Isoperimetric Inequalities and the Structure of Metric Spaces - Part 2

Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality

Isoperimetric inequalities and the structure of metric spaces - Part 2

Stefan Wenger

University of Fribourg

July 20, 2013

Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Overview - Part 2

1 Review and aim of today’s lecture

2 Brief review of currents in metric spaces

3 Asymptotic cones of metric spaces

4 Persistence of quadratic isoperimetric inequality

X metric space, c : S1 → X Lipschitz curve

Fillarea0(c) = inf {Area(ϕ): ϕ: D → X Lip, ϕ|S1 = c} . Filling area function in X :

X FA0 (r) = sup {Fillarea0(c): L(c) ≤ r}. Aim is to prove Theorem (W.) X geodesic metric space. If there exists ε > 0 such that 1 − ε FAX (r) ≤ r 2 ∀r 1 0 4π

X then FA0 (r) r.

. . . and . . . Theorem (W.) G α There exists nilpotent Lie group G of step 2 such that FA0 (r) 6' r for any α ∈ R.

Basic idea in both proofs:

1 Rescale metric d in X by factors rn & 0.

Sequence (X , rn d) has limit asymptotic cones Xω.

2 Would like:

X 2 Xω 2 FA0 (r) r ⇒ FA0 (r) ≤ Cr ∀r ≥ 0.

Xω Xω True? True for homological version FA (r) of FA0 (r).

X complete metric space Idea (De Giorgi): Use (m + 1)-tuples

(f , π1, . . . , πm) of Lipschitz functions on X as substitute for m-forms. N If X = R and f , πi smooth then think

(f , π1, . . . , πm) ! f dπ1 ∧ · · · ∧ dπm.

Note: d(f dπ1 ∧ · · · ∧ dπm) = df ∧ dπ1 ∧ · · · ∧ dπm

If ϕ : RM → RN smooth then ∗ ϕ (f dπ1 ∧ · · · ∧ dπm) = f ◦ ϕ d(π1 ◦ ϕ) ∧ · · · ∧ d(πm ◦ ϕ)

Deﬁnition For X complete metric space and m ≥ 0:

m m D (X ) := Lipb(X ) × Lip(X ) .

Notation: Lip(X ) := {f : X → R : f Lipschitz}

Lipb(X ) := {f : X → R : f Lipschitz and bounded} B∞(X ) := {f : X → R : f Borel and bounded} For f : X → Y set 0 dY (f (x), f (x )) Lip(f ) := sup 0 x6=x0 dX (x, x )

Deﬁnition: A function T : Dm(X ) → R is called m-current if: (i) T is multi-linear. n n n (ii) If πi ∈ Lip(X ) with supn Lip(πi ) < ∞ and πi → πi then n n T (f , π1 , . . . , πm) → T (f , π1, . . . , πm).

(iii) If πi is constant on spt f for some i then

T (f , π1, . . . , πm) = 0.

(iv) ∃ ﬁnite Borel measure µ, concentrated on σ-cpt set, such that m Y Z |T (f , π1, . . . , πm)| ≤ Lip(πi ) |f |dµ i=1 X m for all (f , π1, . . . , πm) ∈ D (X ).

Space of m-currents in X : Mm(X ) := m-currents in X .

Remark: Each T ∈ Mm(X ) extends uniquely to

T : B∞(X ) × Lip(X )m → R. Can show it satisﬁes same properties (i) – (iv) with f ∈ B∞(X ). Proposition

Given T ∈ Mm(X ) there exists a smallest Borel measure on X satisfying (iv), denoted kT k. For U ⊂ X open

n X n n X o kT k(U) = sup |T (fn, π )| : Lip(πi ) ≤ 1, |fn| ≤ 1U . n

Deﬁne mass of T by

M(T ) := kT k(X ).

Lower semi-continuity of mass: Corollary

If (Tn) ⊂ Mm(X ) converges weakly (i.e. pointwise) to T ∈ Mm(X ) then M(T ) ≤ lim inf M(Tn). n→∞

Remark: Mm(X ) becomes a complete metric space with

dM (T , S) := M(T − S).

Constructions for T ∈ Mm(X ): (i) If m ≥ 1 deﬁne

∂T (f , π1, . . . , πm−1) := T (1, f , π1, . . . , πm−1). ⇒ ∂T satisﬁes axioms (i) – (iii) for currents; and ∂∂T = 0.

(ii) If ϕ : X → Y Lipschitz deﬁne

ϕ#T (g, τ1, . . . , τm) := T (g ◦ ϕ, τ1 ◦ ϕ, . . . , τm ◦ ϕ). m ⇒ ϕ#T ∈ Mm(Y ) and kϕ#T k ≤ Lip(ϕ) ϕ#kT k.

(iii) For A ⊂ X Borel deﬁne

(T A)(f , π1, . . . , πm) := T (f 1A, π1, . . . , πm).

⇒ T A ∈ Mm(X ) and kT Ak = kT k A.

Example: For θ ∈ L1(Rm) Z [θ](f , π) := f θ det(dπ)dL m Rm deﬁnes metric m-current in Rm and k[θ]k = |θ|dL m. Space of normal m-currents in X : Nm(X ) := T ∈ Mm(X ): ∂T ∈ Mm−1(X )}.

Theorem

If X is compact and (Tn) ⊂ Nm(X ) satisﬁes

sup [M(Tn) + M(∂Tn)] < ∞ n

then there exists (Tnj ) converging weakly to some T ∈ Nm(X ).

Deﬁnition

T ∈ M0(X ) is called integer rectiﬁable if ∃xi ∈ X and mi ∈ Z, i = 1,..., n, with n X T = mi [xi ]. i=1

Deﬁnition

T ∈ Mm(X ) with m ≥ 1 is called integer rectiﬁable if (i) kT k is concentrated on a countably H m-rectiﬁable set. (ii) kT k vanishes on H m-negligible Borel sets. (iii) ∀ϕ : U → Rm Lipschitz, U ⊂ X open, ∃θ ∈ L1(Rm, Z) with

ϕ#(T U) = [θ].

Space of integer rectiﬁable m-currents in X :

Im(X ) := T ∈ Mm(X ): T integer rectiﬁable}.

Representation of integer rectiﬁable currents: Theorem m If T ∈ Im(X ) with m ≥ 1 then ∃ ψi : Ki → X biLipschitz, Ki ⊂ R 1 cpt, and ∃ θi ∈ L (Ki , Z) such that

∞ ∞ X X T = ψi#[θi ] and M(T ) = M(ψi#[θi ]). i=1 i=1

Space of integral m-currents in X :

Im(X ) := Im(X ) ∩ Nm(X ).

Closure Theorem: Theorem

If T ∈ Nm(X ) is the weak limit of a sequence (Tn) ⊂ Im(X ) with

sup [M(Tn) + M(∂Tn)] < ∞ n

then T ∈ Im(X ).

How to deﬁne limit of (arbitrary) sequence of metric spaces?

Let (Xn) be sequence of metric spaces Xn = (Xn, dn).

For given base points pn ∈ Xn deﬁne n o Xˆ = (xn)n∈N : xn ∈ Xn, sup dn(xn, pn) < ∞ . n

0 ˆ For (xn), (xn) ∈ X want to deﬁne distance by

0 lim dn(xn, x ) n→∞ n if limit exists.

∃ device making consistent "choice" of convergent subsequences:

Let ω be a non-principal ultra-filter on N: ω : 2N → {0, 1} finitely additive; ω(N) = 1 and ω(A) = 0 for all A ⊂ N finite. Existence: Zorn’s lemma.

Proposition

If Z is compact Hausdorﬀ and (zn) ⊂ Z then ∃! z ∈ Z such that for all U ⊂ Z open with z ∈ U

ω({n ∈ N : zn ∈ U}) = 1.

Notation: limω zn := z.

⇒ dω is pseudo-metric on Xˆ. Deﬁnition

Ultralimit of (Xn, dn, pn) with respect to ω: (Xn, dn, pn)ω := Xˆ/∼, dω, [(pn)] ,

0 0 where (xn) ∼ (xn):⇔ dω((xn), (xn)) = 0. Examples: ∼ 1 X proper, p ∈ X ⇒ (X , p)ω = (X , p). pt. GH ∼ 2 Xn proper, (Xn, pn) −→ (X∞, p∞) ⇒ (Xn, pn)ω = (X∞, p∞). ∼ 3 E ∞-dim. Banach ⇒ (E, 0)ω =6 (E, 0).

(X , d) metric space, (pn) ⊂ X , rn & 0, ω non-principal ultraﬁlter

Deﬁnition

Ultralimit of (X , rnd, pn) w.r.t. ω is called asymptotic cone of X .

Notation: Xω := (X , rnd, pn)ω.

Theorem (Gromov) For a geodesic metric space X the following are equivalent: (i) X is Gromov hyperbolic. (ii) Every asymptotic cone of X is a metric tree.

X , Y geodesic metric spaces, Y complete, and X ⊂ Y

For c : S1 → X Lipschitz deﬁne

Y Fillarea (c) = inf M(S): S ∈ I2(Y ), ∂S = c#[1[0,1]] .

X Note: ∃λ ≥ 1 such that Fillarea (c) ≤ λ Fillarea0(c) for all c.

Deﬁnition The generalized homological ﬁlling area function in X w.r.t. Y is n o FAX ,Y (r) = sup FillareaY (c): c : S1 → X Lip., L(c) ≤ r

for all r ≥ 0. Abbreviation: FAX (r) := FAX ,X (r).

For T ∈ I1(X ) with ∂T = 0 deﬁne

Y Fillarea (T ) := inf{M(S): S ∈ I2(Y ), ∂S = T }.

Deﬁnition

Y has quadratic isoperimetric inequality for I1(Y ) if there exists C ≥ 0 with FillareaY (T ) ≤ C M(T )2

for all T ∈ I1(Y ) with ∂T = 0.

The following are equivalent:

(i) Y has quadratic isoperimetric inequality for I1(Y ). (ii) ∃D such that FAY (r) ≤ Dr 2 for all r ≥ 0.

Proposition (W.) X , Y geodesic, X ⊂ Y , Y complete, at bounded distance of X . If

FAX ,Y (r) r 2

then ∃ C ≥ 0 such that for every asymptotic cone Xω of X

FAXω (r) ≤ Cr 2

for all r ≥ 0.

Direct consequence:

X 2 Xω 2 FA0 (r) r ⇒ FA (r) ≤ Cr ∀ r ≥ 0.

Let Y be complete metric space. Suppose Y has quadratic isoperimetric inequality for I1(Y ) with constant C.

Lemma (Ambrosio-Kirchheim)

Let T ∈ I1(Y ) with ∂T = 0 and ε > 0. Then ∃ S ∈ I2(Y ) with (i) ∂S = T (ii) M(S) ≤ (1 + ε) FillareaY (T )

(iii) for each x ∈ spt S and all r ∈ [0, d(x, spt T )] 1 kSk(B(x, r)) ≥ r 2. 4C

Remark: Also works for higher-dimensional integral currents.

Stefan Wenger Isoperimetric inequalities and metric structures