On Uryson widths

ConanLeave a comment

This is a note on parts of Gromov’s paper ‘width and related invariants of Riemannian manifolds’ (1988).

For a compact subset C of \mathbb{R}^n, we define the k-codimensional width (or simply k-width) to be the smallest possible number w where there exists a k-dimensional affine subspace P_k \subseteq \mathbb{R}^n s.t. all points of C is no more than w away from P_k.

i.e.

\displaystyle{W}_k(C) = \inf_{P_k \subseteq \mathbb{R}^n} \sup_{p\in C} \mbox{dist}(p, P_k)
where \mbox{dist}(p, P_k) is the length of the orthogonal segment from p to P_k.

It’s easy to see that, for any C,

\mathcal{W}_0(C) \geq \mathcal{W}_1(C) \geq \cdots \geq \mathcal{W}_n(C) = 0.

At the first glance it may seems that \mathcal{W}_0(C) = \frac{\mbox{diam}(C)}{2}. However it is not the case since for example the equilateral triangle of side length 1 in \mathbb{R}^2 has diameter 1 but 0-width \frac{1}{\sqrt{3}}. In fact, by a theorem of Jung, this is indeed the optimum case, i.e. we have:

\frac{1}{2}\mbox{diam}(C) \leq \mathcal{W}_0(C) \leq \sqrt{\frac{n}{2(n+1)}}\mbox{diam}(C)

At this point one might wonder (at least I did), if we want to invent a notion that captures the ‘diameter’ after we ‘forget the longest k-dimensions’, a more direct way seem to be taking the smallest possible number w' where there is an orthogonal projection of C onto a k dimensional subspace P_k where any point p \in P_k has pre-image with diameter \leq w'.

i.e.

\displaystyle \widetilde{\mathcal{W}_k}(X) = \inf_{P_k \subseteq \mathbb{R}^n} \sup_{p \in P_k} \mbox{diam}(\pi^{-1}_{P_k}(p))

Now we easily have \mbox{diam}(C) = \widetilde{\mathcal{W}_0}(C) \geq \widetilde{\mathcal{W}_1}(C) \geq \cdots \geq \widetilde{\mathcal{W}_n}(C) = 0.

However, the disadvantage of this notion is, for example, there is no reason for a semicircle arc to have 1-width 0 but a three-quarters circular arc having positive 1-width.

Since we are measuring how far is the set from being linear, taking convex hull should not make the set ‘wider’ \widetilde{\mathcal{W}_k}, unlike \widetilde{\mathcal{W}_k} is not invariant under taking convex hulls. Note that for convex sets we do have

\frac{1}{2}\widetilde{\mathcal{W}_k}(C) \leq \mathcal{W}_k(C) \leq \sqrt{\frac{n-k}{2(n-k+1)}}\widetilde{\mathcal{W}_k}(C)

\mathcal{W}_k(C) = 0 iff C is contained in a k-plane.

We now generalize this notion to general metric spaces:

Definition: The Uryson k-width of a compact metric space M is the smallest number w where there exists k dimensional topological space X and a continuous map \pi: M \rightarrow X where any point x \in X has pre-image with diameter \leq w.

i.e. \displaystyle UW_k(M) = \inf \{ \ \sup_{x \in X} \mbox{diam}(\pi^{-1}(x)) \ |

\dim{X} = k, \pi:M \rightarrow X \ \mbox{is continuous} \}

Note: Here dimension is the usual covering dimension for topological spaces: i.e. a topological space X is n dimensional if any finite cover of X has a finite refinement s.t. no point of X is contained in more than n_1 sets in the cover and n is the smallest number with this property.

For compact subsets C of \mathbb{R}^n with induced metric, we obviously we have UW_k(C) \leq \widetilde{\mathcal{W}_k}(C) since the pair (P_k, \pi_{p_k}) is clearly among the pairs we are minimizing over.

Speaking of topological dimensions, one of the classical results is the following:

Lebesgue’s lemma: Let M=[0,1]^n be the solid n-dimensional cube, then for any topological space X with \dim(X)<n and any continuous map p: M \rightarrow X, we have image of at least one pair of opposite (n-1)-faces intersect.

Since the conclusion is purely topological, this applies equally well to rectangles. i.e. for M = [0, L_1] \times [0, L_2] \times \cdots \times [0, L_n], L_1 \geq L_2 \geq \cdots \geq L_n, we have UW_{n-1}(M) \geq L_n; furthermore, UW_k(M) \geq L_{k+1} for all k.

(If the later statement does not hold, we write M as M_1 \times M_2, M_1 being the product of the first (k+1) coordinates. Now UW_k(M) \geq UW_k(M_1) \geq L_{k+1}).

In light of the earlier post about minimax inequality, we should note that if we restrict X to be a homeomorphic copy of \mathbb{R}^k then the notion is the same as the minimax length of fibres. In particular as proved in the post the minimax length of the unit disc to \mathbb{R} is 2.

Exercise: Check that for the unit 2-disk, UW_1(D^2) = \sqrt{3}, i.e. the optimum is obtained by contracting the disc onto a triod.

Hence it can indeed be strictly smaller than merely taking \mathbb{R}^k as the targeting space, even for simply connected sets. This gives a better measurement of ‘width’ in the sense that, for example, the \varepsilon neighborhood of a tree will have 1-width about 2 \varepsilon.

When k looks and smells like the unknot…

Conan5 Comments

Valentine’s day special issue~ ^_^

Professor Gabai decided to ‘do some classical topology before getting into the fancy stuff’ in his course on Heegaard structures on 3-manifolds. So we covered the ‘loop theorem’ by Papakyriakopoulos last week. I find it pretty cool~ (So I started applying it to everything regardless of whether a much simpler argument exists >.<)

Let M be a three dimensional manifold with (non-empty) boundary. In what follows everything is assumed to be in the smooth category.

Theorem: (Papakyriakopoulos, ’58)
If f: \mathbb{D}^2 \rightarrow M extends continuously to \partial \mathbb{D} and the image f(\partial \mathbb{D}) \subseteq \partial M is homotopically non-trivial in \partial M. Then in any neighborhood N(f(\mathbb{D})) we can find embedded disc D \subseteq M such that \partial D is still homotopically non-trivial in \partial M.

i.e. this means that if we have a loop on \partial M that is non-trivial in \partial M but trivial in M, then in any neighborhood of it we can find a simple loop that’s still non-trivial in \partial M and bounds an embedded disc in M.

We apply this to the following:

Corollary: If a knot k \subseteq \mathbb{S}^3 has \pi_1(\mathbb{S}^3 \backslash k) = \mathbb{Z} then k is the unknot.

Proof: Take tubular neighborhood N_\varepsilon(k), consider M=\mathbb{S}^3 \backslash \overline{N_\varepsilon(k)}, boundary of M is a torus.

By assumption we have \pi_1(M) = \pi_1(\mathbb{S}^3 \backslash k) = \mathbb{Z}.

Let k' \subseteq \partial M be a loop homotopic to k in N_\varepsilon(k).

Since \pi_1(M) = \mathbb{Z} and any loop in M is homotopic to a loop in \partial M = \mathbb{T}^2. Hence the inclusion map i: \pi_1(\mathbb{T}^2) \rightarrow \pi_1(M) is surjective.

Let l \subseteq \partial M be the little loop winding around k.

It’s easy to see that i(l) generates \pi_1(M). Hence there exists n s.t. k'-n \cdot l = 0 in \pi_1(M). In other words, after n Dehn twists around l, k' is homotopically trivial in M i.e. bounds a disk in M. Denote the resulting curve k''.

Since k'' is simple, there is small neighborhood of k'' s.t. any homotopically non-trivial simple curve in the neighborhood is homotopic to k''. The loop theorem now implies k'' bounds an embedded disc in M.

By taking a union with the embedded collar from k to k'' in N_\varepsilon(k):

We conclude that k bounds an embedded disc in \mathbb{S}^3 \backslash k hence k is the unknot.

Establishes the claim.

Happy Valentine’s Day, Everyone! ^_^

Intergal geometry and the minimax inequality in a nutshell

Conan1 Comment

The goal for most of the posts in this blog has been to take out some very simple parts of certain papers/subjects and “blow them up” to a point where anybody (myself included) can understand. Ideally the simple parts should give some inspirations and ideas towards the more general subject. This one is on the same vein. This one is based on parts of professor Guth’s minimax paper.

In an earlier post, we talked about the extremal length where one is able to bound the “largest possible minimum length” (i.e. the “maximum minimum length“) of a family of rectifiable curves under conformal transformation. When combined with the uniformization theorem in for surfaces, this becomes a powerful tool for understanding arbitrary Riemannian metrics (and for conformal classes of metrics in higher dimensions).

However, in ‘real life’ we often find what we really want to bound is, instead, the “minimum maximum length” of a family of curves, for example:

Question: Let \mathbb{D} \subseteq \mathbb{R}^2 be the unit disc. Given any family \mathcal{F} of arcs with endpoints on \partial \ \mathbb{D} and \mathcal{F} foliates \mathbb{D}, then how short can the logest arc in \mathcal{F} possibly be?

In other words, let \mathbb{F} be the collection of all possible such foliations \mathcal{F} as above, what is

\displaystyle \inf_{\mathcal{F} \in \mathbb{F}} \ \sup_{A \in \mathcal{F}} \ \ell(A)?

After playing around a little bit with those foliations, we should expect one of the fibres to be at least as long as the diameter ( i.e. no foliation has smaller maximum length leaf than foliating by straight lines ). Hence we should have

\displaystyle \inf_{\mathcal{F} \in \mathbb{F}} \ \sup_{A \in \mathcal{F}} \ \ell(A) = 2.

This is indeed easy to prove:

Proof: Consider the map f: S^1 \rightarrow S^1 where S^1 = \partial \ \mathbb{D}, f switches the end-points of each arc in \mathcal{F}. It is easy to check that f is a continuous, orientation reversing homeomorphism of the circle (conjugate to a reflection). Let p, q be its fixed points, L_1, L_2 be the two arcs in S^1 connecting p to q.

Let

g: z \mapsto -z

be the antipodal map on S^1.

Suppose p \neq g(q) then one of L_1, L_2 is longer than \pi, say it’s L_1.

Then we have

f \circ g (L_1) \subseteq L_1.

Hence f \circ g has a fixed point m in L_1, i.e. f(m) = -m.

There is a fibre A in \mathcal{F} with endpoints m, -m, the fibre must have length

\ell(A) \geq d(-m,m) = 2.

The remaining case is trivial: if p = g(q) then both L_1 and L_2 gets mapped into themselves orientation-reversingly, hence fixed points still exists.

Establishes the claim.

Instead of the disc, we may look at circles that sweep out the sphere (hence to avoid the end-point complications):

Theorem: Any one-parameter family of circles that foliates S^2 (except two points) must have the largest circle being longer than the equator.

This is merely applying the same argument, i.e. one of the circles needs to contain a pair of antipodal points hence must be longer than the equator.

In order for easier generalization to higher dimensions, with slight modifications, this can be formulated as:

Theorem: For any f: T^2 \rightarrow S^2 having non-zero degree, there is \theta \in S^1 where \ell(f(S^1 \times \{ \theta \}) is larger than the equator.

Hence in higher dimensions we can try to prove the same statement for largest image of a lever k-sphere under f: S^k \times S^{n-k} \rightarrow S^n. However before we do that I would like to highlight some intergal geometry machineries that are new to me but seemingly constantly used in proving those kinds of estimates. We shall get some idea of the method by showing:

Theorem: Let \mathbb{R}P^n be equipped with the round metric. p^k \subseteq \mathbb{R}P^n be a ‘flat’ k-dimensional plane. Then any k-chain z^k \subseteq \mathbb{R}P^n in the same k dimensional homology class as p^k must have volume at least as large as p^k.

Proof: Let Gr(\mathbb{R}P^n, n-k) be the set of all (n-k)-planes in \mathbb{R}P^n (i.e. the Grassmannian).

There is a standard way to associate a measure \mu on Gr(\mathbb{R}P^n, n-k):

Let \lambda be the Haar measure on SO(n+1), fix some Q \in Gr(\mathbb{R}P^n, n-k).

Since SO(n+1) acts on \mathbb{R}P^n, for open set S \subseteq Gr(\mathbb{R}P^n, n-k), we set

\mu(S) = \lambda( \{ T \in SO(n+1) \ | \ T(Q) \in S \}).

–The measure of a collection of planes is the measure of linear transformations that takes the given plane to an element of the set.

Now we are able to integrate over all (n-k)-planes!

For almost all Q \in Gr(\mathbb{R}P^n, n-k), since P is k-plane, we have | Q \cap P | = 1. ( not 1 only when they are ‘parallel’ )

Since [z] = [p] in H_k(\mathbb{R}P^n, \mathbb{Z}_2), for almost all Q, z intersects Q at least as much as P does. We conclude that for almost all Q, \ | z \cap Q | \geq 1.

Fact: There exists constant C such that for any k-chain \Sigma^k \in \mathbb{R}P^N,

\mbox{Vol}_k(\Sigma^k) = \mathbb{E}(|\Sigma \cap Q |).

The fact is obtained by diving the chain into fine cubes, observe that both volume and expectation are additive and translation invariant. Therefore we only need to show this for infinitesimal cubes (or balls) near 0. We won’t work out the details here.

Hence in our case, since for almost all Q we have | z \cap Q | \geq 1, the expectation \mathbb{E}(|z \cap Q |) \geq 1.

We therefore deduce

\mbox{Vol}_k(z) = \mathbb{E}(|z \cap Q |) \geq 1.

Establishes the theorem.

Remark: I found this intergal geometry method used here being very handy: in the old days I always try to give lower bounds on volume of stuff by intersecting it with planes and then pretend the ‘stuff’ were orthogonal to the plane, which is the worst case in terms of having small volume. An example of such bound can be found in the knot distorsion post where in order to lower bound the length we look at its intersection number with a family of parallel planes and then integrate the intersection.

This is like looking from one particular direction and record how many times did a curve go through each height, of course one would never get the exact length if we know the curve already. What if we are allowed to look from all directions?

I always wondered if we know the intersection number with not only a set of parallel planes but planes in all directions, then are there anything we can do to better bound the volume? Here I found the perfect answer to my question: by integrating over the Grassmannian, we are able to get the exact volume from how much it intersect each plane!

We get some systolic estimates as direct corollaries of the above theorem, for example:

Corollary: \mbox{Sys}_1(\mathbb{R}P^2) = \sqrt{\pi/2} where \mathbb{R}P^2 carries the round metric with total volume 1.

Back to our minimax problems, we state the higher dimensional version:

Wish: For any C^1 map f: S^k \times S^{n-k} \rightarrow S^n where S^n carries the standard round metric, there exists some \theta \in S^{n-k} with

\mbox{Vol}_k(f(S^k\times \{\theta\})) \geq \mbox{Vol}_k(E^k)

where E^k \subseteq S^n is the k-dimensional equator.

But what we have is that there is a (small) positive constant c(n,k) s.t. \mbox{deg}(f) \neq 0 implies

\displaystyle \sup_{\theta \in S^{n-k}} \mbox{Vol}_k(f(S^k \times \{\theta\})) \geq c(n,k) \mbox{Vol}_k(E^k)

(shown by an inductive application of the isomperimetric inequality on S^N, which is obtained from applying intergal geometry methods)

Recurrence and genericity – a translation from French

Conan1 Comment

To commemorate passing the French exam earlier this week (without knowing any French) and also to test this program ‘latex to wordpress‘, I decided to post my French-translation assignment here.

Last year, I went to Paris and heard a French talk by Crovisier. Strangely enough, although I can’t understand a single word he says, just by looking at the slides and pictures, I liked the talk. That’s why when being asked the question ‘so are there any French papers you wanted to look at?’, I immediately came up with this one which the talk was based on.

Here is a translation of selected parts (selected according to my interest) in section 1.2 taken from the paper `Récurrence et Généricité‘ ( Inventiones Mathematicae 158 (2004), 33-104 ) by C. Bonatti and S. Crovisier. In which they proved a connecting lemma for pseudo-orbits.

Interestingly, just in this short section they referred to two results I have discussed in earlier posts of this blog: Conley’s fundamental theorem of dynamical systems and the closing lemma. In any case, I think it’s a cool piece of work to look at! Enjoy~ (Unfortunately, if one wants to see the rest of the paper, one has to read French >.<)

Precise statements of results

1. Statement of the connecting lemma for pseudo-orbits

In all the following work we consider compact manifold {M} equipped with an arbitrary Riemannian metric and sometimes also with a volume form {\omega} (unrelated to the metric). We write {\mbox{Diff}^1(M)} for the set of diffeomorphisms of class {C^1} on {M} with the {C^1} topology and {\mbox{Diff}^1_\omega(M) \subset \mbox{Diff}^1(M)} the subset preserving volume form {\omega}.

Recall that, in any complete metric space, a set is said to be residual if it contains a countable intersection of open and dense sets. A property is said to be generic if it is satisfied on a residual set. By slight abuse of language, we use the term generic diffeomorphisms: the phase ‘generic diffeomorphisms satisfy property P‘ means that property P is generic.

Let f \in \mbox{Diff}^1(M) be a diffeomorphism of M. For all \varepsilon>0, an \varepsilon-pseudo-orbit of f is a sequence (finite or infinite) of points(x_i) such that for all i, d(x_{i+1},f(x_i)) < \varepsilon. We define the following binary relations for pairs of points (x,y) on M:

– For all \varepsilon > 0, we write x \dashv_\varepsilon y if there exists an \varepsilon-pseudo-orbit (x_0, x_1, \cdots, x_k) where x_0 = x and x_k = y for some k \geq 1.

– We write {x \dashv y} if {x \dashv_\varepsilon y} for all {\varepsilon>0}. We sometimes write {x \dashv_f y} to specify the dynamical system in consideration.

– We write {x \prec y} (or {x \prec_f y}) if for all neighborhoods {U, V} of {x} and {y}, respectively, there exists {n \geq 1} such that {f^n(U)} intersects {V}.

Here are a few elementary properties of these relations.

1. The relations {\dashv} and {\dashv_\varepsilon} are, by construction, transitive. The chain recurrent set {\mathcal{R}(f)} is the set of points {x} in {M} such that {x \dashv x}.

2. The relation {x \prec y} is not a-priori transitive. The non-wandering set {\Omega(f)} is the set of points {x} in {M} such that {x \prec x}.

Marie-Claude Arnaud has shown in [Ar] that the relation {\prec} is transitive for generic diffeomorphisms. By using similar methods we show:

Theorem 1: There exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} (or in {\mbox{Diff}^1_\omega(M)}) such that for all diffeomorphisms {f} in {\mathcal{G}} and all pair of points {(x, y)} in {M} we have:

\displaystyle x \dashv_f y \Longleftrightarrow x \prec_f y.

This theorem is a consequence of the following general perturbation result:

Theorem 2: Let {f} be a diffeomorphism on compact manifold {M}, satisfying one of the following two hypotheses:

1. all periodic orbits of {f} are hyperbolic,

2. {M} is a compact surface and all periodic orbits are either hyperbolic or elliptic with irrational rotation number (its derivative has complex eigenvalues, all of modulus {1}, but are not powers of roots of unity).

Let {\mathcal{U}} be a {C^1}-neighborhood of {f} in {\mbox{Diff}^1(M)} (or in {\mbox{Diff}^1_\omega(M)}, if {f} preserves volume form {\omega}). Then for all pairs of points {(x,y)} in {M} such that {x \dashv y}, there exists a diffeomorphism {g} in {\mathcal{U}} and an integer {n>0} such that {g^n(x) = y}.

Remark: In Theorem 2 above, if the diffeomorphism {f} if of class {C^r} with {r \in (\mathbb{N} \backslash \{0\})\cup \{ \infty \}}, then the {C^1}-perturbation {g} can also be chosen in class {C^r}. Indeed the diffeomorphism {g} is obtained thanks to a finite number of {C^1}-perturbations given by the connecting lemma (Theorem 2.1), each of these perturbations is itself of class {C^r}.

Here are a few consequences of these results:

Corollary: There exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all diffeomorphism {f} in {\mathcal{G}}, the chain recurrent set {\mathcal{R}(f)} coincides with the non-wandering set {\Omega(f)}.

Corollary: Suppose {M} is connected, then there exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that if {f \in \mathcal{G}} satisfies {\Omega(f) = M} then it is transitive. Furthermore, {M} is the unique homoclinic class for {f}.

For volume preserving diffeomorphism {f}, the set {\Omega(f)} always coincide with the whole manifold {M}. We therefore find the analogue of this corollary in the conservative case (see section 1.2.4).

2. Dynamical decomposition of generic diffeomorphisms into elementary pieces

Consider the symmetrized relation {\vdash\dashv } of {\dashv} defined by {x \vdash\dashv y} if {x \dashv y} and {y\dashv x}. This relation then induces an equivalence relation on {\mathcal{R}(f)}, where the equivalence classes are called chain recurrence classes.

We say a compact {f}-invariant set {\Lambda} is weakly transitive if for all {x, y \in \Lambda}, we have {x \prec y}. A set {\Lambda} is maximally weakly transitive if it is maximal under the partial order {\subseteq} among the collection of weakly transitive sets.

Since the closure of increasing union of weakly transitive sets is weakly transitive, Zorn’s lemma implies any weakly transitive set is contained in a maximally weakly transitive set. In the case where the relation {\prec_f} is transitive (which is a generic property), the maximally weakly transitive sets are the equivalence classes of the symmetrized relation induced by {\prec} on the set {\Omega(f)}. Hence we obtain, for generic diffeomorphisms:

Corollary: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}} the chain recurrence classesare exactly the maximally weakly transitive sets of {f}.

The result of Conley (see posts on fundamental theorem of dynamical systems) on the decomposition of {\mathcal{R}(f)} into chain recurrence classes will therefore apply (for generic diffeomorphisms) to the decomposition of {\Omega(f)} into maximally weakly transitive sets.

{\cdots}

3. Chain recurrence classes and periodic orbits

Recall that after the establishment of closing lemma by C. Pugh (see the closing lemma post), it is known that periodic points are dense in {\Omega(f)} for generic diffeomorphisms, we would like to use these periodic orbits to better understand the dynamics of chain recurrence classes.

Recall the homoclinic class {H(p, f)} of a hyperbolic periodic point {p} is the closure of all transversal crossing points of its stable and unstable manifolds. This set is by construction transitive, as we have seen in section 1.2, the results of [CMP] imply that, for generic diffeomorphisms any homoclinic class is maximally weakly transitive. By applying corollary 1.4, we see that:

Remark: For generic diffeomorphisms homoclinic classes are also chain recurrence classes.

However, for generic diffeomorphisms, there are chain recurrence classes which are not homoclinic classes, therefore contains no periodic orbit, we call such chain recurrence class with no periodic points aperiodic class.

Corollary: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}}, any connected component with empty interior of {\Omega(f) = \mathcal{R}(f)} is periodic and its orbit is a homoclinic class.

The closing lemma of Pugh and Remark 5 show:

Remark: For generic {f}, any isolated chain recurrence class in {R(f)} is a homoclinic class. In particular this applies to classes that are topological attractors or repellers.
{\cdots}

For non-isolated classes, a recent work (see [Cr]) specifies how a chain recurrence class is approximated by periodic orbits:

Theorem: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}}, all maximally weakly transitive sets of {f} are Hausdorff limits of sequences of periodic orbits.

More general chain recurrence classes satisfy the upper semi-continuity property: if {(x_i) \subseteq \mathcal{R}(f)} is a sequence of points converging to a point {x} then for large enough {n}, the class of {x_n} is contained in an arbitrary small neighborhood of the class of {x}.

C^1 vs. C^1 volume preserving

Conan4 Comments

One of the things I’ve always been interested in is, for a given compact set say in \mathbb{R}^n, what maps defined on the set into \mathbb{R}^n can be extended to a volume preserving map (of certain regularity) on a larger set (for example, some open set containing the original set).

The analogues extension question without requiring the extended map to be volume preserving is answered by the famous Whitney’s extension theorem. It gives a beautiful necessary and sufficient condition on when the map has C^r extension – See this pervious post for more details.

A simple case of this type of question was discussed in my earlier Moser’s theorem post:

Question: Given a diffeomorphism on the circle, when can we extend it to a volume preserving diffeomorphism on the disc?

In the post, we showed that any C^r diffeomorphism on the circle can be extended to a C^{r-1} volume preserving diffeomorphism on the disc. Some time later Amie Wilkinson pointed out to me that, by using generating function methods, in fact one can avoid losing derivative and extend it to a C^r volume preserving.

Anyways, so we know the answer for the circle, what about for sets that looks very different from the circle? Is it true that whenever we can C^r extend the map, we can also so it volume-preserving? (Of course we need to rule out trivial case such as the map is already not volume-preserving on the original set or the map sends, say a larger circle to a smaller circle.)

Question: Is it true that for any compact set K \subseteq \mathbb{R}^n with connected complement, for any function f: K \rightarrow \mathbb{R}^n satisfying the Whitney condition with all candidate derivatives having determent 1, one can always extend f to a volume preserving F: \mathbb{R}^n \rightarrow \mathbb{R}^n.

Note: requiring the set to have connected complement is to avoid the ‘larger circle to small circle’ case and if some candidate derivative does not have determent 1 then the extended map cannot possibly be volume preserving near the point.

After thinking about this for a little bit, we (me, Charles and Amie) came up with the following simple example where the map can only be C^1 extended but not C^1 volume preserving.

Example: Let K \subset \mathbb{R}^2 be the countable union of segments:

K = \{0, 1, 1/2, 1/3, \cdots \} \times [0,1]

As shown below:

Define f: K \rightarrow K be the map that sends the vertical segment above 1/n to the vertical segment above 1/(n+1), preserves the y-coordinate and fixes the segment \{0\} \times [0,1]:

Claim: f can be extended to a C^1 map F: \mathbb{R}^2 \rightarrow \mathbb{R}^2.

Proof: Define g: \mathbb{R} \rightarrow \mathbb{R} s.t.

1) g is the identity on \mathbb{R}^{\leq 0}

2) g(x) = x-1/2 for x>1

3) g: 1/n \mapsto 1/(n+1)

4) g is increasing and differentiable on each [1/n, 1/(n-1)] with derivative no less than (1-1/n)(n^2-n)/(n^2+n) and the one sided derivative at the endpoints being 1.

It’s easy to check such g exists and is continuous:

Since \lim_{n \rightarrow \infty} (1-1/n)(n^2-n)/(n^2+n) = 1, we deduce g is continuously differentiable with derivative 1 at 0.

Let F = g \otimes \mbox{id}, F: \mathbb{R}^2 \rightarrow \mathbb{R}^2 is a C^1 extension of f.

Establishes the claim.

Hence the pair (K, f) satisfies the Whitney condition for extending to C^1 map. Furthermore, since the F as above has derivative being the identity matrix at all points of K, the determent of candidate derivatives are uniformly 1. In other words, this example satisfies all conditions in the question.

Claim: f cannot be extended to a C^1 volume preserving diffeomorphism of the plane.

Proof: The idea here is to look at rectangles with sides on the set K, if F preserves area, they have to go to regions enclosing the same area as the original rectangles, then apply the isoperimetric inequality to deduce that image of some edges of the rectangle would need to be very long, hence at some point on the edge the derivative of F would need to be large.

Suppose such extension F exists, consider rectangle R_n = [1/n, 1/(n-1)] \times [0,1]. We have

m_2(R_n) = 1/(n^2-n)

m_2(R_n) - m_2(R_{n+1})

=1/(n^2-n)-1/(n^2+n)=2/(n^3-n)

Hence in order for F(R_n) to have the same area as R_n, the image of the two segments

s_{n,0} = [1/n, 1/(n-1)] \times \{ 0\} and

s_{n,1}= [1/n, 1/(n-1)] \times \{ 1\}

would need to enclose an area of 2/(n^3-n) \sim n^{-3} outside of the rectangle R_{n+1}.

By isoparametric inequality, the sum of the length of the two curves must be at least \sim n^{-3/2}, while the length of the original segments is 2/(n^2-n) \sim n^{-2}.

Hence somewhere on the segments F needs to have derivative having norm at least

\ell(F(s_{n,0} \cup s_{n,1}) / \ell(s_{n,0} \cup s_{n,1})

\sim n^{-3/2}/n^{-2} = n^{1/2}

We deduce that there exists a sequence of points (p_n) converging to either (0,0) or (0,1) where

|| F'(p_n) || \sim n^{1/2} \rightarrow \infty.

Hence F cannot be C^1 at the limit point of (p_n).

Remark: In fact we have showed the stronger statement that no volume preserving Lipschitz extension could exist and gave an upper bound 1/2 on the best possible Holder exponent.

From this we know the answer to the above question is negative, i.e. not all C^1 extendable map can me extended in a volume preserving fashion. It would be very interesting to give criteria on what map on which sets can be extended. By applying same methods we are also able to produce an example where the set K is a Cantor set on the plane.