Morse quasiflats I

Definition A.1.

Let K⊂X be a closed subset. A map π:X→K is called a λ-quasi-retraction if it is λ-Lipschitz and the restriction π|K has displacement at most λ.

Lemma A.2.

Let X be a length space and Φ:En→X an L-Lipschitz (L,A)-quasiflat with image Q. Then there exist a constant L¯ depending only on L and A, a metric space X¯, an L¯-bilipschitz embedding X→X¯ and an L-Lipschitz retraction X¯→X with the following additional properties: X¯ contains a L¯-bilipschitz flat Q¯ such that dH⁢(Q,Q¯)<L and dH⁢(X,X¯)<L.

Proof.

We glue 𝔼n×[0,L] to X along 𝔼n×{L} via Φ. The resulting space equipped with the induced length metric is denoted by X¯. We define Q¯=𝔼n×{0}, viewed as a subset of X¯. Note that for x,y∈Q¯, dX¯⁢(x,y)<2⁢L implies dX¯⁢(x,y)=d𝔼n⁢(x,y). Moreover, if d𝔼n⁢(x,y)≥2⁢L⁢(2⁢L+A), then dX¯⁢(x,y)≥12⁢L⁢d𝔼n⁢(x,y). Hence for points x,y∈𝔼n with 2⁢L<d𝔼n⁢(x,y)<2⁢L⁢(2⁢L+A) there holds dX¯⁢(x,y)≥12⁢L+A⁢d𝔼n⁢(x,y) and the canonical embedding 𝔼n↪𝔼n×{0}⊂X¯ is (2⁢L+A)-bilipschitz. The remaining statements follow immediately from the construction. ∎

Corollary A.3.

Let X be a length space and Φ:En→X an L-Lipschitz (L,A)-quasiflat with image Q. Then there exist constants λ1 and λ2 which depend only on L,A and n, and a λ1-Lipschitz quasiretraction π:X→Q such that d⁢(x,π⁢(x))≤λ2 for any x∈Q. Moreover, λ2→0 as A→0.

Proof.

Choose a thickening X¯ as in Lemma A.2 and denote by Q¯⊂X¯ the bilipschitz flat close to Q. By McShane’s extension lemma we obtain a Lipschitz retraction X¯→Q¯, where the Lipschitz constant is controlled by L,A,n. Composing with the natural projection X¯→X we obtain the required map since Q is at distance L from Q¯. ∎

Lemma A.4.

Let X be a metric space with an L′-Lipschitz bicombing and let Q⊂X an n-dimensional (L,A)-quasiflat with λ-Lipschitz quasiretraction π:X→Q. Let σ∈𝐈n⁢(X) and let S=spt⁡(σ). Suppose that h:[0,1]×S→X is the homotopy from S to π⁢(S) induced by the bicombing on X. Let ρ=supx∈S⁡{d⁢(x,π⁢(x))}. Then there exists a constant C>0 depending only on L′,L,A and n such that

Corollary A.5.

Let Q,π,λ1,λ2 be as in Corollary A.3. Suppose X has L′-Lipschitz bicombing. Let σ∈𝐈n⁢(X) such that S=spt⁡(σ)⊂N2⁢ρ⁢(Q)∖Nρ⁢(Q). Suppose that the map h:[0,1]×S→X is the homotopy from S to π⁢(S) induced by the bicombing on X. Suppose ρ>λ2. Then there exists a constant C>0 depending only on L′,L,A and n such that

Proof.

By Lemma A.4, it suffices to show d⁢(x,π⁢(x))≤C⁢ρ for any x∈S. Let z be a point in Q such that ρ≤d⁢(x,z)=d⁢(x,Q)≤2⁢ρ. Then

Definition A.6.

Let 𝒞 be a regular cubulation of ℝn at scale R and let φ:X→ℝn be a Lipschitz map. For a constant C>0, a (C-controlled) rectifiable stratification (at scale R) of an m-cycle S∈𝐙m,c⁢(X) is a collection of currents (SB)B∈𝒞 such that

1. for every B∈𝒞(n-k) holds SB∈𝐈m-k,c⁢(X) and ∥SB∥ is concentrated on φ-1⁢(B∘) where B∘ denotes the interior of B,

2. for every B∈𝒞(n) holds SB=S⁢⌞⁢φ-1⁢(B),

3. for B∈𝒞, let ∂⁡B=∑i=1ϵi⁢Bi be the face decomposition of ∂⁡B, then ∑i=1ϵi⁢SBi is a piece decomposition of ∂⁡SB,

4. ∑B∈𝒞(n-k)𝐌⁢(SB)≤C⋅𝐌⁢(S)Rk.

Lemma A.7.

Let φ:X→Rk be a Lipschitz map and let σ∈Zk⁢(X) be a cycle. Denote by pi:X→R the composition of φ with the projection onto the i-th coordinate axis. Suppose that

1. 〈σ,pi,1〉 and 〈σ,pi,-1〉 exist and are elements in 𝑰k-1⁢(X) for all 1≤i≤k, in particular ∥σ∥⁢(pi-1⁢{±1})=0,

2. for any i≠j hold ∥〈σ,pi,1〉+〈σ,pi,-1〉∥⁢(pj-1⁢{±1})=0.

Let B=[-1,1]k⊂Rk and Bi±=[-1,1]i-1×{±1}×[-1,1]k-i with their interior denoted by B∘ and (Bi±)∘. Then the face decomposition of ∂⁡B=∑i=1k(Bi+-Bi-) induces a piece decomposition

Proof.

We will use the abbreviation 〈σ,pi,s〉|-11=〈σ,pi,1〉-〈σ,pi,-1〉. We want to prove the following slightly more general statement:

Let us introduce the characteristic functions χi:=χ|{-1<pi<1}. Then we can write

We induct on the cardinality of I. The case |I|=1 is clear. Suppose I={i1,…,im}. For 1≤j≤m, let Ij=I∖{ij}. By induction, σ⁢⌞⁢∏i∈I1χi is a normal current:

There are m terms in the above summation, denoted by S1,…,Sm from left to right. Note that for j>1, ∥Sj∥ is concentrated on the set

We claim ∥S1∥ is concentrated on A1. Recall ∥σ∥({pi1=±1})=0, then

by [83, Lemma 4.7]. By induction,

Thus by assumption (2), ∥∂(σ⌞∏i∈I1χi)∥({pi1=±1})=0. Then

As ∥σ⁢⌞⁢∏i∈I1χi∥ is concentrated on ⋂i∈I1pi-1⁢{(-1,1)}, it follows from Lemma 3.3 that ∥〈σ⁢⌞⁢∏i∈I1χi,pi1,1〉∥ is concentrated on ⋂i∈I1pi-1{(-1,1)}∩{pi1=1}. Thus ∥S1∥ is concentrated on A1 as claimed.

Note that Aj1∩Aj2=∅ when j1≠j2, so the above sum is a piece decomposition.

Now we repeat the discussion with I1 replaced by I2 to obtain another piece decomposition. By comparing these, we conclude

which finishes the induction. ∎

Proposition A.8.

Let φ:X→Rn be an L-Lipschitz map. Then there exists a constant C=C⁢(L,n)>0 such that the following holds. For any S∈Zm⁢(X) and R>0 there exists a regular cubulation C of Rn at scale R such that S has a C-controlled rectifiable stratification subordinate to C.

Proof.

The action of R⋅ℤn on ℝn induces a 1-Lipschitz covering map h:ℝn→T, where T is a product of n-circles of length R. For a subset I={i1,i2,…,ik}⊂{1,2,…,n}, let ℝI⊂ℝn and TI⊂T be the associated subspaces. Denote by pI:X→ℝI the composition of φ with the projection ℝn→ℝI. Let πI=h∘pI.

First we claim there exist a point (s1,s2,…,sn)∈T and a constant C=C⁢(L,n) such that for any collection of mutually disjoint subsets I1,I2,…,Ik of {1,…,n}, we have

1. σ:=〈…⁢〈〈S,πI1,sI1〉,πI2,sI2〉⁢…,πIk,sIk〉∈𝐈m-|I|⁢(X),

2. permuting the order of I1,,Ik in the definition of σ will result in the same current,

3. σ=〈S,πI,sI〉, where I=⋃i=1kIi,

4. 𝐌⁢(σ)≤C⋅𝐌⁢(S)R|I|.

Here |I| denotes the cardinality of I. To see the claim, note that by repeatedly applying [83, Theorem 6.5] and Fubini, we can find a full measure subset A0⊂T such that (i), (ii) and (iii) hold. To arrange (iv), by the coarea formula, for each subset I of {1,…,n}, we can find a subset AI⊂A0 such that ℒn⁢(AI)≥(1-ϵ)⁢ℒn⁢(A0) and 𝐌⁢(σ)≤Cϵ⋅𝐌⁢(S)R|I| whenever (s1,…,sn)∈AI. By choosing ϵ sufficiently small (depending on n), the intersection of all AI with I ranging over subsets of {1,…,n} is non-empty and the claim follows.

Let 𝒞 be the cubulation (of scale R) with vertex set h-1⁢((s1,…,sn)). Let B∈𝒞(n-k) be a cube and let I:=IB be the smallest subset of {1,…,n} such that h⁢(B)⊂TI, in particular |I|=n-k. We define SB=〈S,πI,sI〉⁢⌞⁢φ-1⁢(B) and SB∘=〈S,πI,sI〉⁢⌞⁢φ-1⁢(B∘). We claim that the collection (SB)B∈𝒞 satisfies Definition A.6.

Definition A.6 (2) holds by definition and (4) follows from item (iv). We claim SB=SB∘. If B is top-dimensional, then this claim follows from ∥S∥⁢(πi-1⁢(si))=0 for any i∈{1,…,n}. As we slice at regular point when defining iterated slices, a similar argument implies the lower-dimensional case of the claim, which implies the second half of Definition A.6 (1). Now Definition A.6 (3) follows from Lemma A.7 and properties (i)–(iii). Note that (iv) implies SB has finite mass for each B, hence ∂⁡SB has finite mass by (3). The first half of Definition A.6 (1) follows from (3), (i), (ii) and (iii). ∎

Proof of Proposition 5.6.

We are going to use the Federer–Fleming deformation, henceforth referred to as (FFD), with respect to the cubulation 𝒞R0. By Proposition A.8, there exists a second regular cubulation 𝒞 of ℝn at the larger scale R such that S has a rectifiable stratification (SB)B∈𝒞 subordinate to 𝒞.

We will write a≲b if a≤C′⁢b for a constant C′ depending only on L,A,m,n and c.

We only prove the case m≤n and the m>n case is similar.

First we claim that there exists a constant C0=C0⁢(L,A,n) and a family of currents (PB)B∈𝒞 such that

1. for every B∈𝒞(n-k) holds PB∈𝐈m-k,c⁢(ℝn) and spt⁡(PB)⊂NC0⁢(φ⁢(spt⁡(SB))),

2. each PB is a cubical chain with respect to 𝒞R0,

3. for B∈𝒞, let ∂⁡B=∑iϵi⁢Bi be the face decomposition of ∂⁡B, then ∂⁡PB=∑iϵi⁢PBi,

4. ∑B∈𝒞(n-k)𝐌⁢(PB)≤C0⋅𝐌⁢(S)Rk,

5. Fill⁡(φ#⁢S-∑B∈𝒞(n)PB)≤C0⋅𝐌⁢(S).

We define (PB)B∈𝒞 inductively as follows. Let τB=φ#⁢SB. Take SB with B∈𝒞(n-m). Then SB is a 0-dimensional cycle. Applying (FFD) to τB with respect to the cubulation 𝒞R0, we obtain a cubical chain PB and a homotopy hB with ∂⁡hB=PB-τB. Note that

Clearly

Suppose PB and hB with B∈𝒞(n-m+k-1) are already defined such that conditions (a)–(d) hold, ∂⁡hB=PB-τB and

Take B with B∈𝒞n-m+k. Let ∂⁡B=∑iϵi⁢Bi be the face decomposition. Define

Then ∂⁡PB′=∑iϵi⁢PBi. Applying FFD to PB′ with respect to 𝒞R0, we obtain a cubical chain PB and a homotopy hB with ∂⁡hB=PB-τB (note that ∂⁡PB′ is fixed when applying the radial push-out procedure of FFD to PB′). Then

Then the claim follows.

For each PB, define TB=ι⁢(PB), where ι is defined with respect to 𝒞R0 (see Proposition 5.2). Define

Now we construct H∈𝐈m+1⁢(X) such that ∂⁡H=S-T.

We start with B with B∈𝒞(n-m). By the estimates in Definition 5.1, SB-TB is a cycle of diameter ≲R. We apply the strong cone inequality to obtain HB such that

Thus

Suppose HB with B∈𝒞(n-m+k-1) are already defined such that

1. diam⁡(spt⁡(HB))≲R,

2. ∑B∈𝒞(n-m+k-1)𝐌⁢(HB)≲𝐌⁢(S)Rm-k,

3. ∂⁡HB=SB-TB-∑iϵi⁢HBi, where ∂⁡B=∑iBi is the face decomposition.

Take B with B∈𝒞(n-m+k). Consider

Let ∂⁡Bi=∑i⁢jϵi⁢j⁢Bi⁢j be the face decomposition of Bi. Then

by the sign convention. Applying the strong cone inequality to σB we obtain a filling HB. As diam⁡(spt⁡(HBi))≲R for each i, we have diam⁡(spt⁡(σB))≲R, hence diam⁡(spt⁡(HB))≲R. Moreover, 𝐌(HB)≲R⋅(𝐌(SB)+𝐌(TB)+∑Bi𝐌(HBi). Thus by Proposition A.8 and the previous claim,

All items of the conclusions of the proposition now follow. ∎