The sphere eversion project

2 Local theory of convex integration

2.1 Key construction

The goal of this chapter is to explain the local aspects of (Theillière’s implementation of) convex integration, the next chapter will cover global aspects.

The elementary step of convex integration modifies the derivative of a map in one direction. Let $E$ and $F$ be finite dimensional real normed vector spaces. Let $f \! :E → F$ be a smooth map with compact support.

Definition 2.1
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A dual pair on $E$ is a pair $(π, v)$ where $π$ is a linear form on $E$ and $v$ a vector in $E$ such that $π(v) = 1$.

Say we wish $Df(x)v$ could live in some open subset $Ω_x ⊂ F$. Assume there is a smooth compactly supported family of loops $γ \! :E × 𝕊^1 → F$ such that each $γ_x$ takes values in $Ω_x$, and has average value $\int _{𝕊^1} γ_x = Df(x)v$. Obviously such loops can exist only if $Df(x)v$ is in the convex hull of $Ω_x$, and we saw in the previous chapter that this is almost sufficient (and we’ll see this is sufficiently almost sufficient for our purposes).

Definition 2.2
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The map obtained by corrugation of $f$ in direction $(π, v)$ using $γ$ with oscillation number $N$ is

\[ x ↦ f(x) + \frac1N ∫_0^{Nπ(x)} \left[γ_x(s) - Df(x)v\right]ds. \]

In the above definition, we mostly think of $N$ as a large natural number. But we don’t actually requireit, any positive real number will do.

The next proposition implies that, provided $N$ is large enough, we have achieved $Df’(x)v ∈ Ω_x$, almost without modifying derivatives in the other directions of $\ker π$, and almost without moving $f(x)$. In addition, if we assume that $γ_x$ is constant (necessarily with value $Df(x)v$) for $x$ in some closed subset $K$ where $Df(x)v$ was already good, then the modification is relative to $K$.

Lemma 2.3 Theillière 2018

The corrugated function $f’$ satisfies, uniformly in $x$:

  1. $Df’(x)v = γ(x, Nπ(x)) + O\left(\frac1N\right)$, and the error vanishes whenever $γ_x$ is constant.

  2. $Df’(x)w = Df(x)w + O\left(\frac1N\right)$ for $w ∈ \ker π$

  3. $f’(x) = f(x) + O\left(\frac1N\right)$

  4. $f’(x) = f(x)$ whenever $γ_x$ is constant.

Proof
We set $Γ_x(t) = ∫_0^t \left(γ_x(s) - Df(x)v\right)ds$, so that $f’(x) = f(x) + Γ_x(Nπ(x))/N$. Because each $Γ_x$ is $1$-periodic, and everything has compact support in $E$, all derivatives of $Γ$ are uniformly bounded. Item 3 in the statement is then obvious. Item 2 also follows since $∂_if’(x) = ∂_if(x) + ∂_iΓ(x, Nπ(x))/N$. In order to prove Item 1, we compute: \begin{align*} Df’(x)v & = Df(x)v + \frac1N ∂_jΓ(x, Nπ(x)) + \frac NN ∂_tΓ(x, Nπ(x))\\ & = Df(x)v + O\left(\frac1N\right) + γ(x, Nπ(x)) - Df(x)v\\ & = γ(x, Nπ(x)) + O\left(\frac1N\right). \end{align*} Item 4 is obvious since $Γ_x$ vanishes identically when $γ_x$ is constant.

2.2 The main inductive step

Definition 2.4
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Let $E’$ be a linear subspace of $E$. A map $\mathcal{F}= (f, φ) : E → F × \operatorname{Hom}(E, F)$ is $E’$–holonomic if, for every $v$ in $E’$ and every $x$, $Df(x)v = φ(x)v$.

Definition 2.5
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A first order differential relation for maps from $E$ to $F$ is a subset $\mathcal{R}$ of $E × F × \operatorname{Hom}(E, F)$.

Until the end of this section, $\mathcal{R}$ will always denote a first order differential relation for maps from $E$ to $F$.

Definition 2.6
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A formal solution of a differential relation $\mathcal{R}$ over $U ⊂ E$ is a map $\mathcal{F}= (f, φ) \! :E → F × \operatorname{Hom}(E, F)$ such that, for every $x$ in $U$, $(x, f(x), φ(x))$ is in $\mathcal{R}$.

The first component of a map $\mathcal{F}\! :E → F × \operatorname{Hom}(E, F)$ will sometimes be denoted by $\operatorname{bs}\mathcal{F}\! :E → F$ and called the base map of $\mathcal{F}$.

Definition 2.7

A homotopy of formal solutions over $U$ is a map $\mathcal{F}: ℝ × E → F × \operatorname{Hom}(E, F)$ which is smooth over $[0, 1] × U$ and such that each $x ↦ \mathcal{F}(t, x)$ is a formal solution over $U$ when $t$ is in $[0, 1]$.

Typically, $x ↦ \mathcal{F}(t, x)$ will be denoted by $\mathcal{F}_t$.

We’ll use the notation $\operatorname{Conn}_w A$ to denote the connected component of $A$ that contains $w$, or the empty set if $w$ doesn’t belong to $A$.

Definition 2.8

For every $σ = (x, y, φ)$, the slice of $\mathcal{R}$ at $σ$ with respect to $(π, v)$ is:

\[ \mathcal{R}(σ, π, v) = \operatorname{Conn}_{φ(v)}\{ w ∈ F \; |\; (x, y, φ + (w - φ(v)) ⊗ π) ∈ \mathcal{R}\} . \]

Lemma 2.9

The linear map $φ + (w - φ(v)) ⊗ π)$ coincides with $φ$ on $\ker π$ and sends $v$ to $w$. If $\sigma $ belongs to $\mathcal{R}$ then $φ(v)$ belongs to $\{ w ∈ F, (x, y, φ + (w - φ(v)) ⊗ π) ∈ \mathcal{R}\} $.

Proof
This is direct check.

Definition 2.10

A formal solution $\mathcal{F}$ of $\mathcal{R}$ over $U$ is $(π, v)$–short if, for every $x$ in $U$, $Df(x)v$ belonds to the interior of the convex hull of $\mathcal{R}((x, f(x), φ(x)), π, v)$.

Let $\mathcal{F}$ be a formal solution of $\mathcal{R}$ over an open set $U$. Let $K_1 ⊂ U$ be a compact subset, and let $K_0$ be a compact subset of the interior of $K_1$. Let $C$ be a subset of $U$. Let $E’$ be a linear subspace of $E$ contained in $\ker π$. Let $ε$ be a positive real number.

Assume $\mathcal{R}$ is open over $U$. Assume that $\mathcal{F}$ is $E’$–holonomic near $K_0$, $(π, v)$–short over $U$, and holonomic near $C$. Then there is a homotopy $\mathcal{F}_t$ such that:

  1. $\mathcal{F}_0 = \mathcal{F}$ ;

  2. $\mathcal{F}_t$ is a formal solution of $\mathcal{R}$ over $U$ for all $t$ ;

  3. $\mathcal{F}_t(x) = \mathcal{F}(x)$ for all $t$ when $x$ is near $C$ or outside $K_1$ ;

  4. $d(\operatorname{bs}\mathcal{F}_t(x), \operatorname{bs}\mathcal{F}(x)) ≤ ε$ for all $t$ and all $x$ ;

  5. $\mathcal{F}_1$ is $E’ ⊕ ℝv$–holonomic near $K_0$.

Proof

We denote the components of $F$ by $f$ and $φ$. Since $\mathcal{F}$ is short over $U$, proposition 1.2 applied to $g \! :x ↦ Df(x)v$, $β \! :x ↦ φ(x)v$, $Ω_x = \mathcal{R}(\mathcal{F}(x), π, v)$, and $K = C ∩ K_1$ gives us a smooth family of loops $γ \! :E × [0, 1] × 𝕊^1 → F$ such that, for all $x$ in $U$:

  • $∀ t\, s,\; γ^t_x(s) ∈ \mathcal{R}(\mathcal{F}(x), π, v)$

  • $∀ s,\; γ^0_x(s) = φ(x)v$

  • $\barγ^1_x = Df(x)v$

  • if $x$ is near $C$, $∀t\, s,\; γ^t_x(s) = φ(x)v$

Let $ρ: E → ℝ$ be a smooth cut-off function which equals one on a neighborhood of $K_0$ and whose support is contained in $K_1$.

Let $N$ be a positive real number. Let $\bar f$ be the corrugated map constructed from $f$, $γ^1$ and $N$. Lemma 2.3 ensures that, for all $x$ in $U$,

\[ D\bar f(x) = Df(x) + \left[γ^1_x(Nπ(x)) - Df(x)v\right] ⊗ π + \frac1N B_x \]

for some bounded map $B$ which vanishes whenever $γ_x$ is constant, hence vanishes near $C$.

We set $\mathcal{F}_t(x) = \big (f_t(x), φ_t(x)\big )$ where:

\[ f_t(x) = f(x) + \frac{tρ(x)}N \int _0^{Nπ(x)} \left[γ^t_x(s) - Df(x)v\right]\, ds \]

and

\[ φ_t(x) = φ(x) + \left[γ^{tρ(x)}_x(Nπ(x)) - φ(x)v\right] ⊗ π + \frac{tρ(x)}N B_x. \]

We now prove that $\mathcal{F}_t$ has the announced properties, starting with he obvious ones. The fact that $\mathcal{F}_0 = \mathcal{F}$ is obvious since $γ^0_x(s) = φ(x)v$ for all $s$.

When $x$ is near $C$, $Df(x) = φ(x)$ since $\mathcal{F}$ is holonomic near $C$. In addition, $γ^t_x(s) = φ(x)v$ for all $s$ and $t$, hence $B_x$ vanishes. Hence $\mathcal{F}_t(x) = \mathcal{F}(x)$ for all $t$ when $x$ is near $C$.

Outside of $K_1$, $ρ$ vanishes. Hence $f_t(x) = f(x)$ for all $t$, and $γ^{tρ(x)}_x(s) = φ(x)v$ for all $s$ and $t$, and $φ_t(x) = φ(x)$.

The distance between $f(x)$ and $f_t(x)$ is zero outside of $K_1$ which is compact, and $O\left(1/N\right)$, so it is less than $ε$ for $N$ large enough.

We now turn to the interesting parts. The first one is that each $\mathcal{F}_t$ is a formal solution of $\mathcal{R}$ over $U$. We already now that $\mathcal{F}_t$ coincides with $\mathcal{F}$, which is a formal solution, outside of the compact set $K_1$. We set

\[ \mathcal{F}’_t(x) = \left(f(x), φ(x) + \left[γ^{tρ(x)}_x(Nπ(x)) - φ(x)v\right] ⊗ π\right). \]

Since $\mathcal{R}$ is open over $U$, and $K_1 × [0, 1]$ is compact and $\mathcal{F}_t$ is within $O\left(1/N\right)$ of $\mathcal{F}’_t$, it suffices to prove that $\mathcal{F}’_t$ is a formal solution for all $t$. This is guaranteed by the definition of the slice $\mathcal{R}(\mathcal{F}(x), π, v)$ to which $γ^{tρ(x)}_x(Nπ(x))$ belongs.

Finally, let’s prove that $\mathcal{F}_1$ is $E’ ⊕ ℝv$–holonomic near $K_0$. Since $ρ = 1$ near $K_0$, we have, for $x$ near $K_0$,

\[ Df_1(x) = Df(x) + \left[γ^1_x(Nπ(x)) - Df(x)v\right] ⊗ π + \frac1N B_x, \]

and

\[ φ_1(x) = φ(x) + \left[γ^1_x(Nπ(x)) - φ(x)v\right] ⊗ π + \frac{1}N B_x. \]

Let $p$ be the projection of $E$ onto $\ker π$ along $v$, so that $\operatorname{Id}_E = p + v ⊗ π$. We can rewrite the above formulas as

\[ Df_1(x) = Df(x) ∘ p + γ^1_x(Nπ(x)) ⊗ π + \frac1N B_x, \]

and

\[ φ_1(x) = φ(x) ∘ p + γ^1_x(Nπ(x)) ⊗ π + \frac{1}N B_x. \]

So we see the difference is $Df(x) ∘ p - φ(x) ∘ p$ which vanishes on $E’$ since $\mathcal{F}$ is $E’$–holonomic near $K_0$, and vanishes on $v$ since $p(v) = 0$.

2.3 Ample differential relations

Definition 2.12
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A subset $Ω$ of a real vector space $E$ is ample if the convex hull of each connected component of $Ω$ is the whole $E$.

Lemma 2.13

The complement of a linear subspace of codimension at least 2 is ample.

Proof

Let $F$ be subspace of $E$ with codimension at least $2$. Let $F’$ be a complement subspace. Its dimension is at least $2$ since it is isomorphic to $E/F$ and $\dim (E/F) = \operatorname{codim}(F) ≥ 2$. First note the complement of $F$ is path-connected. Indeed let $x$ and $y$ be points outside $F$. Decomposing on $F ⊕ F’$, we get $x = u + u’$ and $y = v + v’$ with $u’ ≠ 0$ and $v’ ≠ 0$. The segments from $x$ to $u’$ and $y$ to $v’$ stay outside $F$, so it suffices to connect $u’$ and $v’$ in $F’ ∖ \{ 0\} $. If the segment from $u’$ to $v’$ doesn’t contains the origin then we are done. Otherwise $v’ = μu’$ for some (negative) $u’$. Since $\dim (F’) ≥ 2$ and $u’ ≠ 0$, there exists $f ∈ F’$ which is linearly independent from $u’$, hence from $v’$. We can then connect both $u’$ and $v’$ to $f$ by a segment away from zero.

We now turn to ampleness. The connectedness result reduces to prove that every $e$ in $E$ is in the convex hull of $E ∖ F$. If $e$ is not in $F$ then it is the convex combination of itself with coefficient $1$ and we are done. Now assume $e$ is in $F$. The codimension assumption guarantees the existence of a subspace $G$ such that $\dim (G) = 2$ and $G ∩ F = \{ 0\} $. Let $(g_1, g_2)$ be a basis of $G$. We set $p_1 = e + g_1$, $p_2 = e + g_2$, $p_3 = e - g_1 - g_2$. All these points are in $E ∖ F$ and $e = p_1/3 + p_2/3 + p_3/3$.

A first order differential relation $\mathcal{R}$ is ample if all it slices are ample.

Lemma 2.15

The relation of immersions in positive codimension is open and ample.

Proof
For every $σ = (x, y, φ)$ in the immersion relation $\mathcal{R}$, and for every dual pair $(π, v)$, the slice $\mathcal{R}(σ, π, v)$ is the set of $w$ which do not belong to the image of $\ker π$ under $φ$. Since $\dim F > \dim E$, this image has codimension at least $2$ in $F$, and lemma 2.13 concludes.

Lemma 2.16

Let $\mathcal{F}$ be a formal solution of $\mathcal{R}$ over an open set $U$. Let $K_1 ⊂ U$ be a compact subset, and let $K_0$ be a compact subset of the interior of $K_1$. Assume $\mathcal{F}$ is holonomic near a subset $C$ of $U$. Let $ε$ be a positive real number.

If $\mathcal{R}$ is open and ample over $U$ then there is a homotopy $\mathcal{F}_t$ such that:

  1. $\mathcal{F}_0 = \mathcal{F}$

  2. $\mathcal{F}_t$ is a formal solution of $\mathcal{R}$ over $U$ for all $t$ ;

  3. $\mathcal{F}_t(x) = \mathcal{F}(x)$ for all $t$ when $x$ is near $C$ or outside $K_1$.

  4. $d(\operatorname{bs}\mathcal{F}_t(x), \operatorname{bs}\mathcal{F}(x)) ≤ ε$ for all $t$ and all $x$ ;

  5. $\mathcal{F}_1$ is holonomic near $K_0$.

Proof

This is a straightforward induction using lemma 2.11. Let $(e_1, \dots , e_n)$ be a basis of $E$, and let $(π_1, \dots , π_n)$ be the dual basis. Let $E’_i$ be the linear subspace of $E$ spanned by $(e_1, \dots , e_i)$, for $1 ≤ i ≤ n$, and let $E’_0$ be the zero subspace of $E$. Each $(π_i, e_i)$ is a dual pair and the kernel of $π_i$ contains $E’_{i - 1}$.

Lemma 2.11 allows to build a sequence of homotopies of formal solutions, each homotopy relating a formal solution which is $E’_i$–holonomic to one which is $E’_{i+1}$–holonomic (always near $K_0$). The shortness condition is always satisfies because $\mathcal{R}$ is ample over $U$. Each homotopy starts where the previous one stopped, stay at $C^0$ distance at most $ε/n$, and is relative to $C$ and the complement of $K_1$.

It then suffices to do a smooth concatenation of theses homotopies. We first pre-compose with a smooth map from $[0, 1]$ to itself that fixes $0$ and $1$ and has vanishing derivative to all orders at $0$ and $1$. Then we precompose by affine isomorphisms from $[0, 1]$ to $[i/n, (i + 1)/n]$ before joining them.