Analysis in Metric Spaces Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, and Jeremy Tyson

study of quasiconformal maps on such boundaries moti- The authors of this piece are organizers of the AMS vated Heinonen and Koskela [HK98] to axiomatize several 2020 Mathematics Research Communities summer aspects of Euclidean quasiconformal geometry in the set- conference Analysis in Metric Spaces, one of five ting of metric measure spaces and thereby extend Mostow’s topical research conferences offered this year that are work beyond the sub-Riemannian setting. The ground- focused on collaborative research and professional breaking work [HK98] initiated the modern theory of anal- development for early-career mathematicians. ysis on metric spaces. Additional information can be found at https://www Analysis on metric spaces is nowadays an active and in- .ams.org/programs/research-communities dependent field, bringing together researchers from differ- /2020MRC-MetSpace. Applications are open until ent parts of the mathematical spectrum. It has far-reaching February 15, 2020. applications to areas as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer sci- The subject of analysis, more specifically, first-order calcu- ence. As a further sign of recognition, analysis on met- lus, in metric measure spaces provides a unifying frame- ric spaces has been included in the 2010 MSC classifica- work for ideas and questions from many different fields tion as a category (30L: Analysis on metric spaces). In this of mathematics. One of the earliest motivations and ap- short survey, we can discuss only a small fraction of areas plications of this theory arose in Mostow’s work [Mos73], into which analysis on metric spaces has expanded. For in which he extended his celebrated rigidity theorem for more comprehensive introductions to various aspects of hyperbolic manifolds to the more general framework of the subject, we invite the reader to consult the monographs manifolds locally modeled on negatively curved symmet- [Hei01,HK00,HKST15,BB11,MT10,AGS08,BS07,Hei07]. ric spaces of rank one. In his proof, Mostow used the theory of quasiconformal mappings on the visual boundaries Poincar´einequalities in metric spaces. Inspired by the of rank-one symmetric spaces. These visual boundaries fundamental theorem of calculus, Heinonen and Koskela are equipped with a sub-Riemannian structure that is lo- proposed the notion of upper gradient as a substitute for the cally non-Euclidean and has a fractal nature. Mostow’s derivative of a function on a metric measure space (푋, 푑, 휇). More precisely, 푔 ≥ 0 is an upper gradient for a real-valued Mario Bonk is a professor of mathematics at the University of California, Los function 푢 on 푋 if Angeles. His email address is [email protected]. Luca Capogna is a professor of mathematical sciences at Worcester Polytechnic Institute. His email address is [email protected]. |푢(훾(1)) − 푢(훾(0))| ≤ ∫푔 푑푠 Piotr Hajłasz is a professor of mathematics at the University of Pittsburgh. His 훾 email address is [email protected]. Nageswari Shanmugalingam is a professor of mathematical sciences at the Uni- for each path 훾∶ [0, 1] → 푋 of finite length. versity of Cincinnati. Her email address is [email protected]. Upper gradients are not unique, but if a function 푢 has Jeremy Tyson is a professor of mathematics at the University of Illinois, Urbana– an upper gradient 푔 ∈ 퐿푝(휇), then there is a unique 푝- 푝 Champaign. His email address is [email protected]. weak upper gradient 푔ᵆ with minimal 퐿 -norm for which For permission to reprint this article, please contact: the preceding inequality holds for “almost every” curve 훾. [email protected]. The metric measure space 푋 is said to support a 푝-Poincar´e DOI: https://doi.org/10.1090/noti2030 inequality for some 푝 ≥ 1 if constants 퐶 > 0 and 휆 ≥ 1 exist

FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 253 so that for every ball 퐵 = 퐵(푥, 푅) ⊂ 푋, the inequality 푛-dimensional Euclidean domains as those homeomor-

1/푝 phisms that preserve the class of 푛-quasiharmonic func- 푝 tions. A similar statement is also true for PI spaces. This ∫− |푢 − 푢퐵|푑휇 ≤ 퐶푅(∫− 푔ᵆ 푑휇) 퐵 휆퐵 generalizes the well-known fact that planar conformal mappings are precisely the orientation-preserving home- holds for all function-upper gradient pairs (푢, 푔 ). Here ᵆ omorphisms that preserve harmonic functions under pull- 푢 = ∫− 푢 푑휇 and 휆퐵 = 퐵(푥, 휆푅). 퐵 퐵 back. Over the past twenty years, many aspects of first-order The further development of potential theory in the set- calculus have been systematically developed in the setting ting of metric measure spaces leads to a classification of PI spaces, that is, metric measure spaces equipped with of spaces as either 푝-parabolic or 푝-hyperbolic. This di- a doubling measure and supporting a Poincar´einequality. chotomy can be seen as a nonlinear analog of the recur- For example, for PI spaces we now have a rich theory of rence/transience dichotomy in the theory of Brownian mo- Sobolev functions which in turn lies at the foundation of tion. This classification is helpful in the development ofa the theory of quasconformal mappings and nonlinear po- quasiconformal uniformization theory or for a deeper un- tential theory. derstanding of the links between the geometry of hyper- A wealth of interesting and important examples of non- bolic spaces and the analysis on their boundaries at infin- Euclidean PI spaces exist, including sub-Riemannian man- ity. ifolds such as the Heisenberg group, Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds, vi- Differentiability of Lipschitz functions. The notion of upper gradient generalizes to metric spaces the norm of the sual boundaries of certain hyperbolic buildings, and frac- 1 tal spaces that are homeomorphic to the Menger curve. gradient of a 퐶 -function. It is a priori unclear how to for- The scope of the theory, however, is not fully explored. mulate a notion of the gradient itself (or of the differential of a function) in the absence of a linear structure. Cheeger Quasiconformal maps and nonlinear potential theory in [Che99] introduced a linear differential structure for real- metric spaces. A homeomorphism between metric spaces valued functions on metric measure spaces and established is said to be quasiconformal if it distorts the geometry of a version of Rademacher’s theorem for Lipschitz functions infinitesimal balls in a controlled fashion. Conformal defined on PI spaces. This differential structure gives rise maps form a special subclass for which infinitesimal balls to a finite-dimensional measurable vector bundle, the gen- are mapped to infinitesimal balls. Since the only confor- eralized cotangent bundle, over the metric space: to each real- mal maps between higher-dimensional Euclidean spaces valued Lipschitz function 푢 corresponds an 퐿∞-section 푑푢 are Möbius transformations, quasiconformal homeomor- of this bundle. Moreover, the pointwise Euclidean norm phisms form a more flexible class for geometric mapping |푑푢| is comparable to the minimal upper gradient 푔 al- problems. For quasiconformal maps on PI spaces, we now ᵆ most everywhere. This structure can be used in turn to in- have a well-developed theory that features many of the as- vestigate second-order PDEs in divergence form as a basis pects of the Euclidean theory, such as Sobolev regularity, for a theory of differential currents in metric spaces and for preservation of sets of measures zero, and global distortion many other purposes. estimates, among other things. A function 푢 on a domain Ω in a metric measure space Bi-Lipschitz embedding theorems. An earlier version of (푋, 푑, 휇) is said to be 푝-quasiharmonic for 푝 ≥ 1 if a constant Rademacher’s differentiation theorem for Lipschitz maps 푄 ≥ 1 exists so that the inequality between Carnot groups was proved by Pansu [Pan89]. Semmes observed that the Pansu–Rademacher theorem 푝 푝 implies that nonabelian Carnot groups do not admit bi- ∫ 푔ᵆ 푑휇 ≤ 푄 ∫ 푔ᵆ+휑 푑휇 spt휑 spt휑 Lipschitz copies in finite-dimensional Euclidean spaces. Moreover, such spaces do not bi-Lipschitz embed into holds whenever 휑 is a Lipschitz function with compact sup- Hilbert space or even into any Banach space with the port spt 휑 in Ω. In case 푄 = 1, we say that 푢 is 푝-harmonic; Radon–Nikod´ymproperty (RNP). Indeed, the algebraic this coincides with the classical Euclidean notion of a 푝- features of sub-Riemannian geometry have direct implica- harmonic function, defined as a weak solution to the 푝- tions for metric questions such as bi-Lipschitz equivalence Laplace equation or embeddability. div(|∇푢|푝−2∇푢) = 0. The bi-Lipschitz embedding problem is intimately re- lated to the existence of suitable differentiation theories Quasiharmonic functions are useful in the study for Lipschitz functions and maps. Roughly speaking, of quasiconformal mappings. For example, one can this relationship proceeds via incompatibility between the characterize quasiconformal homeomorphisms between

254 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 geometry of the cotangent bundles of the source and target 2-sphere. While the conjecture is still open, one can show spaces. In view of Cheeger’s differentiation theorem, one that the desired conclusion is true if 휕∞퐺 (equipped with can allow arbitrary PI space as source spaces here and take a visual metric) has good analytic properties, say, if it is RNP Banach spaces as targets, for example. On the other quasisymmetrically equivalent to a PI space. For more in- hand, there is no effective differentiation theory for maps formation see the ICM lectures [Bon06] and [Kle06]. into ℓ∞, because according to the Fréchetembedding the- The problem of deciding when a metric space is quasi- orem, every separable metric space embeds isometrically symmetrically equivalent to a space with “better” analytic into ℓ∞. properties can be seen as a generalization of classical uni- The target space 퐿1 presents an interesting intermediate formization theorems in complex analysis, at least for low- case. It is not an RNP Banach space, yet deep bi-Lipschitz dimensional fractals such as Sierpi ńskicarpets or fractal nonembedding theorems are available for this target. In 2-spheres. In order to study such problems, one often em- particular, Cheeger and Kleiner [CK10] showed that the ploys concepts from classical complex analysis in a metric Heisenberg group does not bi-Lipschitz embed into 퐿1. In space setting. For example, the modulus of a path fam- concert with results of Lee and Naor, this fact exhibits the ily, originally introduced by Ahlfors and Beurling in the Heisenberg group as a geometrically natural example rele- complex plane, now plays a prominent role in much re- vant for algorithmic questions in computer science. There cent work on mapping theory in general, abstract metric has been significant additional quantitative work along spaces. these lines, culminating in Naor and Young’s sharp lower Conclusion. This brief note barely hints at the breadth bound for the integrality gap of the Goemans–Linial semi- and the depth of the problems of current concern in the definite program for the Sparsest Cut Problem [NY18]. For theory of analysis on metric spaces. The 2020 AMS MRC further information, see Naor’s ICM lecture [Nao10]. Analysis in Metric Spaces will address a number of ques- Geometric measure theory on metric spaces. In yet an- tions that have been the subject of much recent investiga- other direction, Kirchheim’s proof of the almost every- tion but are far from being completely understood. where metric differentiability of Lipschitz mappings into metric spaces led to far-reaching generalizations of the ACKNOWLEDGMENTS. The second author is sup- area and co-area formulas. Subsequently, Ambrosio and ported by the Simons Foundation, and the other au- Kirchheim [AK00] developed an extension of the Federer– thors by the National Science Foundation. Fleming theory of currents in complete metric spaces, thus opening a new chapter in geometric measure theory lead- References ing to a study of (quantitative) rectifiability in metric [AGS08] Ambrosio L, Gigli N, SavaréG. Gradient Flows in Met- spaces. This notion has been further developed in sub- ric Spaces and in the Space of Probability Measures, 2nd ed., Riemannian settings, especially in the Heisenberg group, Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, but many questions remain open. These theories have Basel, 2008. MR2401600 been relevant, for example, in the work of Naor and Young [AK00] Ambrosio L, Kirchheim B. Currents in metric [NY18] mentioned above, where quantitative rectifiability spaces, Acta Math., no. 1 (185):1–80, 2000, DOI of surfaces in the Heisenberg group is prominently fea- 10.1007/BF02392711. MR1794185 tured. [BB11] Björn A, Björn J. Nonlinear Potential Theory on Met- ric Spaces, EMS Tracts in Mathematics, vol. 17, European Dynamics and analysis on metric spaces. An interesting Mathematical Society (EMS), Zürich, 2011. MR2867756 source of examples of spaces that can be studied with the [Bon06] Bonk M. Quasiconformal geometry of fractals. Proc. methods of quasiconformal geometry and analysis on met- Internat. Congr. Math. (Madrid 2006), Vol. II, European ric spaces is fractals that arise from self-similar or dynam- Mathematical Society, 2006, pp. 1349–1373. MR2275649 ical constructions such as limit sets of Kleinian groups, [BS07] Buyalo S, Schroeder V. Elements of Asymptotic Geometry, Julia sets of rational maps, or attractors of iterated func- EMS Monographs in Mathematics, European Mathemati- tion systems. Often the geometry of these spaces is too cal Society (EMS), Zürich, 2007. MR2327160 “rough” to expect finer analytic properties such as the [Che99] Cheeger J. Differentiability of Lipschitz functions on Poincaréinequality to hold. However, if these spaces ad- metric measure spaces, Geom. Funct. Anal., no. 3 (9):428– 517, 1999, DOI 10.1007/s000390050094. MR1708448 mit a good first-order calculus, then striking consequences [CK10] Cheeger J, Kleiner B. Differentiating maps into often emerge. For example, Cannon’s well-known con- 퐿1, and the geometry of BV functions, Ann. of Math. jecture in geometric group theory predicts that a Gromov (2), no. 2 (171):1347–1385, 2010, DOI 10.4007/an- hyperbolic group 퐺 admits a geometric action on hyper- nals.2010.171.1347. MR2630066 bolic 3-space if its boundary at infinity 휕∞퐺 is a topological [HK00] Hajłasz P, Koskela P. Sobolev met Poincaré, Mem. Amer. Math. Soc., no. 688 (145):x+101, 2000. MR1683160

FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 255 AMS Membership [Hei01] Heinonen J. Lectures on Analysis on Metric Spaces, Uni- versitext, Springer-Verlag, New York, 2001. MR1800917 [Hei07] Heinonen J. Nonsmooth calculus, Bull. Amer. for Early-Career Math. Soc. (N.S.), no. 2 (44):163–232, 2007, DOI 10.1090/S0273-0979-07-01140-8. MR2291675 [HK98] Heinonen J, Koskela P. Quasiconformal maps Mathematicians in metric spaces with controlled geometry, Acta Math., The AMS supports early career mathemati- no. 1 (181):1–61, 1998, DOI 10.1007/BF02392747. cians with opportunities for career develop- MR1654771 [HKST15] Heinonen J, Koskela P, Shanmugalingam N, Tyson ment, access to information and job listings, JT. Sobolev Spaces on Metric Measure Spaces, An approach connections with prospective employers, and based on upper gradients, New Mathematical Mono- valuable member bene ts, including: graphs, vol. 27, Cambridge University Press, Cambridge, 2015. MR3363168 Free shipping for members residing in the [Kle06] Kleiner B. The asymptotic geometry of negatively United States of America (including Puerto curved spaces: uniformization, geometrization and rigid- Rico) and Canada ity, Internat. Congr. Math. Vol. II: pp. 743–768, 2006. Discounts on AMS/MAA Press titles MR2275621 Reduced registration at the Joint Mathematics [Mos73] Mostow GD. Strong Rigidity of Locally Symmetric Meetings and AMS Sectional Meetings Spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Free subscriptions to Notices of the AMS Tokyo, 1973. MR0385004 and Bulletin of the AMS [MT10] Mackay JM, Tyson JT. Conformal Dimension, The- ory and application, University Lecture Series, vol. 54, American Mathematical Society, Providence, RI, 2010. Discounted dues rates for MR2662522 early career mathematicians [Nao10] Naor A. 퐿1 embeddings of the Heisenberg group and are available for 2020: fast estimation of graph isoperimetry, Proc. Internat. Congr. Math. Volume III: pp. 1549–1575, 2010. MR2827855 Graduate/Undergraduate Introductory Rate* [NY18] Naor A, Young R. Vertical perimeter versus horizontal Student Rate perimeter, Ann. of Math. (2), no. 1 (188):171–279, 2018, DOI 10.4007/annals.2018.188.1.4. MR3815462 [Pan89] Pansu P. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, $51 $77 (French, with English summary), Ann. of Math. (2), no. 1 (129):1–60, 1989, DOI 10.2307/1971484. MR979599

te Awtrey, Atlanta f Ka Con y o ve tes nti ur on co P o h t ot *Applies per year to the fi rst ho o P gr ap h fi ve consecutive years of y regular membership. Eligibility begins with the fi rst year of membership in any category other than Student, Graduate Student, or Affi liate.

Join OR Renew your AMS membership today by visiting: www.ams.org/membership

256 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2