f (xi ).

(9)

i=1

The collection of 0-dimensional integer-rectifiable currents in X is denoted
by I0 (X).
Definition 5 (Integral currents). Let X be a complete metric space, k ∈ Z≥0
and T ∈ Ik (X). We say that T is a k-dimensional integral current in
X if and only if ∂T ∈ Mk (X) or k = 0. The collection of k-dimensional
integral currents in X is denoted by Ik (X).
For each T ∈ Ik (X), we assign a value S(T ), which we want to discuss
the semicontinuity:
Definition 6 (Canonical set and size). Let X be a complete metric space,
k ∈ Z≥0 and T ∈ Ik (X). Then let

}
{

∥T ∥(Br (x))
>0
(10)
ST := x ∈ X  lim inf k
r↓0
L (Br (0))
where Br (x) is the metric open ball centered at x ∈ X with radius r, and
S(T ) := Hk (ST ).

(11)

We say that ST and S(T ) is the canonical set and size of T , respectively.
In order to discuss the functional S(·) on Ik (X), we use the topology
induced by the following distance:
Definition 7 (Flat distance). Let X be a complete metric space, k ∈ Z≥0
and T1 , T2 ∈ Ik (X). Then let
dX
F (T1 , T2 ) := inf(∥U ∥(X) + ∥V ∥(X)),

(12)

where the infimum is taken over all U ∈ Ik (X) and V ∈ Ik+1 (X) which
satisfies T1 − T2 = U + ∂V . We say that dX
F (T1 , T2 ) is the flat distance
between T1 , T2 ∈ Ik (X).
The following Theorem 8 is the main theorem in this thesis:
Theorem 8 (Lower semicontinuity of the size of integral currents, [9]).
Let H be a separable Hilbert space, k ∈ Z≥1 and {Tj }∞
j=1 be a sequence of
k-dimensional integral currents in H which converges to a k-dimensional
integral current in H with respect to the flat distance. Then we have
lim inf S(Tj ) ≥ S(T ).
j→∞

(13)

The sketch of the proof is as follows: By the nice work of countably
sets in [3], for Hk -a.e. x ∈ ST we can find a “good” Borel
set Sx ⊂ ST and k-dimensional subspace Tank (S, x) ⊂ H, which is called
the approximate tangent space to S at x, and a Lipschitz map πx : H →
Lx which almost preserves the distance. By using slicing theorem (see [2,
Theorem 5.6] for details) for πx , we compare the Hausdorff measure of ST
and STj .
The second purpose is to define the pointed intrinsic flat distance. This is
motivated by the intrinsic flat distance introduced by Sormani and Wenger
in [7]. In order to discuss the convergence of currents defined in other metric
spaces, they constructed the following flamework: Let X be a complete
metric space, k ∈ Z≥0 and T ∈ Ik (X) satisfies ST = X. Then they called
such a triplet (X, dX , T ) a k-dimensional integral current space, and for
two k-dimensional integral current spaces Mi = (Xi , dXi , Ti ), i = 1, 2, they
defined the pointed intrinsic flat distance between M1 and M2 by
Hk -rectifiable

dF (M1 , M2 ) := inf(∥U ∥(Z) + ∥V ∥(Z)),

(14)

where the infimum is taken over all complete metric space Z, isometric
embeddings φi : Xi ,→ Z, U ∈ Ik (Z) and V ∈ Ik+1 (Z) with φ1# T1 −
φ2# T2 = U + ∂V . However, the intrinsic flat distance deals with only
integral current spaces which has finite mass, we can not discuss the space
with infinite mass such as Euclidean spaces. To achieve such an aim, let us
briefly review the theory of locally integral currents, which was introduced
and studied in [5] and [6].
Let X be a complete metric space, then for each k ∈ Z≥0 we define
k (X) by
DLoc
k
DLoc
(X) := LipB (X) × (LipLoc (X))k ,

(15)

where LipB (X) is the collection of Lipschitz functions on X with bounded
support and LipLoc (X) is the collection of functions on X which is Lipschitz
on for any bounded set. Now local currents in metric spaces are defined as
in the following Definitions:
Definition 9 (Local metric functionals, [6, Definition 2.1]). Let X be a
k (X) → R be a map. We say
complete metric space, k ∈ Z≥0 and T : DLoc
that T is a k-dimensional local metric functional in X if and only if
T satisfies the following:
(1) (linearity) T is multilinear, i.e. T (f, π1 , π2 , . . . , πk ) is linear with respect to for each component f, π1 , π2 , . . . , πk .
k (X) and (f, π , π , . . . , π )
(2) (continuity) Let (f, π1 , π2 , . . . , πk ) ∈ DLoc
1,j
2,j
k,j
k
∈ DLoc (X) for j ∈ Z≥1 , and assume that for all i ∈ {1, 2, . . . , k} and
for any bounded set B ⊂ X
sup Lip(πi,j |B ) < ∞

(16)

j

and πi,j pointwisely converges to πi as j → ∞. Then we have
lim T (f, π1,j , π2,j , . . . , πk,j ) = T (f, π1 , π2 , . . . , πk ).

j→∞

(17)

k (X) and assume that there exist
(3) (locality) Let (f, π1 , π2 , . . . , πk ) ∈ DLoc
some i ∈ {1, 2, . . . , k} and δ > 0 such that πi |B(spt f, δ) is constant.
Then we have T (f, π1 , π2 , . . . , πk ) = 0.

For a k-dimensional local metric functional T , we define a set function
∥T ∥, see Section 2.2 of [6] for the detail.
Definition 10 (Local currents). Let X be a complete metric space, k ∈ Z≥0
and T be a k-dimensional local metric functional in X. We say that T is a
k-dimensional local current in X if and only if T satisfies the following:
(1) For any bounded open set O ⊂ X we have ∥T ∥(O) < ∞.
(2) For any bounded open set O ⊂ X and ε > 0 there exists a compact set
K ⊂ O such that ∥T ∥(O \ K) < ε.
The collection of k-dimensional local currents in X is denoted by MLoc,k (X).
Now we are in the place to define locally integer-rectifiable currents and
locally integral currents in metric spaces.
Definition 11 (Locally integer-rectifiable currents). ...