DOCSLIB.ORG

Jxkj,L = 0. R1j, 1Xh11?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Jxkj,L = 0. R1j, 1Xh11? VOL. 45, 1959 MATHEMATICS: R. OSSERMAN 1031 Since XhIJ, I= (XhligU), I= gij)hli, I, the above equation may be given the form Xhl, 1Xkli + Xh|JXkJ,l = 0. By means of this result equation (12) becomes R1j, 1Xh11?k1l + (Ph -Pk)Xkli>g, I = 0, which may be given the form Rij, lXhAlXklj + (Ph - Pk)Xk4Xhl , I = 0. We multiply this equation by Xm., sum for 1 and get Rij, lXh|iXkJIcpm1 + (Ph - Pk)Xk|iXmg Xh~i, I = 0. (13' We require that Vn is such that Ri I = '/2R,1gjl + 1/2Rgj11, (14) where R is the Ricci scalar, that is R = g1JRij. When equation (14) is multiplied by giJ and contracted for i and j, the result is satisfied identically. For Rij, as given in (14) and h, k, 1 different, the first term of equation (13) is equal to zero in accordance with equations (7) and since the p's are different one has Xhli, 1XkJiXmn= 0, (h, k, m ±). Hence the vectors are normal. Since similar results hold for all the principal director vectors, we have In a V,,for which equation (14) holds the Ricci principal direction vectors are normal. 1 Eisenhart, L. P., Ri6mannian Geometry (Princeton University Press, 1926, 1944), equation (34.4), 114. 2 Ibid., equation (21.8), 28. 3 Ibid., equation (13.5), 38. 4 Walker, R. S., Algebraic Curves (Princeton University Press, 1950), 24, 25. 5 Eisenhart, L. P., ibid., equation (30.1), 97, and equation (36.1), 117. AN EXTENSION OF CERTAIN RESULTS IN FUNCTION THEORY TO A CLASS OF SURFACES* BY ROBERT OSSERMAN DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY Communicated by Lars V. Ahifors, May 1, 1959 The object of this paper is to show that a large number of results in function theory can be carried over from plane regions to! a certain well-defined class of differential-geometric surfaces. The definition of this class is not constructive, so that the difficult part is to prove that a given surface belongs to the class. How- ever, we do show that the class includes a large class of minimal surfaces, which in turn includes all simply-connected minimal surfaces representable in the form z = f(x, y), and the results obtained are of interest even for this restricted class of surfaces. An example of the type of theorem obtained for, this case is the follow- ing analogue of the Koebe '-Theorem. Let S be a simply-connected minimal surface of the form z = f(x, y) and let p be a- point of S whose geodesic distance to the boundary of S is at least d. If ds is the ele- Downloaded by guest on September 23, 2021 1032 MATHEMATICS: R. OSSERMAN PROC. N. A. S. ment of arc length on S, and if S is mapped one-to-one conformally onto a region R in the complex w-plane with the normalization IdwI /ds = 1 at p, then every boundary point of R is at a distance at least d/8 from the image of p. Definition: Let S be a surface with a Riemannian metric ds2. We shall say that S belongs to the class 8(m) if there exists a mapping of S into the complex z-plane which is conformal with respect to the metric on S and which satisfies 1< <dz<dlI< m (1) ds- throughout S. The mapping is not required to be globally one-to-one, but the image of S will at any rate be an unbranched Riemann surface lying over the z-plane. LEMMA 1. Let S e 8(m) and let p be a point of S whose distance to the boundary of S is at least d. Then there exists a one-to-one conformal map of the disk zI < d into S such that the point z = 0 maps into p, and m-< d8 l <~o Proof: By the definition of 8(m), S may be mapped conformally onto an un- branched Riemann surface R over the z-plane, and we may assume that the image q of p lies over the origin. Let D be the largest open disk about q lying in R. Then D projects onto a disk IzI < r < X . We wish to show that r > d. If r =c, there is nothing more to prove. If not, there must exist a point z1 with z1| = r such that the relative boundary of D contains no point over zi. Let L be the seg- ment in D projecting onto the line segment from 0 to z1. Then L corresponds to an arc C in S which goes from p to the boundary of S. Hence, using (1), we have d < cds <fL dz =r, which proves the lemma. COROLLARY. A complete simply-connected surface S e 3(m) is conformally equiv- alent to the plane. Proof: The image Riemann surface R over the z-plane has no algebraic branch points and no boundary points at finite distance and must therefore coincide with the z-plane. THEOREM 1. Let S e 8(m) and let p be a point of S whose distance to the boundary of S is at least d. Let S be mapped one-to-one conformally into the w-plane with the normalization dw I ds = 1 at p. Then the image region includes a disk of radius d/4m about the image of p. Proof: By Lemma 1, we get a one-to-one conformal map w = f(z) of the disk Iz < d into the w-plane such that If'(0) = [ldwI/ds],. [ds/IdzI lo 2 1/m. By the Koebe '-Theorem the image region must contain a disk about f(o) of radius d/4m. We note that any plane region, considered as a surface S in the euclidean metric, belongs to 3(1) by virtue of the identity map. Theorem 1 therefore reduces in the case of plane regions to the classical Koebe Theorem. Definition: The class = (a) is defined to be the class of all simply-connected minimal surfaces S having the property that all normals to S make an angle of at least a > 0 with some fixed direction. Downloaded by guest on September 23, 2021 VOL. 45, 1959 MATHEMATICS: R. OSSERMAN 1033 We note that a minimal surface of the form z = f(x, y) belongs to the class Sfl(w/2). a THEOREM 2. =(a) C 8(m) for m = cscC2 Proof: Let S e =(a). Since S is a minimal surface, the map of S by parallel normals into the unit sphere E is conformal, and if do is the line element on we have da2/d&2 = KI, where K is the Gauss curvature of S. We project E stereographically from the point corresponding to the distinguished direction in space onto the complex w-plane. Since the image of S in E is bounded away from the distinguished direction by at least a, the image of S in the w-plane will lie in the disk IwI < R, where R = cot a. We may write the line element do on E as do = XIdwvI where X = 2/(1 + | wJ 2). Let h(p) be the function on S defined to have at each point the value of X at the corresponding point in the w-plane. Then 2 I + R2 < h(p) < 2 (2) on S. We shall show that there exists a locally conformal map of S into the com- plex c-plane satisfying [dr]= h(p) (3) throughout S. We give the proof in the case that K is nowhere zero on S. This restriction can be lifted by a slight additional argument' at points where K = 0. For K # 0 the map of S into I wj < R is locally one-to-one, and the image of S will be a simply-connected unbranched Riemann surface W over the tv-plane. In the neighborhood of each point of S we may write ds = PI dwf, where ds ds da- (4) Jdw, do- Idwi The formula for the Gauss curvature in isothermal coordinates is K =-(A log p)/ p2, where A is the Laplacian with respect to w. This gives A log p = -Kp2. (5) The same formula applied to E yields - (A log X)/X2 = 1, or A log X = _X2 = _Pp21 K = Kp2 (6) where we have used (4) and the fact that K < 0 since S is a minimal surface. Combining (5) and (6) we see that the function p = log pX is a harmonic function on W. By the simple-connectivity of W we can find a harmonic conjugate 4#. We then define =fev"+idw (7) where the integral is taken along paths on W from a fixed point to a variable point. Then I dr/dw =e = pX, and since d8/IdwI = p, we obtain fdfI/ds = X, which proves (3). Downloaded by guest on September 23, 2021 1034 MATHEMATICS: R. OSSERMAN PROC.'N. A. S. Finally, if we let z = P(1 + R2)/2 and use (2) we obtain (1) with m e 1 + R2. This proves the theorem. In particular, we see that every simply-connected minimal surface of the form z = f(x, y) is contained in 8(2), which combined with Theorem 1 gives the result stated at the beginning of the paper. We note further, combining Theorem 2 with the corollary to Lemma 1, that every complete surface S in l(a) is conformally equivalent to the plane. By Liouville's Theorem, the map of S into w < R must be a constant.
Load more

Recommended publications
  • Math, Physics, and Calabi–Yau Manifolds
    Math, Physics, and Calabi–Yau Manifolds Shing-Tung Yau Harvard University October 2011 Introduction I’d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory. This happens to be the subject of the new book I coauthored, THE SHAPE OF INNER SPACE It also tells some of my own story and a bit of the history of geometry as well. 2 In that spirit, I’m going to back up and talk about my personal introduction to geometry and how I ended up spending much of my career working at the interface between math and physics. Along the way, I hope to give people a sense of how mathematicians think and approach the world. I also want people to realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but is instead crucial to our understanding of the physical world. 3 There are several major contributions of mathematicians to fundamental physics in 20th century: 1. Poincar´eand Minkowski contribution to special relativity. (The book of Pais on the biography of Einstein explained this clearly.) 2. Contributions of Grossmann and Hilbert to general relativity: Marcel Grossmann (1878-1936) was a classmate with Einstein from 1898 to 1900. he was professor of geometry at ETH, Switzerland at 1907. In 1912, Einstein came to ETH to be professor where they started to work together. Grossmann suggested tensor calculus, as was proposed by Elwin Bruno Christoffel in 1868 (Crelle journal) and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita (1901).
    [Show full text]
  • Kepler's Laws, Newton's Laws, and the Search for New Planets
    Integre Technical Publishing Co., Inc. American Mathematical Monthly 108:9 July 12, 2001 2:22 p.m. osserman.tex page 813 Kepler’s Laws, Newton’s Laws, and the Search for New Planets Robert Osserman Introduction. One of the high points of elementary calculus is the derivation of Ke- pler’s empirically deduced laws of planetary motion from Newton’s Law of Gravity and his second law of motion. However, the standard treatment of the subject in calcu- lus books is flawed for at least three reasons that I think are important. First, Newton’s Laws are used to derive a differential equation for the displacement vector from the Sun to a planet; say the Earth. Then it is shown that the displacement vector lies in a plane, and if the base point is translated to the origin, the endpoint traces out an ellipse. This is said to confirm Kepler’s first law, that the planets orbit the sun in an elliptical path, with the sun at one focus. However, the alert student may notice that the identical argument for the displacement vector in the opposite direction would show that the Sun orbits the Earth in an ellipse, which, it turns out, is very close to a circle with the Earth at the center. That would seem to provide aid and comfort to the Church’s rejection of Galileo’s claim that his heliocentric view had more validity than their geocentric one. Second, by placing the sun at the origin, the impression is given that either the sun is fixed, or else, that one may choose coordinates attached to a moving body, inertial or not.
    [Show full text]
  • Stable Minimal Hypersurfaces in Four-Dimensions
    STABLE MINIMAL HYPERSURFACES IN R4 OTIS CHODOSH AND CHAO LI Abstract. We prove that a complete, two-sided, stable minimal immersed hyper- surface in R4 is flat. 1. Introduction A complete, two-sided, immersed minimal hypersurface M n → Rn+1 is stable if 2 2 2 |AM | f ≤ |∇f| (1) ZM ZM ∞ for any f ∈ C0 (M). We prove here the following result. Theorem 1. A complete, connected, two-sided, stable minimal immersion M 3 → R4 is a flat R3 ⊂ R4. This resolves a well-known conjecture of Schoen (cf. [14, Conjecture 2.12]). The corresponding result for M 2 → R3 was proven by Fischer-Colbrie–Schoen, do Carmo– Peng, and Pogorelov [21, 18, 36] in 1979. Theorem 1 (and higher dimensional analogues) has been established under natural cubic volume growth assumptions by Schoen– Simon–Yau [37] (see also [45, 40]). Furthermore, in the special case that M n ⊂ Rn+1 is a minimal graph (implying (1) and volume growth bounds) flatness of M is known as the Bernstein problem, see [22, 17, 3, 45, 6]. Several authors have studied Theorem 1 under some extra hypothesis, see e.g., [41, 8, 5, 44, 11, 32, 30, 35, 48]. We also note here some recent papers [7, 19] concerning stability in related contexts. It is well-known (cf. [50, Lecture 3]) that a result along the lines of Theorem 1 yields curvature estimates for minimal hypersurfaces in R4. Theorem 2. There exists C < ∞ such that if M 3 → R4 is a two-sided, stable minimal arXiv:2108.11462v2 [math.DG] 2 Sep 2021 immersion, then |AM (p)|dM (p,∂M) ≤ C.
    [Show full text]
  • ON the INDEX of MINIMAL SURFACES with FREE BOUNDARY in a HALF-SPACE 3 a Unit Vector on ℓ [MR91]
    ON THE INDEX OF MINIMAL SURFACES WITH FREE BOUNDARY IN A HALF-SPACE SHULI CHEN Abstract. We study the Morse index of minimal surfaces with free boundary in a half-space. We improve previous estimates relating the Neumann index to the Dirichlet index and use this to answer a question of Ambrozio, Buzano, Carlotto, and Sharp concerning the non-existence of index two embedded minimal surfaces with free boundary in a half-space. We also give a simplified proof of a result of Chodosh and Maximo concerning lower bounds for the index of the Costa deformation family. 1. Introduction Given an orientable Riemannian 3-manifold M 3, a minimal surface is a critical point of the area functional. For minimal surfaces in R3, the maximum principle implies that they must be non-compact. Hence, minimal surfaces in R3 are naturally studied under some weaker finiteness assumption, such as finite total curvature or finite Morse index. We use the word bubble to denote a complete, connected, properly embedded minimal surface of finite total curvature in R3. As shown by Fischer-Colbrie [FC85] and Gulliver–Lawson [GL86, Gul86], a complete oriented minimal surface in R3 has finite index if and only if it has finite total curvature. In particular, bubbles have finite Morse index. Bubbles of low index are classified: the plane is the only bubble of index 0 [FCS80, dCP79, Pog81], the catenoid is the only bubble of index 1 [LR89], and there is no bubble of index 2 or 3 [CM16, CM18]. Similarly, given an orientable Riemannian 3-manifold M 3 with boundary ∂M, a free boundary minimal surface is a critical point of the area functional among all submanifolds with boundaries in ∂M.
    [Show full text]
  • Differential Geometry Proceedings of Symposia in Pure Mathematics
    http://dx.doi.org/10.1090/pspum/027.1 DIFFERENTIAL GEOMETRY PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME XXVII, PART 1 DIFFERENTIAL GEOMETRY AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1975 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT STANFORD UNIVERSITY STANFORD, CALIFORNIA JULY 30-AUGUST 17, 1973 EDITED BY S. S. CHERN and R. OSSERMAN Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-37243 Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics, Stanford University, UE 1973. Differential geometry. (Proceedings of symposia in pure mathematics; v. 27, pt. 1-2) "Final versions of talks given at the AMS Summer Research Institute on Differential Geometry." Includes bibliographies and indexes. 1. Geometry, Differential-Congresses. I. Chern, Shiing-Shen, 1911- II. Osserman, Robert. III. American Mathematical Society. IV. Series. QA641.S88 1973 516'.36 75-6593 ISBN0-8218-0247-X(v. 1) Copyright © 1975 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. CONTENTS Preface ix Riemannian Geometry Deformations localement triviales des varietes Riemanniennes 3 By L. BERARD BERGERY, J. P. BOURGUIGNON AND J. LAFONTAINE Some constructions related to H. Hopf's conjecture on product manifolds ... 33 By JEAN PIERRE BOURGUIGNON Connections, holonomy and path space homology 39 By KUO-TSAI CHEN Spin fibrations over manifolds and generalised twistors 53 By A. CRUMEYROLLE Local convex deformations of Ricci and sectional curvature on compact manifolds 69 By PAUL EWING EHRLICH Transgressions, Chern-Simons invariants and the classical groups 72 By JAMES L.
    [Show full text]
  • Mathematics of the Gateway Arch Page 220
    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice.
    [Show full text]
  • Chern Flyer 1-3-12.Pub
    TAKING THE LONG VIEW The Life of Shiing-shen Chern A film by George Csicsery “He created a branch of mathematics which now unites all the major branches of mathematics with rich structure— which is still developing today.” —C. N. Yang Shiing-shen Che rn (1911–2004) Taking the Long View examines the life of a remarkable mathematician whose classical Chinese philosophical ideas helped him build bridges between China and the West. Shiing-shen Chern (1911-2004) is one of the fathers of modern differential geometry. His work at the Institute for Advanced Study and in China during and after World War II led to his teaching at the University of Chicago in 1949. Next came Berkeley, where he created a world-renowned center of geometry, and in 1981 cofounded the Mathematical Sciences Research Institute. During the 1980s he brought talented Chinese scholars to the United States and Europe. By 1986, with Chinese government support, he established a math institute at Nankai University in Tianjin. Today it is called the Chern Institute of Mathematics. Taking the Long View was produced by the Mathematical Sciences Research Institute with support from Simons Foundation. Directed by George Csicsery. With the participation of (in order of appearance) C.N. Yang, Hung-Hsi Wu, Bertram Kostant, James H. Simons, Shiing-shen Chern, Calvin Moore, Phillip Griffiths, Robert L. Bryant, May Chu, Yiming Long, Molin Ge, Udo Simon, Karin Reich, Wentsun Wu, Robert Osserman, Isadore M. Singer, Chuu-Lian Terng, Alan Weinstein, Guoding Hu, Paul Chern, Rob Kirby, Weiping Zhang, Robert Uomini, Friedrich Hirzebruch, Zixin Hou, Gang Tian, Lei Fu, Deling Hu and others.
    [Show full text]
  • Emissary Mathematical Sciences Research Institute
    Spring 2012 Emissary Mathematical Sciences Research Institute www.msri.org Randomness, Change, and All That A word from Director Robert Bryant statistical mechanics conformal invariance The spring semester here at MSRI has been a remarkably active one. Our jumbo scientific program, Random Spatial Processes, has attracted an enthusiastic group evolution equation of mathematicians who keep the building humming with activities, both in their (five!) mixing times scheduled workshops and their weekly seminars. dimer model In February, we had the pleasure of hosting the Hot Topics Workshop, “Thin Groups and Super-strong Approximation”, which brought together experts in the recent developments random sampling in “thin groups” (i.e., discrete subgroups of semisimple Lie groups that are Zariski dense lattice models and yet have infinite covolume). These subgroups, delicately balanced between being “big” and “small”, have turned out to have surprising connections with dynamics on random clusters Teichmüller space, the Bourgain–Gamburd–Sarnak sieve, arithmetic combinatorics, and Schramm–Loewner the geometry of monodromy groups and covering. It was an exciting workshop, and the talks and notes (now online) have proved to be quite popular on our website. critical exponent Also in February, MSRI hosted the playwrights of PlayGround for a discussion of how Markov chains mathematicians think about randomness and physical processes. Our research members percolation James Propp and Dana Randall, together with two of our postdoctoral fellows, Adrien Kassel and Jérémie Bettinelli, described how they think about their work and what fascinates scaling limits them about it, and the playwrights responded with a lively and stimulating set of questions. The Ising model playwrights each then had 9 days to write a 10-minute play around the theme “Patterns of Chaos”.
    [Show full text]
  • The Shape of Inner Space Provides a Vibrant Tour Through the Strange and Wondrous Possibility SPACE INNER
    SCIENCE/MATHEMATICS SHING-TUNG $30.00 US / $36.00 CAN Praise for YAU & and the STEVE NADIS STRING THEORY THE SHAPE OF tring theory—meant to reconcile the INNER SPACE incompatibility of our two most successful GEOMETRY of the UNIVERSE’S theories of physics, general relativity and “The Shape of Inner Space provides a vibrant tour through the strange and wondrous possibility INNER SPACE THE quantum mechanics—holds that the particles that the three spatial dimensions we see may not be the only ones that exist. Told by one of the Sand forces of nature are the result of the vibrations of tiny masters of the subject, the book gives an in-depth account of one of the most exciting HIDDEN DIMENSIONS “strings,” and that we live in a universe of ten dimensions, and controversial developments in modern theoretical physics.” —BRIAN GREENE, Professor of © Susan Towne Gilbert © Susan Towne four of which we can experience, and six that are curled up Mathematics & Physics, Columbia University, SHAPE in elaborate, twisted shapes called Calabi-Yau manifolds. Shing-Tung Yau author of The Fabric of the Cosmos and The Elegant Universe has been a professor of mathematics at Harvard since These spaces are so minuscule we’ll probably never see 1987 and is the current department chair. Yau is the winner “Einstein’s vision of physical laws emerging from the shape of space has been expanded by the higher them directly; nevertheless, the geometry of this secret dimensions of string theory. This vision has transformed not only modern physics, but also modern of the Fields Medal, the National Medal of Science, the realm may hold the key to the most important physical mathematics.
    [Show full text]
  • Lars Ahlfors Entered the University of Helsinki, Where His Teachers Were Two Internationally Known Math- Ematicians, Ernst Lindelöf and Rolf Nevanlinna
    NATIONAL ACADEMY OF SCIENCES LARS VALERIAN AHLFORS 1907–1996 A Biographical Memoir by FREDERICK GEHRING Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoirs, VOLUME 87 PUBLISHED 2005 BY THE NATIONAL ACADEMIES PRESS WASHINGTON, D.C. LARS VALERIAN AHLFORS April 18, 1907–October 11, 1996 BY FREDERICK GEHRING PERSONAL AND PROFESSIONAL HISTORY ARS AHLFORS WAS BORN in Helsinki, Finland, on April 18, L 1907. His father, Axel Ahlfors, was a professor of mechanical engineering at the Institute of Technology in Helsinki. His mother, Sievä Helander, died at Lars’s birth. As a newborn Lars was sent to the Åland Islands to be taken care of by two aunts. He returned to his father’s home in Helsinki by the age of three. At the time of Lars’s early childhood, Finland was under Russian sovereignty but with a certain degree of autonomy, and civil servants, including professors, were able to enjoy a fairly high standard of living. Unfortunately, all this changed radically during World War I, the Russian Revolution, and the Finnish civil war that followed. There was very little food in 1918, and Lars’s father was briefly imprisoned by the Red Guard. For historical reasons the inhabitants of Finland are divided into those who have Finnish or Swedish as their mother tongue. The Ahlfors family was Swedish speaking, so Lars attended a private school where all classes were taught in Swedish. He commented that the teaching of math- ematics was mediocre, but credited the school with helping 3 4 BIOGRAPHICAL MEMOIRS him become almost fluent in Finnish, German, and English, and less so in French.
    [Show full text]
  • Leroy P. Steele Prize for Mathematical Exposition 1
    AMERICAN MATHEMATICAL SOCIETY LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION The Leroy P. Steele Prizes were established in 1970 in honor of George David Birk- hoff, William Fogg Osgood, and William Caspar Graustein and are endowed under the terms of a bequest from Leroy P. Steele. Prizes are awarded in up to three cate- gories. The following citation describes the award for Mathematical Exposition. Citation John B. Garnett An important development in harmonic analysis was the discovery, by C. Fefferman and E. Stein, in the early seventies, that the space of functions of bounded mean oscillation (BMO) can be realized as the limit of the Hardy spaces Hp as p tends to infinity. A crucial link in their proof is the use of “Carleson measure”—a quadratic norm condition introduced by Carleson in his famous proof of the “Corona” problem in complex analysis. In his book Bounded analytic functions (Pure and Applied Mathematics, 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981, xvi+467 pp.), Garnett brings together these far-reaching ideas by adopting the techniques of singular integrals of the Calderón-Zygmund school and combining them with techniques in complex analysis. The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs. The book has educated a whole generation of mathematicians with backgrounds in complex analysis and function algebras. It has had a great impact on the early careers of many leading analysts and has been widely adopted as a textbook for graduate courses and learning seminars in both the U.S.
    [Show full text]
  • Chern Flyer 2-10-15Rev.Pub
    TAKING THE LONG VIEW The Life of Shiing-shen Chern A film by George Csicsery “He created a branch of mathematics which now unites all the major branches of mathematics with rich structure— which is still developing today.” —C. N. Yang Shiing-shen Che rn (1911–2004) Taking the Long View examines the life of a remarkable mathematician whose classical Chinese philosophical ideas helped him build bridges between China and the West. Shiing-shen Chern (1911-2004) is one of the fathers of modern differential geometry. His work at the Institute for Advanced Study and in China during and after World War II led to his teaching at the University of Chicago in 1949. Next came Berkeley, where he created a world-renowned center of geometry, and in 1981 cofounded the Mathematical Sciences Research Institute. During the 1980s he brought talented Chinese scholars to the United States and Europe. By 1986, with Chinese government support, he established a math institute at Nankai University in Tianjin. Today it is called the Chern Institute of Mathematics. Taking the Long View was produced by the Mathematical Sciences Research Institute with support from Simons Foundation. Directed by George Csicsery. With the participation of (in order of appearance) C.N. Yang, Hung-Hsi Wu, Bertram Kostant, James H. Simons, Shiing-shen Chern, Calvin Moore, Phillip Griffiths, Robert L. Bryant, May Chu, Yiming Long, Molin Ge, Udo Simon, Karin Reich, Wentsun Wu, Robert Osserman, Isadore M. Singer, Chuu-Lian Terng, Alan Weinstein, Guoding Hu, Paul Chern, Rob Kirby, Weiping Zhang, Robert Uomini, Friedrich Hirzebruch, Zixin Hou, Gang Tian, Lei Fu, Deling Hu and others.
    [Show full text]