Proc. Natl. Acad. Sci. USA Vol. 93, pp. 554-556, January 1996 Mathematics

From local to global in quasiconformal structures JUHA HEINONEN* AND PEKKA KosKELAt *Department of Mathematics, University of Michigan, Ann Arbor, MI 48109; and tDepartment of Mathematics, University of Jyvaskyla, FIN-40351 Jyvaskyla, Finland Communicated by F. W. Gehring, University ofMichigan, Ann Arbor, MI, October 26, 1995 (received for review August 10, 1995)

ABSTRACT We exhibit a large class of metric spaces above sense. In contrast to this, Euclidean n-space Rn for n 2 whose infinitesimal quasiconformal structure is strong 2 exhibits the following remarkable (quasi-)conformal rigidity: enough to capture the global quasiconformal structure. A THEOREM 1. If d' is the pull-back ofthe standard metric d of sufficient condition for this to happen is described in terms of Rn under a homeomorphism f: Rn -> Rn and if d' is locally a Poincare-type inequality. quasiconformally related to d, then d' is globally quasiconfor- mally related to d. Suppose that in a spaceXtwo metric structures d and d' are given, Theorem I in its most general form is due to Gehring (1). It and suppose that locally the structure d' is related to the structure is also true, and proved by Gehring (2), that if the two metrics d in such a manner that, around each point x in the space, the in Theorem 1 are conformally related, then f is a smooth shape of each sufficiently small metric ball in the structure d' is conformal transformation. Observe that no a priori smooth- boundedly distorted when viewed in the structure d. In more ness assumption is placed on f precise terms, we suppose that, for each x in X, each sufficiently Gehring's theorem has two consequences that are not at all small metric ball B'(x, s) contains a metric ball B(x, t) and is immediate from the definition: in the situation of Theorem 1, contained in the metric ball B(x, Ht) for some constant H 2 1 the metric d' is quasiconformally related to d if and only if the independent of the pointx. Notice that no relation is imposed on metric d is quasiconformally related to d'; moreover, if d" is a the radii s and t, which can be vastly different from each other and third metric in Rn which is induced by a self-homeomorphism which are allowed to depend on the pointx. [Here and hereafter, of Rn and which is quasiconformally related to d, then it is B'(x, r) and B(x, r) denote open balls in the metrics d' and d, quasiconformally related to d' as well. There are no known respectively, centered at x with radius r > 0.] easy ways to establish these corollaries. See Vaisala (3) for a We say then that d' is (locally) quasiconformally related to the thorough discussion on these matters. metric d in the space X. If the constant H deserves to be In this paper, we announce a result that identifies a large mentioned, we shall attach it to the description. The relation family of metric spaces where a similar passage from local to is conformal if H can be taken to be any constant larger than global in quasiconformal structures is possible-the details one. In the conformal case, the condition means that at the will appear elsewhere. Until recently, not many spaces of this infinitesimal level the balls in the structure d' become balls in genre were known. It followed from the results of Mostow (4, the structure d, albeit possibly of quite different size. In the 5) and Pansu (6) that the boundaries of rank one symmetric general case, we allow the d'-balls to be distorted in the spaces (of negative curvature) belong to this family. The structure d, but only by some fixed amount. Generally, one boundary of such a space can be realized as [the one point should think ofd as the fixed reference metric in X, and d' will compactification of] a simply connected, nilpotent Lie group be a metric whose distortion with respect to d is under scrutiny. equipped with so-called Carnot metric in which the automor- A standard example is obtained when X is a Riemannian phisms of the symmetric space act as conformal transforma- manifold with d its Riemannian metric obtained from a metric tions. Heinonen and Koskela (7) proved the same to be true tensorg. Then a new Riemannian metric d' onXis conformally for all Carnot groups. These examples are interesting in that related to d if and only if the corresponding tensor g' is the Carnot metric in these groups cannot come from any obtained from g by multiplication by a positive function. A Riemannian metric; in fact, the Hausdorff dimension of a quasiconformal relation is obtained if the ratio of the maxi- Carnot group always exceeds its topological dimension. The mum and the minimum values of the quadratic function case of complex hyperbolic space has been thoroughly exam- corresponding to g is bounded on the unit tangent bundle of ined by Koranyi and Reimann (8); they have shown that in this (X,g'). case there is a rich theory of quasiconformal structures, or In this note, we shall address the following question: mappings. Under what conditions does the above local quasiconformal It is known in general that the quasi-isometry type of the relation imply a global quasiconformal relation? fundamental group of a compact negatively curved manifold is By a global quasiconformal relation we mean that any metric determined by the (global) quasiconformal type of its sphere at ball in the structure d' has a boundedly distorted shape when infinity. This fact was the gist of Mostow's work, and it is the viewed in the structure d-i.e., there is a constant K 2 1 such main motivation to study quasiconformal structures on general that any metric ball B'(x, s) contains a metric ball B(x, t) and spaces. See Gromov and Pansu (9) for an exposition of these is contained in B(x, Kt). ideas; see also Cannon (10). A quantitative form of the question asks that the constant K To describe our depend only on H and on some natural parameters associated main results, some notation and terminol- with both X and the metrics. ogy need to be fixed. We assume thatXcarries a Borel measure Our first example is a counterexample: take X to be the real ,u that gives each ball B(x, r) in the metric d a mass that is line with d its standard metric; then the pull-back of d under comparable to rQ for some real number Q > 1. More precisely, any diffeomorphism of X is conformally related to d, but it is we assume that there is a constant C 2 1 such that easy to see that there need not be any global relation in the C - 'rQ c ,pB(x,r) c CrQ [1] The publication costs of this article were defrayed in part by page charge for each d-ball B(x, r) with 0 < r < diamX. We call the metric payment. This article must therefore be hereby marked "advertisement" in space (X, d) (Ahlfors-)regular of dimension Q if condition 1 accordance with 18 U.S.C. §1734 solely to indicate this fact. holds. In this case, (X, d) has Hausdorff dimension Q and 1 554 Downloaded by guest on September 26, 2021 Mathematics: Heinonen and Koskela Proc. Natl. Acad. Sci. USA 93 (1996) 555 holds (possibly with a slightly different constant C 2 1) when , is replaced with the Hausdorff Q-measure of (X, d). Indeed, I -uBIdVfl - C(n)diamB fI'VuldVn it is understood from now on that the measure ,u is the Hausdorff Q-measure of the ambient Q-regular space, and for this reason it often is ignored in notation. for all smooth bounded functions u in a ball B. Here UB denotes The most important conformal invariant in Euclidean n- the mean value of u in B. space is the modulus, or capacity. This is a numerical invariant To make sense of a Poincare inequality in a metric space, we associated with any pair of two disjoint continua E and F in Rn, need a concept of a gradient. Given an open set U in a metric defined as the infimum of the volumes space (X, d) and a continuous (real-valued) function u in U, we say that a Borel function p: U -> [0, oo] is a very weak gradient of u in U if I p'dVn [2] lu(x) - uOI ' Jpds over all (Borel-measurable) nonnegative metrics p such that Yxy for each pair of points x and y in U and for each rectifiable curve yx joining x and y in U. Jpds > 1 As an example, every Lipschitz function u on U has a very weak gradient, namely the function that is identically the Lipschitz constant of u. for all paths y joining E and F. In 2, dVn denotes Lebesgue We say that a metric measure space (X, d, ,) admits a (weak) measure in Rn. In 1959, Loewner (11) made the crucial (1,p)-Poincare inequality for some p 2 1 if there are constants observation that the modulus between any two nondegenerate C - 1 and 8 E (0, 1] such that continua in Rn is positive (in the plane n = 2 this property of modulus was already known to Grotzsch and Teichmuller in the 1920s), and one can justly say that it is exactly this property gt SB) lu - u8Bldu s CdiamB( JPPdl) [3] of Rn that is responsible for Theorem 1. The modulus can be defined in any Q-regular space; the exponent n in 2 is being replaced by Q and Lebesgue measure for each ball B in (X, d), for all continuous functions u in B, and by ,u. We term a Q-regular space (X, d) to be a Loewner space for all very weak gradients p of u. Here SB denotes the ball with if the modulus between any two disjoint nondegenerate con- same center as B but with radius multiplied by 8, and u5B tinuaE and F inXhas a positive lower bound that depends only denotes the mean value of u in SB. Notice that the bigger p becomes, the weaker condition 3 becomes, by Holder's in- on the relative position equality. dist(E, F) If each pair of points in a metric space (X, d) can be joined by a curve whose length equals the distance between the min{diamE, diamF} points, the space is called geodesic. We also call (X, d) proper if its closed balls are compact. between the continua; to avoid possible pathologies, we also The next theorem characterizes Loewner spaces among include the assumption that each pair of two points in (X, d) proper and geodesic metric spaces in terms of a Poincare can be joined by a rectifiable curve. Note, however, that we do inequality. not assume (X, d) to be complete or locally compact. THEOREM 3. Suppose that (X, d) is a proper and geodesic We can prove the following theorem: Q-regular metric spacefor some Q > 1. Then (X, d) is a Loewner THEOREM 2. Let (X, d) be a Q-regularLoewnerspaceforsome space if and only if it admits a (1, Q)-Poincare inequality. Q > 1. Suppose that d' is a metric in X such that (X, d') is a The statement in Theorem 3 is quantitative in the sense that Q-regular Loewner space, that the balls in either metric are all the relevant constants and parameters depend only on each bounded sets in the other, and that d' is quasiconformally related other and not on the particular space X or metric d. to d. Then d' is globally quasiconformally related to d. There is a natural connection between Poincare inequalities Four remarks need to be made here. First, Theorem 2 and curves in a space. It is not hard to see that a metric space admittedly looks too abstract to be useful, unless one can (X, d), where every pair of points can be joined by a rectifiable provide examples of Q-regular Loewner spaces; this will be curve, is quasiconvex if and only if every continuous function discussed below. Second, even as such, Theorem 2 is non- on X with bounded very weak gradient is Lipschitz. By trivial, for it contains Gehring's theorem 1 as a special case. quasiconvex we mean that every pair of points x andy in X can More importantly, the proof of Theorem 2 does not use any be joined by a curve whose length does not exceed a constant analysis special to Rn-its framework is simply too general for times the distance between the points (the constant should be it. The that proves Theorem 2 is largely combinato- independent of the points). The Loewner condition in itself argument can be shown to imply quasiconvexity, and therefore the rial and based on an idea of a discretization of the modulus. assumption that (X, d) be geodesic in Theorem 3 is not a The third remark is that Q-regularity is not generally preserved restrictive one. But more is true, for the Loewner condition, or under a global quasiconformal change of the metric; for our equivalently the validity of a (1, Q)-Poincare inequality in a proof of the theorem, it has to be included in the hypotheses. Q-regular space, implies in fact that there is not only one short Finally, the assumption that the two metrics share the same curve joining given two points, but a plenitude. bounded sets is necessary, as can be seen by considering a By the aid of Theorem 3 and known results about Poincare conformal map of a disk onto a half-plane. inequalities, we can give a list of Loewner spaces that satisfy the Now we discuss the ubiquity of Loewner spaces. It turns out assumptions, and hence the conclusion, in Theorem 2. This list that the Loewner condition is interestingly linked to Poincare- includes, for example, all Carnot groups and all Riemannian type inequalities, and hence to the isoperimetric profile of the manifolds with nonnegative Ricci curvature and maximal space. Recall that [a form of] a Poincare inequality in Rn states volume growth; it also includes the interesting A.-geometry that studied recently by David and Semmes (12). Semmes has also Downloaded by guest on September 26, 2021 556 Mathematics: Heinonen and Koskela Proc. Natl. Acad. Sci. USA 93 (1996) shown (IHES preprint, 1995) that if a (connected and ori- deduced with the aid of Theorem 5. Among these are important ented) topological n-manifold X has a metric d that makes it compactness properties as well as Holder continuity. See Tukia n-regular, and if (X, d) satisfies a certain linear contractibility and Vaisala (13). condition, then (X, d) admits a (1, 1)-Poincare inequality; ergo, The problem about quasiconformal or quasisymmetric it is a Loewner space. equivalence of spaces is one of the oldest and hardest in the The next theorem gives yet another indication that the subject. Even in a fairly concrete situation of a (finite) sim- Loewner property is far from being uncommon. plicial complex not much is known. The problem about the THEOREM 4. Suppose that X is a connected, finite simplicial existence of quasiconformal coordinates for simplicial com- complex ofpure dimension n > 1 such that the link ofeach vertex plexes was first raised by Siebenmann and Sullivan (14), who is connected. Then X is an n-regular Loewner space (either in the studied the same question for Lipschitz maps. It follows from barycentric metric orin the metric X inherits by sitting inside some the results in this paper that in many cases a global obstruction RN). to the equivalence implies a local obstruction as well. To finish, let us phrase our main problem in terms of mappings, in the spirit of Theorem 1. Suppose that f is a We thank Stephen Semmes for his pointed remarks on this essay. homeomorphism of a metric space (Y, dy) onto a metric space Both authors acknowledge the financial support of the U.S. National (X, dx). Then f is called quasiconformal if the push-forward Science Foundation and the Academy Finland. J.H. is supported by the metric A. P. Sloan Foundation. d'(x, y) = dy(f - '(x),f- '(y)) [4] 1. Gehring, F. W. (1960) Ann. Acad. Sci. Fenn. Ser. A I Math. 281, 1-28. in X is quasiconformally related to dx. This condition means 2. Gehring, F. W. (1962) Trans. Am. Math. Soc. 103, 353-393. that infinitesimal balls in Y are carried onto "ellipsoids" in X 3. Vaisala, J. (1971) Lect. Notes Math. 229. whose eccentricity is uniformly bounded. Next, f is called 4. Mostow, G. D. (1973) Strong Rigidity ofLocally Symmetric Spaces quasisymmetric if the push-forward metric 4 is globally quasi- (Princeton Univ. Press, Princeton). conformally related to dx. Thus, in the language of mappings, 5. Mostow, G. D. (1994) Michigan Math. J. 41, 31-37. our question asks under what conditions quasiconformal maps 6. Pansu, P. (1989) Ann. Math. 129, 1-60. are quasisymmetric. 7. Heinonen, J. & Koskela, P. (1995) Invent. Math. 120, 61-79. THEOREM 5. Quasiconformal maps between Q-regular 8. Koranyi, A. & Reimann, H. M. (1995) Adv. Math. 111, 1-87. 9. Gromov, M. & Pansu, P. (1991) Lect. Notes Math. 1504. Loewner spaces are quasisymmetric if map sets to they bounded 10. Cannon, J. W. (1994) Acta Math. 173, 155-234. bounded sets and if Q > 1. Moreover, the inverse of a quasi- 11. Loewner, C. (1959) J. Math. Mech. 8, 411-414. conformal map is quasiconformal in this case. 12. David, G. & Semmes, S. (1990) Lect. Notes PureAppl. Math. 122. The first assertion of Theorem S follows from Theorem 2. The 13. Tukia, P. & Vaisala, J. (1980)Ann. Acad. Sci. Fenn. Ser. A IMath. second assertion needs, in addition, the fact that Loewner 5, 97-114. spaces are quasiconvex. Many other strong properties of 14. Siebenmann, L. & Sullivan, D. (1979) in Geometric Topology, ed. quasiconformal maps between regular Loewner spaces can be Cantrell, J. C. (Academic, New York), pp. 503-525. Downloaded by guest on September 26, 2021