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.