Real analysis, quantitative topology, and geometric complexity
Abstract
In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings.
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Publ. Mat. 45 (2001), 265–333 REAL ANALYSIS, QUANTITATIVE TOPOLOGY, AND GEOMETRIC COMPLEXITY Stephen Semmes Abstract In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings. Contents 1. Finite polyhedra and combinatorial parameterization problems 266 2. Mappings and distortion 275 3. The mathematics of good behavior much of the time, and the BMO frame of mind 282 4. Quantitative topology, and calculus on singular spaces 288 5. Uniform rectifiability 298 5.1. Smoothness of Lipschitz and bilipschitz mappings 304 5.2. Smoothness and uniform rectifiability 309 5.3. A class of variational problems 313 Appendices Appendix A. Fourier transform calculations 315 Appendix B. Mappings with branching 318 References 320 In general, there can be significant complications involved with homeomorphisms and their behavior. Some aspects of this are reviewed in Section 1; see also [Sem9]. On the other hand, there are ways in which looking at what happens on average, and variations of this, can have useful features, as in real analysis. Here we discuss some topics related to these themes. In Section 2 we consider a basic geometric question about 2000 Mathematics Subject Classification. 42B99. Key words. BMO, bilipschitz mappings, uniform rectifiability. The author was supported by the U.S. National Science Foundation.
266 S. Semmes mappings in the plane and the distortion of distances. The analytic notion of “bounded mean oscillation” arises from this, and is described in Section 3. Section 4 concerns relations between functions and integrals of their derivatives, as on Euclidean spaces. If one is working on something like a surface with a well-behaved parameterization by a Euclidean space, then properties of functions on the surface can be reduced to analogous questions on the Euclidean space, for which there are numerous classical results; here we consider situations in which this may not be available. Finally, Section 5 deals with “uniform rectifiability”, in which a surface can have nice properties on average, but not at all points. The appendices contain some supplements to the earlier sections. This survey originated with the John J. Gergen Memorial Lectures at Duke University in January, 1998. The author would like to thank the Mathematics Department at Duke University for the opportunity to give these lectures. 1. Finite polyhedra and combinatorial parameterization problems Fix a positive integer d, and let Pbe a d-dimensional polyhedron. We assume that Pis a finite union of d-dimensional simplices, so that Phas “pure” dimension d. Problem 1.1. How can one tell if Pis a PL (piecewise-linear) manifold? In other words, when is Plocally PL-equivalent to Rdat each point? To be precise, Pis locally PL-equivalent to Rdat a point x∈Pif there is a neighborhood of xin Pwhich is homeomorphic to an open set in Rdthrough a mapping which is piecewise-linear. This is really just a particular example of a general issue, concerning existence and complexity of parameterizations of a given set. Problem 1.1 has the nice feature that finite polyhedra and piecewise-linear mappings between them can, in principle, be described in finite terms. Before we try to address Problem 1.1 directly, let us review some preliminary matters. It will be convenient to think of Pas being like a simplicial complex, so that it is made up of simplices which are always either disjoint or meet in a whole face of some (lower) dimension. Thus we can speak about the vertices of P, the edges, the 2-dimensional faces, and so on, up to the d-dimensional faces. Since Pis a finite polyhedron, its local structure at any point is pretty simple. Namely, Plooks like a cone over a (d−1)-dimensional polyhedron at every point. To make this precise, imagine that Qis some
Real Analysis, Quantitative Topology, etc. 267 finite polyhedron in some Rn, and let zbeapointinRnwhich is affinelyindependent of Q, i.e., which lies in the complement of an (affine) plane that contains Q. (We can always replace Rnwith Rn+1, if necessary, to ensure that there is such a point.) Let c(Q) denote the set which consists of all rays in Rnwhich emanate from zand pass through an element of Q. We include zitself in each of these rays. This defines the “cone over Qcentered at z”. It does not really depend on the choice of z,in the sense that a different choice of zleads to a set which is equivalent to the one just defined through an invertible affine transformation. If xis a “vertex” of P, in the sense described above, then there is a natural way to choose a (d−1)-dimensional polyhedron Qso that Pis the same as the cone over Qcentered at xin a neighborhood of x. Let us call Qthe link of Pat x. (Actually, with this description Qis only determined up to piecewise-linear equivalence, but this is adequate for our purposes.) Now suppose that xis not a vertex. One can still realize Pas a cone over a (d−1)-dimensional polyhedron near x, but one can also do something more precise. If xis not a vertex, then there is a positive integer kand a k-dimensional face Fof Psuch that xlies in the interior of F. In this case there is a (d−k−1)-dimensional polyhedron Qsuch that Pis locally equivalent to Rk×c(Q) near x, with xin Pcorresponding to a point (y,z)inRk×c(Q), where zis the center of c(Q). This same polyhedron Qworks for all the points in the interior of F, and we call Qthe link of F. Basic Fact 1.2. Pis everywhere locally equivalent to Rdif and only if all of the various links of P(of all dimensions) are piecewise-linearly equivalent to standard spheres (of the same dimension). Here the “standard sphere of dimension m” can be taken to be the boundary of the standard (m+ 1)-dimensional simplex. Basic Fact 1.2 is standard and not hard to see. The “if” part is immediate, since one knows exactly what the cone over a standard sphere looks like, but for the converse there is a bit more to check. A useful observation is that if Qis a j-dimensional polyhedron whose cone c(Q) is piecewise-linearly equivalent to Rj+1 in a neighborhood of the center of c(Q), then Qmust be piecewise-linearly equivalent to a standard j-dimensional sphere. This is pretty easy to verify, and one can use it repeatedly for the links of Pof codimension larger than 1. (A well-known point here is that one should be careful not to use radial projections to investigate links around vertices, but suitable pseudo-radial projections,
268 S. Semmes to fit with the piecewise-linear structure, and not just the topological structure.) A nice feature of Basic Fact 1.2 is that it sets up a natural induction in the dimensions, since the links of Palways have dimension less than P. This leads to the following question. Problem 1.3. If Qis a finite polyhedron which is a k-dimensional PL manifold, how can one tell if Qis a PL sphere of dimension k? It is reasonable to assume here that Qis a PL-manifold, because of the way that one can use Basic Fact 1.2 and induction arguments. Problem 1.3 is part of the matter of the Poincar´e conjecture, which would seek to say that Qis a PL sphere as soon as it is homotopyequivalent to a sphere. This has been established in all dimensions except 3 and 4. (Compare with [RouS].) In dimension 4 the Poincar´e conjecture was settled by M. Freedman [Fre] in the “topological” category (with ordinary homeomorphisms (continuous mappings with continuous inverses) and topological manifolds), but it remains unknown in the PL case. The PL case is equivalent to the smooth version in this dimension, and both are equivalent to the ordinary topological version in dimension 3. (A brief survey related to these statements is given in Section 8.3 of [FreQ].) Although the Poincar´e conjecture is known to hold in the PL category in all higher dimensions (than 4), it does not always work in the smooth category, because of exotic spheres (as in [Mil1], [KerM]). If the PL version of the Poincar´e conjecture is true in all dimensions, then this would give one answer to the question of recognizing PL manifolds among finite polyhedra in Problem 1.1. Specifically, our polyhedron Pwould be a PL manifold if and only if its links are all homotopy-equivalent to spheres (of the correct dimension). This might seem like a pretty good answer, but there are strong difficulties concerning complexity for matters of homotopy. In order for ak-dimensional polyhedron Qto be a homotopy sphere, it has to be simply connected in particular, at least when k≥2. In other words, it should be possible to continuously deform any loop in Qto a single point, or, equivalently, to take any continuous mapping from a circle into Qand extend it to a continuous mapping from a closed disk into Q. This extension can entail enormous complexity, in the sense that the filling to the disk might have to be of much greater complexity than the original loop itself. This is an issue whose geometric significance is often emphasized by Gromov. To describe it more precisely it is helpful to begin with some related algebraic problems, concerning finitely-presented groups.
Real Analysis, Quantitative Topology, etc. 269 Let Gbe a group. A finite presentation of Gis given by a finite list g1,g 2,...,g nof generators for Gtogether with a finite set r1,r 2,...,r m of “relations”. The latter are (finite) words made out of the gi’s and their inverses. Let us assume for convenience that the set of relations includes the inverses of all of its elements, and also the empty word. The rj’s are required to be trivial, in the sense that they represent the identity element of G. This implies that arbitrary products of conjugates of the rj’s also represent the identity element, and the final requirement is that if wis any word in the gi’s and their inverses which represents the identity element in G, then it should be possible to obtain wfrom some product of conjugates of the rj’s through cancellations of subwords of the form g−1 igiand gig−1 i. For instance, the group Z2can be described by two generators a, band one relation, aba−1b−1. As another concrete example, there is the (Baumslag-Solitar) group with two generators x,yand one relation x2yx−1y−1. Suppose that a group Gand finite presentation of Gare given and fixed, and let wbe a word in the generators of Gand their inverses. Given this information, how can one decide whether wrepresents the identity element in G? This is called “the word problem” (for G). It is a famous result that there exist finite presentations of groups for which there is no algorithm to solve the word problem. (See [Man].) To understand what this really means, let us first notice that the set of trivial words for the given presentation is “recursively enumerable”. This means that there is an algorithm for listing all of the trivial words. To do this, one simply has to have the algorithm systematically generate all possible conjugates of the relations, all possible products of conjugates of relations, and all possible words derived from these through cancellations as above. In this way the algorithm will constantly generate trivial words, and every trivial word will eventually show up on the list. However, this does not give a finite procedure for determining that a given word is not trivial. A priori one cannot conclude that a given word is not trivial until one goes through the entire list of trivial words. The real trouble comes from the cancellations. In order to establish the triviality of a given word w, one might have to make derivations through words which are enormously larger, with a lot of collapsing at the end. If one had a bound for the size of the words needed for at least one derivation of the triviality of a given word w, a bound in terms of an effectively computable (or “recursive”) function of the length of w, then the word problem would be algorithmically solvable. One could simply search through all derivations of at most a computable size.
270 S. Semmes This would not be very efficient, but it would be an algorithm. As it is, even this does not always work, and there are finitely-presented groups for which the derivations of triviality may need to involve words of nonrecursive size compared to the given word. One should keep in mind that for a given group and a given presentation there is always some function f(n) on the positive integers so that trivial words of length at most nadmit derivations of their triviality through words of size no greater than f(n). This is true simply because there are only finitely many words of size at most n, and so one can take f(n) to be the maximum size incurred in some finite collection of derivations. The point is that such a function fmay not be bounded by a recursive function. This means that fcould be really huge, larger than any tower of exponentials, for instance. The same kind of phenomenon occurs geometrically, for deciding whether a loop in a given polyhedron can be continuously contracted to a point. This is because any finite presentation of a group Gcan be coded into a finite polyhedron, in such a way that the group Gis represented by the fundamental group of the polyhedron. This is a well-known construction in topology. Note that while the fundamental group of a space is normally defined in terms of continuous (based) loops in the space and the continuous deformations between them, in the case of finite polyhedra it is enough to consider polygonal loops and deformations which are piecewise-linear (in addition to being continuous). This is another standard fact, and it provides a convenient way to think about complexity for loops and their deformations. Although arbitrary finite presentations can be coded into finite polyhedra, as mentioned above, this is not the same as saying that they can be coded into compact manifolds. It turns out that this does work when the dimension is at least 4, i.e., for each n≥4 it is true that every finite presentation can be coded into a compact PL manifold of dimension n. This type of coding can be used to convert algorithmic unsolvability results for problems in group theory into algorithmic unsolvability statements in topology. For instance, there does not exist an algorithm to decide when a given finite presentation for a group actually defines the trivial group, and, similarly, there does not exist an algorithm for deciding when a given manifold (of dimension at least 4) is simply-connected. See [BooHP], [Mar1], [Mar2], [Mar3] for more information and results. Let us mention that in dimensions 3 and less, it is not true that arbitrary finitely-presented groups can be realized as fundamental groups
Real Analysis, Quantitative Topology, etc. 271 of compact manifolds. Fundamental groups of manifolds are very special in dimensions 1 and 2, as is well known. The situation in dimension 3 is more complicated, but there are substantial restrictions on the groups that can arise as fundamental groups. As an aspect of this, one can look at restrictions related to Poincar´e duality. In a different vein, the fundamental group of a 3-dimensional manifold has the property that all of its finitely-generated subgroups are finitely-presented. See [Sco], and Theorem 8.2 on p. 70 of [Hem1]. See also [Jac]. In another direction, there are relatively few abelian groups which can arise as subgroups of fundamental groups of 3-dimensional manifolds. See [Eps], [EvaM], Theorems 9.13 and 9.14 on p. 84f of [Hem1], and p. 67–69 of [Jac]. At any rate, it is a large open problem to know exactly what groups arise as fundamental groups of 3-dimensional manifolds. See also [Thu] and Chapter 12 of [Eps+] concerning these groups. The book [Eps+] treats a number of topics related to computability and groups, and not just in connection with fundamental groups of 3-manifolds. This includes broad classes of groups for which positive results and methods are available. See [Far] as well in this regard. Beginning in dimension 5, it is known that there is no algorithm for deciding when a compact PL manifold is piecewise-linearly equivalent to a standard (PL) sphere. This is a result of S. Novikov. See Section 10 of [VolKF], and also the appendix to [Nab]. Note that in dimensions less than or equal to 3, such algorithms do exist. This is classical for dimensions 1, 2; see [Rub1], [Rub2], [Tho] concerning dimension 3, and related problems and results. Imagine that we have a connected PL manifold Mof some dimension n≥5 whose equivalence to a standard sphere is true but “hard” to check. According to the solution of the Poincar´e conjecture in these dimensions, Mwill be equivalent to an n-sphere if it is homotopyequivalent to Sn. For standard reasons of algebraic topology, this will happen exactly when Mis simply-connected and has trivial homology in dimensions 2 through n−1. (Specifically, this uses Theorem 9 and Corollary 24 on pages 399 and 405, respectively, of [Spa]. It also uses the existence of a degree-1 mapping from Mto Snto get started (i.e., to have a mapping to which the aforementioned results can be applied), and the fact that the homology of Mand Snvanish in dimensions larger than n, and are equal to Zin dimension n. To obtain the degree-1 mapping from Mto Sn, one can start with any point in Mand a neighborhood of that point which is homeomorphic to a ball. One then collapses the complement of that neighborhood to a point, which gives rise to the
272 S. Semmes desired mapping.) The vanishing of homology can be determined algorithmically, and so if the equivalence of Mwith an n-sphere is “hard” for algorithmic verification, then the problem must occur already with the simple-connectivity of M. To determine whether Mis simply-connected it is enough to check that a finite number of loops in Mcan be contracted to a point, i.e., some collection of generators for the fundamental group. If this is “hard”, then it means that the complexity of the contractions should be enormous compared to the complexity of M. For if there were a bound in terms of a recursive function, then one could reverse the process and use this to get an algorithm which could decide whether Mis PL equivalent to a sphere, and this is not possible. If Mis a hard example of a PL manifold which is equivalent to an n-sphere, then any mapping from Mto the sphere which realizes this equivalence must necessarily be of very high complexity as well. Because of the preceding discussion, this is also true for mappings which are homotopy-equivalences, or even which merely induce isomorphisms on π1, if one includes as part of the package of data enough information to justify the condition that the induced mapping on π1be an isomorphism. (For a homotopy equivalence, for instance, one could include the mapping ffrom Mto the n-sphere, a mapping gfrom the n-sphere to M which is a homotopy-inverse to f, and mappings which give homotopies between f◦gand g◦fto the identity on the n-sphere and M, respectively.) This is because one could use the mapping to reduce the problem of contracting a loop in Mto a point to the corresponding problem for the n-sphere, where the matter of bounds is straightforward. Similar considerations apply to the problem of deciding when a finite polyhedron Pis a PL manifold. Indeed, given a PL manifold Mwhose equivalence to a sphere is in question, one can use it to make a new polyhedron Pby taking the “suspension” of M. This is defined by taking two points yand zwhich lie outside of a plane that contains M, and then taking the union of all of the (closed) line segments that go from either of yor zto a point in M. One should also be careful to choose yand zso that these line segments never meet, except in the trivial case of line segments from yand zto the same point xin M, with xbeing the only point of intersection of the two segments. (One can imagine yand zas lying on “opposite sides” of an affine plane that contains M.) If Mis equivalent to a sphere, then this operation of suspension produces a PL manifold equivalent to the sphere of 1 larger dimension, as one can easily check. If Mis not PL equivalent to a sphere, then the
Real Analysis, Quantitative Topology, etc. 273 suspension Pof Mis not a PL manifold at all. This is because Mis the link of Pat the vertices yand z, by construction, so that one is back to the situation of Basic Fact 1.2. Just as there are PL manifolds Mwhose equivalence with a sphere is hard, the use of the suspension shows that there are polyhedra Pfor which the property of being a PL manifold is hard to establish. Through the type of arguments outlined above, when PL coordinates exist for a polyhedron P, they may have to be of enormous complexity compared to the complexity of Pitself. This works more robustly than just for PL coordinates, i.e., it applies to objects which are precise enough to deal with simple-connectivity of the links of P(which are local, as opposed to more global aspects of P). Again, this follows the discussion above. We have focussed on piecewise-linear coordinates for finite polyhedra for the sake of simplicity, but similar themes of complexity come up much more generally, and in a number of different ways. In particular, existence and complexity of parameterizations is often related in a strong manner to the behavior of something like π1, sometimes in a localized form, as with the links of a polyhedron. For topology of manifolds in high dimensions, π1and the filling of loops with disks come up in the Whitney lemma, for instance. This concerns the separation of crossings of submanifolds through the use of embedded 2-dimensional disks, and it can be very useful for making some geometric constructions. (A very nice brief review of some of these matters is given in Section 1.2 of [DonK].) Localized π1-type conditions play a crucial role in taming theorems in geometric topology. Some references related to this are [Bin1], [Bin2], [Bin3], [Bur], [BurC], [Can1], [Can2], [Dave1], [Dave2], [Edw1], [Moi], [Rus1], [Rus2]. As another type of example, one has the famous “double suspension” results of Edwards and Cannon [Can1], [Can3], [Dave2], [Edw2]. Here one starts with a finite polyhedron Hwhich is a manifold with the same homology as a sphere of the same dimension, and one takes the suspension (described above) of the suspension of Hto get a new polyhedron K. The result is that Kis actually homeomorphic to a sphere. A key point is that His not required to be simply-connected. When π1(H)=0, it is not possible for the homeomorphism from Kto a standard sphere to be piecewise-linear, or even Lipschitz continuous (which means that the distance between two points in the image is always bounded by a constant times the distance between their preimages in K). Concerning the latter, see [SieS]. Not much is known about the complexity of the homeomorphisms in this case. (We shall say a bit more about this in Section 5.)
280 S. Semmes different, and one keeps constants like those from the linear decay condition. This comes out clearly in the proof, and we shall see more about it later. This type of exponential decay occurs in a simple way in the example above, in (2.10). (This also comes up in [Joh].) One can obtain this from the presence of 'log rin the angle coordinate in the image. The use of the logarithm here is not accidental, but fits exactly with the requirements on the mapping. For instance, if one differentiates log r in ordinary Cartesian coordinates, then one gets a quantity of size 1/r, and this is balanced by the rin the first part of the polar coordinates in (2.10), to give a result which is bounded. It may be a bit surprising, or disappointing, that uniform approximation to the differential of fdoes not work here. After all, we did have “uniform” (or “supremum”) bounds in the hypothesis (2.1), and so one might hope to have the same kind of bounds in the conclusion. This type of failure of supremum bounds is quite common, and in much the same manner as in the present case. We shall return to this in Section 3. How might one prove (2.11), or the exponential decay bounds for P(λ)? Let us start with a slightly simpler situation. Imagine that we have a rectifiable curve γin the plane whose total length is only slightly larger than the distance between its two endpoints. If the length of γ were equal to the distance between the endpoints, then γwould have to be a straight line segment, and nothing more. If the length is slightly larger, then γhas to stay close to the line segment that joins its endpoints. In analogy with (2.11), we would like to say that the tangents to γare nearly parallel, on average, to the line that passes through the endpoints of γ. In order to analyze this further, let z(t), t∈R,a≤t≤b,bea parameterization of γby arclength. This means that z(t) should be 1-Lipschitz, so that |z(s)−z(t)|≤|s−t|(2.15) for all s, t ∈[a, b], and that |z(t)|= 1 for almost all t, where z(t) denotes the derivative of z(t). Set ζ=z(b)−z(a) b−a=1 b−ab a z(t)dt.(2.16) Let us compute 1 b−ab a|z(s)−ζ|2ds,(2.17)
Real Analysis, Quantitative Topology, etc. 281 which controls the average oscillation of z(s). Let ·,· denote the standard inner product on R2, so that |x−y|2=x−y,x −y=x, x−2x, y+y,y =|x|2−2x, y+|y|2 (2.18) for all x, y ∈R2. Applying this with x=z(s), y=ζ, we get that 1 b−ab a|z(s)−ζ|2ds =1−21 b−ab az(s),ζds +|ζ|2,(2.19) since |z(s)|= 1 a.e., and ζdoes not depend on s. The middle term on the right side reduces to 2ζ,ζ,(2.20) because of (2.16). Thus (2.19) yields 1 b−ab a|z(s)−ζ|2ds =1−2|ζ|2+|ζ|2=1−|ζ|2.(2.21) On the other hand, |z(b)−z(a)|is the same as the distance between the endpoints of γ, and b−ais the same as the length of γ, since z(t)is the parameterization of γby arclength. Thus |ζ|is exactly the ratio of the distance between the endpoints of γto the length of γ, by (2.16), and 1 −|ζ|2is a dimensionless quantity which is small exactly when the length of γand the distance between its endpoints are close to each other (proportionately). In this case (2.21) provides precise information about the way that z(s) is approximately a constant on average. (These computations follow ones in [CoiMe2].) One can use these results for curves for looking at mappings from R2(or Rn) to itself, by considering images of segments under the mappings. This does not seem to give the proper bounds in (2.11), in terms of dependence on δ, though. In this regard, see John’s paper [Joh]. (Compare also with Appendix A.) Note that for curves by themselves, the computations above are quite sharp, as indicated by the equality in (2.21). See also [CoiMe2]. The exponential decay of P(λ) requires more work. A basic point is that exponential decay bounds can be derived in a very general way once one knows (2.11) for all disks Din the plane. This is a famous result of John and Nirenberg [JohN], which will be discussed further in Section 3. In the present situation, having estimates like (2.11) for all disks D (and with uniform bounds) is quite natural, and is essentially automatic, because of the invariances of the condition (2.1) under translations and dilations. In other words, once one has an estimate like (2.11) for some
282 S. Semmes fixed disk Dand all mappings fwhich satisfy (2.1), one can conclude that the same estimate works for all disks D, because of invariance under translations and dilations. 3. The mathematics of good behavior much of the time, and the BMO frame of mind Let us start anew for the moment, and consider the following question in analysis. Let hbe a real-valued function on R2. Let ∆ denote the Laplace operator, given by ∆= ∂2 ∂x2 1 +∂2 ∂x2 2 ,(3.1) where x1,x2are the standard coordinates on R2. To what extent does the behavior of ∆hcontrol the behavior of the other second derivatives of h? Of course it is easy to make examples where ∆hvanishes at a point but the other second derivatives do not vanish at the same point. Let us instead look for ways in which the overall behavior of ∆hcan control the overall behavior of the other second derivatives. Here is a basic example of such a result. Let us assume (for simplicity) that his smooth and that it has compact support, and let us write ∂1 and ∂2for ∂/∂x1and ∂/∂x2, respectively. Then R2|∂1∂2h(x)|2dx ≤R2|∆h(x)|2dx.(3.2) This is a well-known fact, and it can be derived as follows. We begin with the identity R2 ∂1∂2h(x)∂1∂2h(x)dx =R2 ∂2 1h(x)∂2 2h(x)dx,(3.3) which uses two integrations by parts. On the other hand, (3.4) R2|∆h(x)|2dx =R2 (∂2 1h(x)+∂2 2h(x))2dx =R2 (∂2 1h(x))2+2∂2 1h(x)∂2 2h(x)+(∂2 2h(x))2dx.
Real Analysis, Quantitative Topology, etc. 283 Combining this with (3.3) we get that (3.5) R2|∆h(x)|2dx −2R2|∂1∂2h(x)|2dx =R2 (∂2 1h(x))2+(∂2 2h(x))2dx. This implies (3.2), and with an extra factor of 2 on the left-hand side, because the right side of (3.5) is nonnegative. (One can improve this to get a factor of 4 on the left side of (3.2), using the right-hand side of (3.5).) In short, the L2norm of ∆halways bounds the L2norm of ∂1∂2h. There are similar bounds for Lpnorms when 1 <p<∞. Specifically, for each pin (1,∞), there is a constant C(p) such that R2|∂1∂2h(x)|pdx ≤C(p)R2|∆h(x)|pdx(3.6) whenever his a smooth function with compact support. This is a typical example of a “Calder´on-Zygmund inequality”, as in [Duo], [GarcR], [Garn], [Jou], [Sar], [Ste1], [Ste2], [SteW], [StrT], [Torc]. Such inequalities do not work for p=1or∞, and the p=∞case is like the question of supremum estimates in Section 2. Note that the p= 1 and p=∞cases are closely connected to each other, because of duality (of spaces and operators); the operators ∆ and ∂1∂2here are equal to their own transposes, with respect to the standard bilinear form on functions on R2(defined by taking the integral of the product of two given functions). In a modestly different direction, there are classical results which give bounds in terms of the norm for H¨older continuous (or Lipschitz) functions of order α, for every α∈(0,1), instead of the Lpnorm. To be explicit, given α, this norm for a function gon R2can be described as the smallest constant Asuch that |g(x)−g(y)|≤A|x−y|α (3.7) for all x, y ∈R2. One can view this as a p=∞situation, like the L∞ norm for g, but with a positive order αof smoothness, unlike L∞. There is a variety of other norms and spaces which one can consider, and for which there are results about estimates along the lines of (3.6), but for the norm in question instead of the Lpnorm. The p=∞version of (3.6) would say that there is a constant Csuch that sup x∈R2|∂1∂2h(x)|≤Csup x∈R2|∆h(x)|(3.8)
284 S. Semmes whenever his smooth and has compact support. In order to see that this is not the case, consider the function h(x) given by h(x)=x1x2log(x2 1+x2 2),(3.9) x=(x1,x 2). It is not hard to compute ∆hand ∂1∂2hexplicitly, and to see that ∆his bounded while ∂1∂2his not. Indeed, ∂1∂2h(x) = log(x2 1+x2 2) + bounded terms,(3.10) while the logarithm does not survive in ∆h, because ∆(x1x2)≡0. This choice of his neither smooth nor compactly supported, but these defects can be corrected easily. For smoothness we can consider instead h(x)=x1x2log(x2 1+x2 2+'),(3.11) where '>0, and then look at what happens as '→0. To make the support compact we can simply multiply by a fixed cut-off function that does not vanish at the origin. With these modifications we still get a singularity at the origin as '→0, and we see that (3.8) cannot be true (with a fixed constant Cthat does not depend on h). This is exactly analogous to what happened in Section 2, i.e., with a uniform bound going in but not coming out. Instead of a uniform bound for the output, we also have a substitute in terms of “mean oscillation”, just as before. To be precise, let Dbe any disk in R2of radius r, and consider the quantity 1 πr2D|∂1∂2h(x)−AverageD(∂1∂2h)|dx,(3.12) where “AverageD∂1∂2h” is the average of ∂1∂2hover the disk D, i.e., AverageD(∂1∂2h)= 1 πr2D ∂1∂2h(u)du.(3.13) Instead of (3.8), it is true that there is a constant C>0 so that 1 πr2D|∂1∂2h(x)−AverageD(∂1∂2h)|dx ≤Csup x∈R2|∆h(x)|(3.14) for every disk Din R2of radius rand every smooth function hwith compact support. This is not too hard to prove; roughly speaking, the point is to “localize” the L2estimate that we had before. (Results of this nature are discussed in [Duo], [GarcR], [Garn], [Jou], [Sar], [Ste2], [Torc]. There are also replacements for (3.6) for p= 1, and even for 0<p<1 (!); in this connection, see [Ste1], [SteW], [StrT], in addition to the references just mentioned.)
Real Analysis, Quantitative Topology, etc. 285 Let us formalize this estimate by defining a new space of functions, namely the space BMO of functions of bounded mean oscillation, introduced by John and Nirenberg in [JohN]. A locally-integrable function g on R2is said to lie in BMO if there is a nonnegative number ksuch that 1 πr2D|g(x)−AverageD(g)|dx ≤k(3.15) for every disk Din R2of radius r. In this case we set g∗=sup D 1 πr2D|g(x)−AverageD(g)|dx,(3.16) with the supremum taken over all disks Din R2. This is the same as saying that g∗is the smallest number kthat satisfies (3.15). One refers to g∗as the “BMO norm of g”, but notice that g∗= 0 when gis equal to a constant almost everywhere. (The converse is also true.) This definition may look a little crazy, but it works quite well in practice. Let us reformulate (3.14) by saying that there is a constant C so that ∂1∂2h∗≤C∆h∞,(3.17) where φ∞denotes the L∞norm of a given function φ. In other words, although the L∞norm of ∂1∂2his not controlled (for all h)bytheL∞ norm of ∆h, the BMO norm of ∂1∂2his controlled by the L∞norm of ∆h. Similarly, one of the main points in Section 2 can be reformulated as saying that if a mapping f:R2→R2distorts distances by only a small amount, as in (2.1), then the BMO norm df ∗of the differential of fis small (and with precise estimates being available). In Section 2 we mentioned a stronger estimate with exponential decay in the measure of certain “bad” sets. This works for all BMO functions, and can be given as follows. Suppose that gis a BMO function on R2 with g∗≤1, and let Dbe a disk in R2with radius r. As in (2.12), consider the “distribution function” P(λ) defined by P(λ) = Probability({x∈D:|g(x)−AverageD(g)|≥λ}),(3.18) where “Probability” means Lebesgue measure divided by the area πr2 of D. Under these conditions, there is a universal bound for P(λ) with exponential decay, i.e., an inequality of the form P(λ)≤B−λfor λ≥1,(3.19) where Bis a positive number greater than 1, and Bdoes not depend on gor D. This is a theorem of John and Nirenberg [JohN].
286 S. Semmes Although we have restricted ourselves to R2here for simplicity, everything goes over in a natural way to Euclidean spaces of arbitrary dimension. In fact, there is a much more general framework of “spaces of homogeneous type” in which basic properties of BMO (and other aspects of real-variable harmonic analysis) carry over. See [CoiW1], [CoiW2], and compare also with [GarcR], [Ste2]. This framework includes certain Carnot spaces that arise in several complex variables, like the unit sphere in Cnwith the appropriate (noneuclidean) metric. The exponential decay bound in (3.19) helps to make precise the idea that BMO functions are very close to being bounded (which would correspond to having P(λ) = 0 for all sufficiently large λ). The exponential rate of decay implies that BMO functions lie in Lplocally for all finite p, but it is quite a bit stronger than that. A basic example of a BMO function is log |x|. This is not hard to check, and it shows that exponential decay in (3.19) is sharp, i.e., one does not have superexponential decay in general. This example also fits with (3.10), and with the “rotational” part of the differential of the mapping fin (2.10). In general, BMO functions can be much more complicated than the logarithm. Roughly speaking, the total “size” of the unboundedness is no worse than for the logarithm, as in (3.19), but the arrangement of the singularities can be more intricate, just as one can make much more complex singular examples than in (3.9) and (2.10). There are a lot of tools available in harmonic analysis for understanding how BMO functions behave. (See [Duo], [GarcR], [Garn], [Jou], [Sar], [Ste2], [StrT], [Torc], for instance.) BMO functions show up all over the place. One can reformulate the basic scenario in this section with the Laplacian and ∂1∂2by saying that the pseudodifferential or singular integral operator ∂1∂2 ∆ (3.20) maps L∞to BMO, and this holds for similar operators (of order 0) much more generally (as in the references above). This will be discussed a bit further in Appendix A. Note that the nonlinear problem in Section 2 has a natural linearization which falls into this rubric. (See Appendix A.) Sobolev embeddings provide another class of linear problems in which BMO comes up naturally. One might wish that a function gon Rnthat satisfies ∇g∈Ln(Rn) (in the sense of distributions) were bounded or continuous, but neither of these are true in general, when n>1. However, such a function gis always in BMO, and in the subspace VMO
Real Analysis, Quantitative Topology, etc. 287 (“vanishing mean oscillation”), in which the measurements of mean oscillation (as in the left side of (3.15) when n= 2) tend to 0 as the radius r goes to 0. This is a well-known analogue of continuity in the context of BMO. (See [BreN], [GarcR], [Garn], [Sar], [Sem8], [Ste2], [Torc].) BMO arises in a lot of nonlinear problems, in addition to the one in Section 2. For instance, there are circumstances in which one might wish that the derivative of a conformal mapping in the complex plane were bounded, and it is not, but there are natural estimates in terms of BMO. More precisely, it is BMO for the logarithm of the derivative that comes up most naturally. This is closely related to BMO conditions for tangents to curves under certain geometric conditions. See [CoiMe1], [CoiMe2], [CoiMe3], [Davi1], [JeK1], [Pom1], [Pom2], [Pom3], [Sar], for instance. Some basic computations related to the latter were given in Section 2, near the end. In general dimensions (larger than 1), BMO shows up naturally as the logarithm of the density for harmonic measure for Lipschitz domains, and for the logarithm of Jacobians of quasiconformal mappings. See [Dah1], [Dah2], [JeK2], [Geh2], [Rei], [Ste2] and the references therein. In all dimensions, there are interesting classes of “weights”, positive functions which one can use as densities for modifications of Lebesgue measure, whose logarithms lie in BMO, and which in fact correspond to open subsets of BMO (for real-valued functions). These weights have good properties concerning Lpboundedness of singular integral and other operators, and they also show up in other situations, in connection with conformal mappings in the plane, harmonic measure, and Jacobians of quasiconformal mappings in particular, as above. See [Duo], [GarcR], [Garn], [Jou], [Sar], [Ste2], [StrT], [Torc] for information about these classes of weights. There is a simple reason for BMO functions to arise frequently as some kind of logarithm. In many nonlinear problems there is a symmetry which permits one to multiply some quantity by a constant without changing anything in a significant way. (E.g., think of rescaling or rotating a domain, or a mapping, or multiplying a weight by a positive constant.) At the level of the logarithm this invariance is converted into a freedom to add constants, and this is something that BMO accommodates automatically. To summarize a bit, there are a lot of situations in which one has some function that one would like to be bounded, but it is not, and for which BMO provides a good substitute. One may not expect at first to have to take measure theory into account, but then it comes up on its own, or works in a natural or reasonable way.
288 S. Semmes Before leaving this section, let us return to the John-Nirenberg theorem, i.e., the exponential decay estimate in (3.19). How might one try to prove this? The first main point is that one cannot prove (3.19) for a particular disk Dusing only a bound like (3.15) for that one disk. That would only give a rate of decay on the order of 1/λ. Instead one uses (3.15) over and over again, for many different disks. Here is a basic strategy. Assume that gis a BMO function with g∗≤1. First use (3.15) for Ditself (with k= 1) to obtain that the set of points xin Dsuch that |g(x)−AverageD(g)|≥10,(3.21) is pretty small (in terms of probability). On the bad set where this happens, try to make a good covering by smaller disks on which one can apply the same type of argument. The idea is to then show that the set of points xin Dwhich satisfy |g(x)−AverageD(g)|≥10+10(3.22) is significantly smaller still, and by a definite proportion. If one can repeat this forever, then one can get exponential decay as in (3.19). More precisely, at each stage the size of the deviation of g(x) from AverageD(g) will increase by the addition of 10, while the decrease in the measure of the bad set will decrease multiplicatively. This strategy is roughly correct in spirit, but to carry it out one has to be more careful in the choice of “bad” set at each stage, and in the transition from one stage to the next. In particular, one should try to control the difference between the average of gover one disk and over one of the smaller disks created in the next step of the process. As a practical matter, it is simpler to work with cubes instead of disks, for the way that they can be decomposed evenly into smaller pieces. The actual construction used is the “Calder´on-Zygmund decomposition”, which itself has a lot of other applications. See [JohN], [Duo], [GarcR], [Garn], [Jou], [Sar], [Sem8], [Ste2], [StrT], [Torc] for more information. 4. Quantitative topology, and calculus on singular spaces One of the nice features of Euclidean spaces is that it is easy to work with functions, derivatives, and integrals. Here is a basic example of this. Let fbe a real-valued function on Rnwhich is continuously differentiable and has compact support, and fix a point x∈Rn. Then |f(x)|≤ 1 νnRn 1 |x−y|n−1|∇f(y)|dy,(4.1)
Real Analysis, Quantitative Topology, etc. 289 where νndenotes the (n−1)-dimensional volume of the unit sphere in Rn, and dy refers to ordinary n-dimensional volume. This inequality provides a way to say that the values of a function are controlled by averages of its derivative. In this respect it is like Sobolev and isoperimetric inequalities, to which we shall return in a moment. To prove (4.1) one can proceed as follows (as on p. 125 of [Ste1]). Let vbe any element of Rnwith |v|= 1. Then f(x)=−∞ 0 ∂ ∂tf(x+tv)dt,(4.2) by the fundamental theorem of calculus. Thus |f(x)|≤∞ 0|∇f(x+tv)|dt.(4.3) This is true for every vin the unit sphere of Rn, and by averaging over these v’s one can derive (4.1) from (4.3). To put this into perspective, it is helpful to look at a situation where analogous inequalities make sense but fail to hold. Imagine that one is interested in inequalities like (4.1), but for 2-dimensional surfaces in R3 instead of Euclidean spaces themselves. Let Sbe a smoothly embedded 2-dimensional submanifold of R3which looks like a 2-plane with a bubble attached to it. Specifically, let us start with the union of a 2-plane P and a standard (round) 2-dimensional sphere Σ which is tangent to P at a single point z. Then cut out a little neighborhood of z, and glue in a small “neck” as a bridge between the plane and the sphere to get a smooth surface S. If the neck in Sis very small compared to the size of Σ, then this is bad for an inequality like (4.1). Indeed, let xbe the point on Σ which is exactly opposite from z, and consider a smooth function fwhich is equal to 1 on most of Σ (and at xin particular) and equal to 0 on most of P. More precisely, let us choose fso that its gradient is concentrated near the bridge between Σ and P.Iffmakes the transition from vanishing to being 1 in a reasonable manner, then the integral of |∇f|on Swill be very small. This is not hard to check, and it is bad for having an inequality like (4.1), since the left-hand side would be 1 and the righthand side would be small. In particular, one could not have uniform bounds that would work for arbitrarily small bridges between Pand Σ. The inequality (4.1) is a relative of the usual Sobolev and isoperimetric inequalities, which say the following. Fix a dimension nagain, and an exponent pthat satisfies 1 ≤p<n. Define qby 1/q =1/p −1/n, so that p<q<∞. The Sobolev inequalities assert the existence of a
296 S. Semmes would like to find a mapping πx:M\{x}→Sn−1which is topologically nondegenerate and satisfies |dπx(u)|≤Kd(u, x)−1 (4.18) for some constant Kand all u∈M\{x}. Note that now the norm of the differential of πxinvolves the Riemannian metric on M. For the topological nondegeneracy of πx, let us ask that it have nonzero degree on small spheres in Mthat surround xin a standard way. This makes sense, because of the a priori assumption that Mbe smooth. If one can produce such a mapping πx, then one can derive (4.8) as a consequence, using the same kind of argument with differential forms as above. One can also find enough curves in the fibers of πx, with control on the way that their arclength measures are distributed in M, through the use of co-area estimates. For this the topological nondegeneracy of πxis needed for showing that the fibers of πxconnect xto infinity in M. In the context of conformal deformations of Rn,asin[DaviS1], such mappings πxcan be obtained as perturbations to the standard mapping in (4.13). This is described in [Sem7]. For Theorem 4.11, the method of [Sem6] does not use mappings quite like πx, but a “stabilized” version from which one can draw similar conclusions. In this stabilized version one looks for mappings from Mto Sn(instead of Sn−1) which are constant outside of a (given) ball, topologically nontrivial (in the sense of nonzero degree), and which satisfy suitable bounds on their differentials. These mappings are like snapshots of pieces of M, and one has to move them around in a controlled manner. This means moving them both in terms of location (the center of the supporting ball) and scale (the radius of the ball). At this stage the hypotheses of Theorem 4.11 may make more sense. Existence of mappings like the ones described above is a standard matter in topology, except for the question of uniform bounds. The hypotheses of Theorem 4.11 (the doubling condition and local linear contractability) are also in the nature of quantitative topology. Note, however, that the kind of bounds involved in the hypotheses of the theorem and the construction of mappings into spheres are somewhat different from each other, with bounds on the differentials being crucial for the latter, while control over moduli of continuity does not come up in the former. (The local linear contractability condition restricts the overall distances by which points are displaced in the contractions, but not the sizes of the smaller-scale oscillations, as in a modulus of continuity.) In the end the
Real Analysis, Quantitative Topology, etc. 297 bounds for the differentials come about because the hypotheses of Theorem 4.11 permit one to reduce various constructions and comparisons to finite models of controlled complexity. In the proof of Theorem 4.11 there are three related pieces of information that come out, namely (1) estimates for the behavior of functions on our space Min terms of their derivatives, as in (4.8), (2) families of curves in Mwhich are well-distributed in terms of arclength measure, and (3) mappings to spheres with certain estimates and nondegeneracy properties. These three kinds of information are closely linked, through various dualities, but to some extent they also have their own lives. Each would be immediate if Mhad a bilipschitz parameterization by Rn,but in fact they are more robust than that, and much easier to verify. Indeed, one of the original motivations for [DaviS1] was the problem of determining which conformal deformations of Rnlead to metric spaces (through the geodesic distance) which are bilipschitz equivalent to Rn. The deformations are allowed to be nonsmooth here, but this does not matter too much, because of the natural scale-invariance of the problem, and because one seeks uniform bounds. This problem is the same in essence as asking which (positive) functions on Rnarise as the Jacobian of a quasiconformal mapping, modulo multiplication by a positive function which is bounded and bounded away from 0. Some natural necessary conditions are known for these questions, with a principal ingredient coming from [Geh2]. It was natural to wonder whether the necessary conditions were also sufficient. As a test for this, [DaviS1] looked at the Sobolev and related inequalities that would follow if the necessary conditions were sufficient. These inequalities could be stated directly in terms of the data of the problems, the conformal factor or prospective Jacobian. The conclusion of [DaviS1] was that these inequalities could be derived directly from the conditions on the data, independently of whether these conditions were sufficient for the existence of bilipschitz/quasiconformal mappings as above. In [Sem5] it was shown that the candidate conditions are not sufficient for the existence of such mappings, at least in dimensions 3 and higher. (Dimension 2 remains open.) The simplest counterexamples involved considerations of localized fundamental groups, in much the same fashion as in Section 1. (Another class of counterexamples were based on a different mechanism, although these did not start in dimension 3.) These counterexamples are all perfectly well-behaved in terms of the doubling and local linear contractability properties, and in fact are much better than that.
298 S. Semmes Part of the bottom line here is that spaces can have geometry which behaves quite well for many purposes even if they do not behave so well in terms of parameterizations. For some other aspects of “quantitative topology”, see [Ale], [AleV1], [AleV2], [Att1], [Att2], [BloW], [ChaF], [Che], [Fer1], [Fer2], [Fer3], [Fer4], [Geh1], [Gro1], [Gro2], [Gro3], [HeiY], [HeiS], [Luu], [Pet1], [Pet2], [TukV], [V¨ai2], [V¨ai3], [V¨ai4]. Related matters of Sobolev and other inequalities on non-smooth spaces come up in [HeiKo2], [HeiKo3], [HeiKo+], in connection with the behavior of quasiconformal mappings. 5. Uniform rectifiability A basic fact in topology is that there are spaces which are manifold factors but not manifolds. That is, there are topological spaces Msuch that M×Ris a manifold (locally homeomorphic to a Euclidean space) but Mis not. This can even happen for finite polyhedra, because of the double-suspension results of Edwards and Cannon. See [Dave2], [Edw2], [Kir] for more information. Uniform rectifiability is a notion of controlled geometry that trades topology for estimates. It tolerates some amount of singularities, like holes and crossings, and avoids some common difficulties with homeomorphisms, such as manifold factors. The precise definition is slightly technical, and relies on measure theory in a crucial way. In many respects it is analogous to the notion of BMO from Section 3. The following is a preliminary concept that helps to set the stage. Definition 5.1 (Ahlfors regularity).Fix nand d, with na positive integer and 0 <d≤n. A set Econtained in Rnis said to be (Ahlfors) regular of dimension dif it is closed, and if there is a positive Borel measure µsupported on Eand a constant C>0 such that C−1rd≤µ(B(x, r)) ≤Cr d (5.2) for all x∈Eand 0 <r≤diam E. Here B(x, r) denotes the (open) ball with center xand radius r. Roughly speaking, this definition asks that Ebehave like ordinary Euclidean space in terms of the distribution of its mass. Notice that d-planes satisfy this condition automatically, with µequal to the ordinary d-dimensional volume. The same is true for compact smooth manifolds, and finite polyhedra which are given as unions of d-dimensional simplices (i.e., with no lower-dimensional pieces sticking off in an
Real Analysis, Quantitative Topology, etc. 299 isolated manner). There are also plenty of “fractal” examples, like selfsimilar Cantor sets and snowflake curves. In particular, the dimension d can be any (positive) real number. A basic fact is that if Eis regular and µis as in Definition 5.1, then µis practically the same as d-dimensional Hausdorff measure Hdrestricted to E. Specifically, µand Hdare each bounded by constant multiples of the other when applied to subsets of E. This is not hard to prove, and it shows that µis essentially unique. Definition 5.1 could have been formulated directly in terms of Hausdorff measure, but the version above is a bit more elementary. Let us recall the definition of a bilipschitz mapping. Let Abe a set in Rn, and let fbe a mapping from Ato some other set in Rn.Wesay that fis k-bilipschitz, where kis a positive number, if k−1|x−y|≤|f(x)−f(y)|≤k|x−y|(5.3) for all x, y ∈A. Definition 5.4 (Uniform rectifiability).Let Ebe a subset of Rnwhich is Ahlfors regular of dimension d, where dis a positive integer, d<n, and let µbe a positive measure on Eas in Definition 5.1. Then Eis uniformly rectifiable if there exists a positive constant kso that for each x∈Eand each r>0 with r≤diam Ethere is a closed subset Aof E∩B(x, r) such that µ(A)≥9 10 ·µ(E∩B(x, r))(5.5) and there is a k-bilipschitz mapping ffrom Ainto Rd.(5.6) In other words, inside of each “snapshot” E∩B(x, r)ofEthere should be a large subset, with at least 90% of the points, which is bilipschitz equivalent to a subset of Rd, and with a uniform bound on the bilipschitz constant. This is like asking for a controlled parameterization, except that we allow for holes and singularities. Definition 5.4 should be compared with the classical notion of (countable) rectifiability, in which one asks that Ebe covered, except for a set of measure 0, by a countable union of sets, each of which is bilipschitz equivalent to a subset of Rd. Uniform rectifiability implies this condition, but it is stronger, because it provides quantitative information at definite scales, while the classical notion really only gives asymptotic information as one zooms in at almost any point. See [Fal], [Fed], [Mat] for more information about classical rectifiability.
300 S. Semmes Normally one would be much happier to simply have bilipschitz coordinates outright, without having to allow for bad sets of small measure where this does not work. In practice bilipschitz coordinates simply do not exist in many situations where one might otherwise hope to have them. This is illustrated by the double-suspension spheres of Edwards and Cannon [Can1], [Can3], [Dave2], [Edw2], and the observations about them in [SieS]. Further examples are given in [Sem4], [Sem5]. The use of arbitrary scales and locations is an important part of the story here, and is very similar to the concept of BMO. At the level of a single snapshot, a fixed ball B(x, r) centered on E, the bad set may seem pretty wild, as nothing is said about what goes on there in (5.5) or (5.6). However, uniform rectifiability, like BMO, applies to all snapshots equally, and in particular to balls in which the bad set is concentrated. Thus, inside the bad set, there are in fact further controls. We shall see other manifestations of this later, and the same basic principle is used in the John-Nirenberg theorem for BMO functions (discussed in Section 3). Uniform rectifiability provides a substitute for (complete) bilipschitz coordinates in much the same way that BMO provides a substitute for L∞bounds, as in Section 3. Note that L∞bounds and bilipschitz coordinates automatically entail uniform control over all scales and locations. This is true just because of the way they are defined, i.e., a bounded function is bounded in all snapshots, and with a uniform majorant. With BMO and uniform rectifiability the scale-invariance is imposed by hand. It may be a little surprising that one can get anything new through concepts like BMO and uniform rectifiability. For instance, suppose that fis a locally-integrable function on Rk, and that the averages 1 ωktkB(z,t)|f(w)|dw(5.7) are uniformly bounded, independently of zand t. Here ωkdenotes the volume of the unit ball in Rk, so that ωktkis the volume of B(z,t). This implies that fmust itself be bounded by the same amount almost everywhere on Rk, since f(u) = lim t→0 1 ωktkB(z,t) f(w)dw(5.8) almost everywhere on Rk. Thus a uniform bound for the size of the snapshots does imply a uniform bound outright. For BMO the situation is different because one asks only for a uniform bound on the mean oscillation in every ball. In other words, one also has the freedom to make renormalizations by additive constants when moving from place to
Real Analysis, Quantitative Topology, etc. 301 place, and this gives enough room for some unbounded functions, like log |x|. Uniform rectifiability is like this as well, although with different kinds of “renormalizations” available. These remarks might explain why some condition like uniform rectifiability could be useful or natural, but why the specific version above in particular? Part of the answer to this is that nearly all definitions of this nature are equivalent to the formulation given above. For instance, the 9/10 in (5.5) can be replaced by any number strictly between 0 and 1. See [DaviS3], [DaviS5] for more information. Another answer lies in a theme often articulated by Coifman, about the way that operator theory can provide a good guide for geometry. One of the original motivations for uniform rectifiability came from the “Calder´on program” [Cal2], concerning the Lp-boundedness of certain singular operators on curves and surfaces of minimal smoothness. David [Davi2], [Davi3], [Davi5] showed that uniform rectifiability of a set Eimplies Lp-boundedness of wide classes of singular operators on E. (See [Cal1], [Cal2], [CoiDM], [CoiMcM] and the references therein for related work connected to the Calder´on program.) In [DaviS3], a converse was established, so that uniform rectifiability of an Ahlforsregular set Eis actually equivalent to the boundedness of a suitable class of singular integral operators (inherited from the ambient Euclidean space Rn). See also [DaviS2], [DaviS5], [MatMV], [MatP]. Here is a concrete statement about uniform rectifiability in situations where well-behaved parameterizations would be natural but may not exist. Theorem 5.9. Let Ebe a subset of Rnwhich is regular of dimension d. If Eis also a d-dimensional topological manifold and satisfies the local linear contractability condition (Definition 4.10), then Eis uniformly rectifiable. Note that Ahlfors-regularity automatically implies the doubling condition (Definition 4.9). Theorem 5.9 has been proved by G. David and myself. Now-a-days we have better technology, which allows for versions of this which are localized to individual “snapshots”, rather than using all scales and locations at once. See [DaviS8] (with some of the remarks in Section 12.3 of [DaviS8] helping to provide a bridge to the present formulation). We shall say a bit more about this, near the end of Subsection 5.3. The requirement that Ebe a topological manifold is convenient, but weaker conditions could be used. For that matter, there are natural variations of local linear contractability too.
302 S. Semmes One can think of Theorem 5.9 and related results in the following terms. Given a compact set K, upper bounds for the d-dimensional Hausdorff measure of Ktogether with lower bounds for the d-dimensional topology of Kshould lead to strong information about the geometric behavior of K. See [DaviS6], [DaviS8], [Sem3] for more on this. To understand better what Theorem 5.9 means, let us begin by observing that the hypotheses of Theorem 5.9 would hold automatically if Ewere bilipschitz equivalent to Rd,orifEwere compact and admitted bilipschitz local coordinates from Rd. Under these conditions, a test of the hypotheses of Theorem 5.9 on Ecan be converted into a similar test on Rd, where it can then be resolved in a straightforward manner. A similar argument shows that the hypotheses of Theorem 5.9 are “bilipschitz invariant”. More precisely, if Fis another subset of Rn which is bilipschitz equivalent to E, and if the hypotheses of Theorem 5.9 holds for one of Eand F, then it automatically holds for the other. Since the existence of bilipschitz coordinates implies the hypotheses of Theorem 5.9, we cannot ask for more than that in the conclusions. In other words, bilipschitz coordinates are at the high end of what one can hope for in the context of Theorem 5.9. The hypotheses of Theorem 5.9 do in fact rule out a lot of basic obstructions to the existence of bilipschitz coordinates, like cusps, fractal behavior, self-intersections and approximate self-intersections, and bubbles with very small necks. (Compare with Section 4, especially Theorem 4.11 and the discussion of its proof and consequences.) Nonetheless, it can easily happen that a set Esatisfies the hypotheses of Theorem 5.9 but does not admit bilipschitz local coordinates. Double-suspension spheres provide spectacular counterexamples for this (using the observations of [SieS]). Additional counterexamples are given in [Sem4], [Sem5]. We should perhaps emphasize that the assumption of being a topological manifold in Theorem 5.9 does not involve bounds. By contrast, uniform rectifiability does involve bounds, which is part of the point. In the context of Theorem 5.9, the proof shows that the uniform rectifiability constants for the conclusion are controlled in terms of the constants that are implicit in the hypotheses, i.e., in Ahlfors-regularity, the linear contractability condition, and the dimension. If bilipschitz coordinates are at the high end of what one could hope for, what happens if one asks for less? What if one asks for homeomorphic local coordinates with some control, but not as much? For instance, instead of bounding the “rate” of continuity through Lipschitz conditions
Real Analysis, Quantitative Topology, etc. 303 like |f(x)−f(y)|≤C|x−y|(5.10) (for some Cand all x,yin the domain of f), one could work with H¨older continuity conditions, which have the form |f(x)−f(y)|≤C|x−y|γ.(5.11) Here γis a positive number, sometimes called the H¨older “exponent”. As usual, (5.11) is supposed to hold simultaneously for all xand yin the domain of f, and with a fixed constant C. When xand yare close to each other and γis less than 1, this type of condition is strictly weaker than that of being Lipschitz. Just as f(x)=|x|is a standard example of a Lipschitz function that is not differentiable at the origin, g(x)=|x|γ is a basic example of a function that is H¨older continuous of order γ, γ≤1, but not of any order larger than γ, in any neighborhood of the origin. Instead of local coordinates which are bilipschitz, one could consider ones that are “bi-H¨older”, i.e., H¨older continuous and with H¨older continuous inverse. It turns out that double-suspension spheres do not admit bi-H¨older local coordinates when the H¨older exponent γlies above an explicit threshold. Specifically, if Pis an n-dimensional polyhedron which is the double-suspension of an (n−2)-dimensional homology sphere that is not simply connected, then there are points in P(along the “suspension circle”) for which bi-H¨older local coordinates of exponent γ>1/(n−2) do not exist. This comes from the same argument as in [SieS]. More precisely, around these points in P, there do not exist homeomorphic local coordinates from subsets of Rnfor which the inverse mapping is H¨older continuous of order γ>1/(n−2) (without requiring a H¨older condition for the mapping itself). Given any positive number a, there are examples in [Sem5] so that local coordinates (at some points) cannot have their inverses be H¨older continuous of order a. These examples do admit bi-H¨older local coordinates (with a smaller exponent), and even “quasisymmetric” [TukV] coordinates, and they satisfy the hypotheses of Theorem 5.9. In [Sem4] there are examples which satisfy the hypotheses of Theorem 5.9, but for which no uniform modulus of continuity for local coordinate mappings and their inverses is possible (over all scales and locations).
304 S. Semmes 5.1. Smoothness of Lipschitz and bilipschitz mappings. Another aspect of uniform rectifiability is that it provides the same amount of “smoothness” as when there is a global bilipschitz parameterization. To make this precise, let us first look at the smoothness of Lipschitz and bilipschitz mappings. A mapping f:Rd→Rnis Lipschitz if there is a constant Cso that (5.10) holds for all x,yin Rd. The space of Lipschitz mappings is a bit simpler than the space of bilipschitz mappings, because the former is a vector space (and even a Banach space) while the latter is not. For the purposes of “smoothness” properties, though, there is not really any difference between the two. Bilipschitz mappings are always Lipschitz, and anything that can happen with Lipschitz mappings can also happen with bilipschitz mappings (by adding new components, or considering x+h(x) when h(x) has Lipschitz norm less than 1 to get a bilipschitz mapping). One should also not worry too much about the difference between Lipschitz mappings which are defined on all of Rd, and ones that are only defined on a subset. Lipschitz mappings into Rnthat are defined on a subset of Rdcan always be extended to Lipschitz mappings on all of Rd. This is a standard fact. There are also extension results for bilipschitz mappings, if one permits oneself to replace the image Rnwith a Euclidean space of larger dimension (which is not too serious in the present context). For considerations of “smoothness” we might as well restrict our attention to functions which are real-valued, since the Rn-valued case can always be reduced to that. Two basic facts about Lipschitz functions on Rdare that they are differentiable almost everywhere (with respect to Lebesgue measure), and that for each η>0 they can be modified on sets of Lebesgue measure less than η(depending on the function) in such a way as to become continuously differentiable everywhere. See [Fed]. These are well-known results, but they do not tell the whole story. They are not quantitative; they say a lot about the asymptotic behavior (on average) of Lipschitz mappings at very small scales, but they do not say anything about what happens at scales of definite size. To make this precise, let a Lipschitz mapping f:Rd→Rbe given, and fix a point x∈Rdand a radius t>0. We want to measure how well fis approximated by an affine function on the ball B(x, t). To do
Real Analysis, Quantitative Topology, etc. 305 this we define the quantity α(x, t)by α(x, t) = inf A∈A sup y∈B(x,t) t−1|f(y)−A(y)|.(5.12) Here Adenotes the (vector space) of affine functions on Rd. The part on the right side of (5.12) with just the supremum (and not the infimum) measures how well the particular affine function Aapproximates finside B(x, t), and then the infimum gives us the best approximation by any affine function for a particular choice of xand t. The factor of t−1 makes α(x, t) scale properly, and be dimensionless. In particular, α(x, t) is uniformly bounded in xand twhen fis Lipschitz, because we can take A(y) to be the constant function equal to the value of fat x. The smallness of α(x, t) provides a manifestation of the smoothness of f. For functions which are twice-continuously differentiable one can get estimates like α(x, t)=O(t),(5.13) using Taylor’s theorem. If 0 <δ<1, then estimates like α(x, t)=O(tδ)(5.14) (locally uniformly in x) correspond to H¨older continuity of the gradient of fof order δ. Differentiability almost everywhere of fimplies that lim t→0α(x, t) = 0 for almost every x.(5.15) This does not say anything about any particular t, because one does not know how long one might have to wait before the limiting behavior kicks in. Here is a simple example. Let us take d= 1, and consider the function gρ(x)=ρ·sin(x/ρ)(5.16) on R. Here ρis any positive number. Now, gρ(x) is Lipschitz with norm 1 no matter how ρis chosen. This is not hard to check; for instance, one can take the derivative to get that g ρ(x) = cos(x/ρ)(5.17) so that |g ρ(x)|≤1 everywhere. This implies that |gρ(u)−gρ(v)|≤|u−v|(5.18) for all uand v(and all ρ), because of the mean-value theorem, or the fundamental theorem of calculus. (One also has that |g ρ(x)|= 1 at some points, so that the Lipschitz norm is always equal to 1.)
312 S. Semmes sufficient to imply the uniform rectifiability of the set E, at least if Eis Ahlfors-regular of dimension d. This was proved in [DaviS5]. In the context of functions, this type of thresholding condition is too weak, in that one can have the α(x, t)’s going to 0 uniformly as t→0 for functions which are differentiable almost nowhere, as mentioned in Subsection 5.1. Similarly, there are Ahlfors-regular sets which are “totally unrectifiable” (in the sense of [Fal], [Fed], [Mat]) and have the β(x, t)’s tending to 0 uniformly as t→0. (See [DaviS3].) For the bβ’s the story is simply different. On the other hand, the fact that suitable thresholding conditions on the bβ’s are sufficient to imply uniform rectifiability relies heavily on the assumption that Ebe Ahlfors-regular, while mass bounds are part of the conclusion (rather than the hypothesis) in Jones’ results, and no counterpart to the mass bounds are included in the abovementioned examples for functions. One does have mass bounds for the examples in [DaviS3] (of totally-unrectifiable Ahlfors-regular sets for which the β(x, t)’s tend to 0 uniformly as t→0), and there the issue is more in the size of the holes in the set. The bβ’s, by definition, control the sizes of holes. Note that this result for the bβ’s does have antecedents for the classical notion of (countable) rectifiability, as in [Mat]. There are a number of variants of the bβ’s, in which one makes comparisons with other collections of sets besides d-planes, like unions of d-planes, for instance. See [DaviS5]. Perhaps the strongest formulation of smoothness for uniformly rectifiable sets is the existence of a “Corona decomposition”. This is a geometric version of the information that one can get about a Lipschitz function from the methods of Carleson’s Corona construction (as mentioned in Subsection 5.1). Roughly speaking, in this condition one controls not only how often Eis well-approximated by a d-plane, but how fast the d-planes turn as well. This can also be formulated in terms of good approximations of Eby flat Lipschitz graphs. Although a bit technical, the existence of a Corona decomposition is perhaps the most useful way of managing the complexity of a uniformly rectifiable set. Once one has a Corona decomposition, it is generally pretty easy to derive whatever else one would like to know. Conversely, in practice the existence of a Corona decomposition can be a good place to start if one wants to prove that a set is uniformly rectifiable. In fact, there is a general procedure for finding a Corona decomposition when it exists, and one which is fairly simple (and very similar to Carleson’s Corona construction). The difficult part is to show that this procedure works in the right way, with the correct estimates. Specifically, it is a stopping-time argument, and one does not want to have to
Real Analysis, Quantitative Topology, etc. 313 stop too often. This is a nice point, because in general it is not so easy to build something like a good parameterization of a set, even if one knows a priori that it exists. In this context, there are in principle methods for doing this. See [DaviS2], [DaviS3], [DaviS5], [DaviS7], [Sem1] for more information about Corona decompositions of uniformly rectifiable sets and the way that they can be used. The paper [DaviS7] is written in such a way as to try to convey some of the basic concepts and constructions without worrying about why the theorems are true (which is much more complicated). In particular, the basic procedure for finding Corona decompositions when they exist is discussed. See [GarnJ], [Jon1], [Jon3] for some other situations in which Carleson’s Corona construction is used geometrically. 5.3. A class of variational problems. Uniform rectifiability is a pretty robust condition. If one has a set which looks roughly as though it ought to be uniformly rectifiable, then there is a good chance that it is. This as opposed to sets which look roughly as though they should admit a well-behaved (homeomorphic) parameterization, and do not (as discussed before). In this subsection we would like to briefly mention a result of this type, concerning a minimal surface problem with nonsmooth coefficients. Let g(x) be a Borel measurable function on Rn, and assume that gis positive, bounded, and bounded away from 0, so that 0<m≤g(x)≤M(5.28) for some constants m,Mand all x∈Rn. Let Q0,Q1be a pair of (closed) cubes in Rn, with sides parallel to the axes, and assume that Q0is contained in the interior of Q1. Let Ube an open subset of Q1which contains the interior of Q0. Consider an integral like ∂U g(x)dνU(x),(5.29) where dνUdenotes the measure that describes the (n−1)-dimensional volume of subsets of ∂U. This would be defined as in calculus when ∂U is at least a little bit smooth (like C1), but in general one has to be more careful. One can simply take for dνUthe restriction of (n−1)-dimensional Hausdorff measure to ∂U, but for technical reasons it is often better to define dνUusing distributional derivatives of the characteristic function of U,asin[Giu]. For this one would work with sets Uwhich have “finite
314 S. Semmes perimeter”, which means exactly that the distributional first derivatives of the characteristic function of Uare measures of finite mass. Here is one way in which this kind of functional, and the minimization of this kind of functional, can come up. Let Fbe a closed subset of Rn. Imagine that one is particularly interested in domains Uwhich have their boundary contained in F, or very nearly so. On the other hand, one might also wish to limit irregularities in the behavior of the boundary of U. For this type of situation one could choose gso that it is much smaller on Fthan on the complement of F, and then look for minimizers of (5.29) to find domains with a good balance between the behavior of ∂U and the desire to have it be contained (as much as possible) in F. (See [DaviS6] for an example of this.) When do minimizers of (5.29) exist, and how do they behave? If one works with sets of finite perimeter, and if the function gis lower semicontinuous, then one can obtain the existence of minimizers through standard techniques (as in [Giu]). That is, one takes limits of minimizing sequences for (5.29) using weak compactness, and one uses the lower semi-continuity of gto get lower semi-continuity of (5.29) with respect to suitable convergence of the U’s. The latter ensures that the limit of the minimizing sequence is actually a minimum. Note that the “obstacle” conditions that Ucontain the interior of Q0and be contained in Q1 prevents the minimization from collapsing into something trivial. As to the behavior of minimizers of (5.29), one cannot expect much in the way of smoothness in general. For instance, if the boundary of U can be represented locally as the graph of a Lipschitz function, then U in fact minimizes (5.29) for a suitable choice of g. Specifically, one can take gto be a sufficiently small positive constant on ∂U, and to be equal to 1 everywhere else. That such a choice of gworks is not very hard to establish, and more precise results are given in [DaviS6]. Conversely, minimizers of (5.29) are always Ahlfors-regular sets of dimension n−1, and uniformly rectifiable. This is shown in [DaviS6], along with some additional geometric information which is sufficient to characterize the class of sets Uwhich occur as minimizers for functionals of the form (5.29) (with gbounded and bounded away from 0). If a set U arises as the minimizer for some g, it is also a minimizer with gchosen as above, i.e., a small positive constant on ∂U and equal to 1 everywhere else. The same regularity results work for a suitable class of “quasiminimizers” of the usual area functional, and one that includes minimizers for (5.29) as a special case.
Real Analysis, Quantitative Topology, etc. 315 Uniform rectifiability provides a natural level of structure for situations like this, where stronger forms of smoothness cannot be expected, but quantitative bounds are reasonable to seek. Note that properties of ordinary rectifiability always hold for boundaries of sets of finite perimeter, regardless of any minimizing or quasiminimizing properties. See [Giu]. Analogous results about regularity work for sets of higher codimension as well, although this case is more complicated technically. See [DaviS8] for more information. One can use this framework of minimization (with respect to nonsmooth coefficient functions g) as a tool for studying the structure of sets in Rnwith upper bounds on their d-dimensional Hausdorff measure and lower bounds for their d-dimensional topology. This brings one back to Theorem 5.9 and related questions, and in particular more “localized” versions of it. To put it another way, minimization of functionals like these can provide useful means for obtaining “existence results” for approximate parameterizations with good behavior, through uniform rectifiability. See [DaviS6], [DaviS8]. Part of the motivation for this came from an earlier argument of Morel and Solimini [MoreS]. Their argument concerned the existence of curves containing a given set, with good properties in terms of the distribution of the arc-length measure of these curves, under more localized conditions on the given set (at all locations and scales). See Lemma 16.27 on p. 207 of [MoreS]. Appendix A. Fourier transform calculations If φ(x) is an integrable function on Rn, then its Fourier transform φ(ξ) is defined (for ξ∈Rn)by φ(ξ)=Rn eix,ξφ(x)dx.(A.1) Here x, ξdenotes the usual inner product for x, ξ ∈Rn, and i=√−1. Often one makes slightly different conventions for this definition —with some extra factors of πaround, for instance— but we shall not bother with this. A key feature of the Fourier transform is that it diagonalizes differential operators. Specifically, if ∂kdenotes the operator ∂/∂xkon Rn, then (∂kφ) (ξ)=iξ k φ(ξ),(A.2) i.e., differentiation is converted into mere multiplication. For this one should either make some differentiability assumptions on φ, so that the
316 S. Semmes left side can be defined in particular, or one should interpret this equation in the sense of tempered distributions on Rn. The Fourier transform also carries out this diagonalization in a controlled manner. That is, there is an explicit inversion formula (which looks a lot like the Fourier transform itself), and the Fourier transform preserves the L2norm of the function φ, except for a multiplicative constant, by the Plancherel theorem. See [Duo], [SteW], [Torc] for these and other basic facts about the Fourier transform. Using Plancherel’s theorem, it is very easy to give another proof of the L2estimate (3.2) from Section 3, and to derive many other inequalities of a similar nature. One can also use the Fourier transform to give a precise definition of the operator R=∂j∂k/∆, where ∆ is the Laplace operator n =1 ∂2 . Specifically, one can define it through the equation (Rφ) (ξ)=ξjξk |ξ|2 φ(ξ).(A.3) If m(ξ) is any bounded function on Rn, then (Tφ) (ξ)=m(ξ) φ(ξ)(A.4) defines a bounded operator on L2(Rn). In general these operators are not bounded on Lpfor any other value of p, but this is true for many of the operators that arise naturally in analysis. For instance, suppose that m(ξ) is homogeneous of degree 0, so that m(tξ)=m(ξ) when t>0,(A.5) and that m(ξ) is smooth away from the origin. Then the associated operator Tis bounded on Lpfor all pwith 1 <p<∞. See [Duo], [GarcR], [Jou], [Ste1], [SteW], [StrT], [Torc]. Note that this criterion applies to the specific choice of m(ξ) in (A.3) above. For a multiplier operator as in (A.4) to be bounded on L1or L∞ is even more exceptional than for Lpboundedness when 1 <p<∞. (See [SteW], [Torc].) For instance, if mis homogeneous, as above, and not constant, then the corresponding operator cannot be bounded on L1or L∞. However, if mis homogeneous and smooth away from the origin, then the operator Tin (A.4) does determine a bounded operator from L∞into BMO. See [Duo], [GarcR], [Garn], [Jou], [Sar], [Ste2], [Torc]. In fact, Tdetermines a bounded operator from BMO to itself.
Real Analysis, Quantitative Topology, etc. 317 Here is another example. Let φnow be a mapping from R2to itself, with components φ1,φ2. Consider the differential dφ of φas a matrixvalued function, namely, ∂1φ1∂1φ2 ∂2φ1∂2φ2.(A.6) (Let us assume that φis smooth enough that the differential is at least some kind of function when taken in the sense of distributions, although one can perfectly well think of dφ as a matrix-valued distribution.) Let A and Sdenote the antisymmetric and symmetric parts of dφ, respectively, so that A=dφ −dφt 2,S=dφ +dφt 2,(A.7) where dφtdenotes the transpose of dφ. In this case of 2×2 matrices, the antisymmetric part Areally contains only one piece of information, namely ∂1φ2−∂2φ1.(A.8) It is not hard to check that this function can be reconstructed from the entries of Sthrough operators of the form (A.4), using functions mwhich are homogeneous of degree 0 and smooth away from the origin. For this one should add some mild conditions on φ, like compact support, to avoid the possibility that Svanishes identically but Adoes not. Under these conditions, we conclude that the Lpnorm of Ais always bounded by a constant multiple of the Lpnorm of S,1<p<∞, and that the BMO norm of Ais controlled by the L∞(or BMO) norm of S. (For the case of BMO norms, the possibility that Svanishes but Adoes not causes no trouble, because Awill be constant in that case.) This example is really a “linearized” version of the problem discussed in Section 2. Specifically, let us think of f:R2→R2as being of the form f(x)=x+'φ(x),(A.9) where 'is a small parameter. The extent to which fdistorts distances is governed by the matrix-valued function df tdf , which we can write out as df tdf =I+4'S+'2dφtdφ.(A.10) Thus the linear term in 'is governed by S, while Acontrols the leading behavior in 'of the “rotational” part of df .
318 S. Semmes Appendix B. Mappings with branching In general, there can be a lot of trouble with existence and complexity of homeomorphisms (with particular properties, like specified domain and range). If one allows mappings with branching, then the story can be very different. As a basic example of this, there is a classical result originating with Alexander to the effect that any oriented pseudomanifold of dimension n admits an orientation-preserving branched covering over the n-sphere. Let us state this more carefully, and then see how it is proved. Let Mbe a finite polyhedron. We assume that Mis given as a finite union of n-dimensional simplices that meet only in their boundary faces (so that Mis really a simplicial complex). To be a pseudomanifold means that every (n−1)-dimensional face in Marises as the boundary face of exactly two n-dimensional simplices. In effect this says that Mlooks like a manifold away from its codimension-2 skeleton (the corresponding statement for the codimension-1 skeleton being automatic). For the present purposes it would be enough to ask that every (n−1)-dimensional face in Marise as the boundary face of at most two n-dimensional simplices, which would be like a “pseudomanifold with boundary”. An orientation for an n-dimensional pseudomanifold Mmeans a choice of orientation (in the usual sense) for each of the constituent n-dimensional simplices in M, with compatibility of orientations of adjacent n-dimensional simplices along the common (n−1)-dimensional face. In terms of algebraic topology, this means that the sum of the n-dimensional simplices in M, with their orientations, defines an n-dimensional cycle on M. For the purposes of the Alexander-type result, it will be convenient to think of the n-sphere as consisting of two standard simplices S1and S2glued together along the boundary. This is not quite a polyhedron in the usual (affine) sense, but one could easily repair this by subdividing S1or S2. We also assume that S1and S2have been oriented, and have opposite orientations relative to their common boundary. To define a mapping from Mto the n-sphere one would like to simply identify each of the constituent n-dimensional simplices in Mwith S1or S2in a suitable manner. Unfortunately, this does not work, even when n= 2, but the problem can be fixed using a barycentric subdivision of M. Recall that the barycenter of a simplex (embedded in some vector space) is the point in the interior of the simplex which is the average of the vertices of the simplex. The set of barycenters for Mmeans the set of barycenters of all of the constituent simplices in M(viewed as a
Real Analysis, Quantitative Topology, etc. 319 simplicial complex), of all dimensions, including 0. In particular, the set of barycenters for Mincludes the vertices of M(which are themselves 0-dimensional simplices, and their own barycenters). The barycentric subdivision of Mis a refinement of Mas a simplicial complex whose vertices are exactly the set of barycenters of M. In other words, the set Mas a whole does not change, just its decomposition into simplices, which is replaced by a finer decomposition. Here is a precise description of the simplices in the barycentric subdivision of M. Let s0,s 1,...,s kbe a finite sequence of simplices in M, with each sian i-dimensional simplex which is a face of si+1 (when i<k). Let b(si) denote the barycenter of si. Then b(s0),b(s1),...,b(sk) are affinely independent, and hence determine a k-dimensional simplex. The simplices that arise in this manner are precisely the ones used for the barycentric subdivision of M. (See p. 123 of [Spa] for more details.) Let Vdenote the set of all vertices in the barycentric subdivision of M. This is the same as the set of points which arise as barycenters of simplices in the original version of M, and in particular we have a natural mapping from Vto the integers {0,1,...,n}, defined by associating to each point bin Vthe dimension of the simplex from which it was derived. If Tis a k-dimensional simplex in the barycentric subdivision of M, then the mapping from Vto {0,1,...,n}just described induces a one-to-one correspondence between the k+ 1 vertices of Tand the set {0,1,...,k}. This follows easily from the definitions. We are now ready to define our mapping from Mto the n-sphere. There are exactly n+ 1 vertices in our realization of the n-sphere as the gluing of S1and S2. Let us identify these vertices with the integers from 0 to n. Thus our mapping from Vto {0,1,...,n}can now be interpreted as a mapping from the vertices of the barycentric subdivision of Mto the vertices of the n-sphere. This mapping between vertices admits a canonical linear extension to each k-dimensional simplex, k<n, in the barycentric subdivision of M. For the n-dimensional simplices the extension is uniquely determined once one chooses S1or S2for the image of the simplex. Because of the orientations, there is only one natural choice of S1or S2for each n-dimensional simplex T, namely the one so that the linear mapping from Tonto Sjis orientation-preserving. In the end we get a mapping from the barycentric subdivision of M to the n-sphere which preserves orientations and which defines an affine isomorphism from each n-dimensional simplex Tin the domain onto one of S1and S2. This uses the fact that our initial mapping between vertices
320 S. Semmes was always one-to-one on the set of vertices in any given simplex in the domain, by construction. This completes the proof. We should emphasize that the singularities of the mapping from Mto the n-sphere —i.e., the places where it fails to be a local homeomorphism— are confined to the codimension-2 skeleton of the barycentric subdivision of M. This is because of the orientation and pseudomanifold conditions, which ensure that if a point in Mlies in the interior of an (n−1)-dimensional simplex in the barycentric subdivision of M, then the (two) adjacent n-simplices at that point are not sent to the same Sjin the image. The idea of branching also makes sense for mappings that are not piecewise-linear, and there are well-developed notions of “controlled geometry” in this case, as with the classes of quasiregular mappings and mappings of bounded length distortion. See [HeiKiM], [MarRiV1], [MarRiV2], [MarRiV3], [MarV], [Res], [Ric1], [V¨ai1], [Vuo], for instance. In [HeiR1], [HeiR2] there are examples where branching maps of controlled geometry can be constructed but suitable homeomorphisms either do not exist or must distort distances more severely. Sullivan [Sul2], [Sul3] has proposed some mechanisms by which the existence of local (controlled) branching maps can be deduced, and some ideas for studying obstructions to controlled homeomorphic coordinates. See [Gut+], [HeiKi], [MarRyV] for some recent results about branching and regularity conditions under which it does not occur. A broader and more detailed discussion of mappings with branching can be found in [HeiR2]. For some real-variable considerations of mappings which may branch but enjoy substantial geometric properties, see [Davi4], [Jon2], [DaviS4]. References [Ale] P. Alestalo, Uniform domains of higher order, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 94 (1994), 48. [AleV1] P. Alestalo and J. V¨ ais¨ al¨ a, Uniform domains of higher order II, Ann. Acad. Sci. Fenn. Ser. A I Math. 21(2) (1996), 411–437. [AleV2] P. Alestalo and J. V¨ ais¨ al¨ a, Uniform domains of higher order III, Ann. Acad. Sci. Fenn. Ser. A I Math. 22(2) (1997), 445–464. [As1] P. Assouad, Espaces m´etriques, plongements, facteurs, Th`ese de doctorat, U.E.R. Math´ematique, Universit´e Paris XI, Orsay, France, 1977.
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