Jacqueline Ferrand and Her Oeuvre

COMMUNICATION Jacqueline Ferrand and Her Oeuvre Pierre Pansu Communicated by Allyn Jackson Jacqueline Ferrand was born in Alès, in the south of 퐷′ is smooth enough. The problem becomes hard if the France, one hundred years ago, in 1918. At the time, she boundary is irregular. was one of the very few women ever admitted to the In 1913, Constantin Carathéodory made a decisive step École Normale Supérieure (ENS), and she succeeded at when he introduced the notion of prime end. Carathéodory the (male) agrégation in 1939. She immediately won a considers nested cuts in 퐷′. These are nested sequences teaching assistant position at École Normale Supérieure of subdomains cut by simple arcs joining two boundary de Jeunes Filles. The director of the school, Mrs. Cotton, points, and whose lengths tend to 0. Two sequences are was convinced that women should nourish the same equivalent if each can be nested in the other. A prime intellectual ambition as men, and she reckoned that this end is an equivalence class of nested cuts. This definition, brilliant young mathematician would raise the level of which involves lengths, is not obviously a conformal teaching at Jeunes Filles to the same level as the ENS. invariant. Carathéodory gives a proof of invariance that Several people witnessed the energy Jacqueline Ferrand relies on fine properties of holomorphic functions. devoted to this task despite poor material conditions. In her thesis, published in the Annales de l’École Nor- With the same energy, she began to do research, under male Supérieure de Paris in 1942, Ferrand gives a new Arnaud Denjoy’s supervision from afar. After her thesis proof of conformal invariance of prime ends that high- defense, on June 12, 1942, she was awarded the Girbal lights the role played by the area of a conformal map Barral Prize in 1943 and the Fondation Peccot Prize in 푓 in estimating lengths of images of almost all curves. 1946. Her career developed quickly: assistant professor Furthermore, the fact that 푓 is open allows one to pass in Bordeaux in 1943, professor in Caen in 1945, in Lille in from almost all curves to all curves, and hence to estimate 1948, in Paris from 1956 until her retirement in 1984. the modulus of continuity of 푓. Precise estimates lead to sufficient conditions on the First Works domain 퐷′ for the conformal mapping to admit limits Jacqueline Ferrand’s thesis deals with boundary values of along curves contained in the disk and having a higher planar conformal mappings. Since the time of Bernhard order contact with the boundary of the disk. Under Riemann, it has been known that every simply connected stronger assumptions, Ferrand shows, in a 1942 paper domain 퐷′ admits a conformal mapping 푓 from the in the Bulletin de la Société Mathématique de France, the unit disk 퐷, i.e. a geographic map in which angles are existence of angular derivatives preserved. One may view 푓 as an analytic function of one 푓(푧) − 푎 lim variable defined on the disk, which is a bijection from 푧→푎 푧 − 푎 ′ 퐷 onto 퐷 . The issue is whether 푓 has a limit at every when 푎 is a boundary point. boundary point. The answer is positive if the boundary of For a conformal (or holomorphic) bijection, the area of the image is given by the Dirichlet integral Pierre Pansu is professor of mathematics at Université Paris-Sud. His email address is [email protected]. ∫ |푓′(푧)|2푑푧. This article originally appeared in French in the Gazette des Math- 퐷 ématiciens, number 141, July 2014. Pansu prepared this English For a harmonic function 푢 (the real part of a holomorphic version, which appears here with the kind permission of the So- function 푓), the Dirichlet integral replaces area and the ciété Mathématique de France. maximum principle provides an ersatz for the openness of For permission to reprint this article, please contact: 푓. Methods developed for conformal mappings in her 1944 [email protected]. paper in the SMF Bulletin hence extend to the boundary DOI: http://dx.doi.org/10.1090/noti1641 study of harmonic and superharmonic functions. February 2018 Notices of the AMS 201 COMMUNICATION The real and imaginary parts of 푓 are preharmonic functions on sub-lattices ′ 푍ℎ = {푧 ∈ 푍ℎ ; (푅푒(푧) + 퐼푚(푧))/ℎ is even} and ″ 푍ℎ = {푧 ∈ 푍ℎ ; (푅푒(푧) + 퐼푚(푧))/ℎ is odd}. Conversely, every preharmonic function on 푍ℎ is the real part of a preholomorphic function on 푍ℎ. Ferrand shows that when the mesh ℎ tends to 0, preholomorphic functions on 푍ℎ converge to holomorphic functions on the domain 퐷′. A priori estimates on the modulus of continuity play a crucial role again. In 1955, Ferrand inferred a pretty and simple proof of the conformal mapping theorem for nonsimply connected domains. Let 퐷′ be a planar domain whose boundary has at least one isolated connected component that is not a single point. Then there exists an essentially unique conformal mapping of 퐷′ onto a rectangle with segments parallel to one side removed. The notion of preholomorphic function has led to numerous developments. Group Actions During a visit to Institute for Advanced Study, Princeton, Ferrand investigated when a Lie algebra action integrates into a Lie group action. The upshot is a functional analytic characterization of completeness of vector fields, which she published in the SMF Bulletin in 1942. Let 휉 be a locally Lip- schitz vector field with This elegant vanishing divergence, on a result does not manifold 푀 equipped with Jacqueline Ferrand, shown here in a photo dated a volume element 휔. Then 1988, was an outstanding French geometer whose seem to be well 휉 is complete if and only works continue to influence research today. She if the differential operator wrote her last mathematical papers at the age of 푖휉 extends to a selfadjoint 1 known. eighty and died in 2014, at the age of ninety-six. operator of 퐿2(휔). Assume that 휉 generates a one-parameter group of isometries of 푀. This group is periodic if and only if the operator 푖휉 Preholomorphic Functions has a closed image. In that same paper, Ferrand introduced a discretization This elegant result does not seem to be well known. of the notion of holomorphic function. Given ℎ > 0, she replaces the plane ℂ (or a bounded domain 퐷′ of ℂ) Teaching Books ′ with the finite set 푍ℎ of points of 퐷 whose real and Ferrand’s mathematical production decreased between imaginary parts are integer multiples of ℎ. Classically, 1958 and 1968. At this time, Jacqueline was the mother of one defines a function 푢 on 푍ℎ to be harmonic (Ferrand four young children. She put a great deal of effort into her writes preharmonic) if teaching and wrote a series of course notes that, with a lot of work, became textbooks. A basic differential geometry 4푢(푧) = 푢(푧 + ℎ) + 푢(푧 − ℎ) + 푢(푧 + 푖ℎ) + 푢(푧 − 푖ℎ). course was published by Masson in 1963. Her freshman Discretizing holomorphicity equations is less classical. courses were published by Armand Colin in 1964 and Ferrand calls preholomorphic a complex-valued function Dunod in 1967. In the 1970s Dunod published the series of books co-authored with Jean-Marie Arnaudiès, which 푓 that satisfies, for all 푧 ∈ 푍ℎ, covered the whole curriculum of the first two years of 푓(푧 + 푖ℎ) − 푓(푧 + ℎ) = 푖(푓(푧 + ℎ + 푖ℎ) − 푓(푧)). university (including exercises). It is still in use today in the classes préparatoires aux grandes écoles, which prepare 1This photo was taken at École Normale Supérieure and kindly students for the competitive examinations to enter the provided to Notices by Ms. Ferrand’s children. École Normale Supérieure. 202 Notices of the AMS Volume 65, Number 2 COMMUNICATION Riemannian Geometry is a conformal diffeomorphism and that the convergence At the end of this period, Ferrand obtained her most is in 퐶∞. This is a nonlinear elliptic regularity theorem, famous results, leading to an invitation to the 1974 due to Frederick Gehring and Yuri Reshetnyak in the International Congress of Mathematicians in Vancouver. Euclidean case, and extended by Ferrand in 1976 to She solved a problem in Riemannian geometry posed by general Riemannian manifolds. André Lichnerowicz in 1964, on which several partial More can be drawn from the 푗 invariant. If a sequence answers had been published. 푓푘 of conformal mappings diverges, then for all sequences A conformal mapping of an open set of Euclidean 푥푘, 푧푘, 푡푘, if 푓푘(푧푘) converges to 푧 and 푓푘(푡푘) converges 푛 space ℝ is a diffeomorphism whose differential preserves to 푡 ≠ 푧, then 푓푘(푥푘) converges to 푧 or to 푡. This angles. This notion extends to open sets of ℝ푛 equipped means that at most two limits are possible, 푧 and 푡. If with a Riemannian metric, i.e. an inner product depending 푧 ≠ 푡, then, up to extracting a subsequence, 푓푘 maps the on position, to submanifolds of Euclidean spaces, and complement of every neighborhood of 푧 into arbitrarily to abstract Riemannian manifolds. The prototype of a small neighborhoods of 푡. This implies that the manifold Riemannian manifold is the sphere 푀 is simply connected and that its metric is conformally 푛+1 2 2 flat, hence 푀 is conformal to the sphere. {푥 ∈ ℝ ; 푥0 + ⋯ + 푥푛 = 1}. Stereographic projection achieves a conformal diffeo- Quasiconformal Mappings morphism of the sphere, with one point deleted, onto How did Ferrand discover her four-point invariants? Euclidean space. By transport of structure, Euclidean simil- The idea of using a conformally invariant metric itudes become conformal diffeomorphisms of the sphere.

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