Incomplete Yamabe Flows and Removable Singularities
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Annual Report for the Fiscal Year
The Institute for Advanced Study nual Report 1979/80 lis Annual Report has been made possible by a generous grant from the Union Carbide Corporation. The Institute for Advanced Study Annual Report for the Fiscal Year July 1, 1979 - June 30, 1980 The Institute for Advanced Study Princeton, New Jersey 08540 Printed by Princeton University Press Designed by Bruce Campbell It is fundamental to our purpose, and our Extract from the letter addressed by the express desire, that in the appointments to Founders to the Institute's Trustees, the staff and faculty, as well as in the dated June 6, 1930, admission of workers and students, no Newark, New Jersey. account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all, the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed or sex. 9;2 3f Table of Contents Trustees and Officers 9 Administration 10 The Institute for Advanced Study: Background and Purpose 11 Report of the Chairman 13 Report of the Director 15 Reports of the Schools 21 Publications of the Faculty, Professors Emeriti and Members with Long-term Appointments: A Selection 69 Record of Events, 1979-80 73 Report of the Treasurer 107 Donors 116 Founders Caroline Bamberger Fuld Louis Bamberger Board of Trustees Daniel Bell Howard C. Kauffmann Professor of Sociologi) President Harvard University Exxon Corporation Charles L. -
METRICS of CONSTANT SCALAR CURVATURE CONFORMAL to RIEMANNIAN PRODUCTS 1. Introduction Any Closed Manifold Admits Metrics of Cons
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 8, August 2010, Pages 2897–2905 S 0002-9939(10)10293-7 Article electronically published on March 29, 2010 METRICS OF CONSTANT SCALAR CURVATURE CONFORMAL TO RIEMANNIAN PRODUCTS JIMMY PETEAN (Communicated by Richard A. Wentworth) Abstract. We consider the conformal class of the Riemannian product g0 +g, m where g0 is the constant curvature metric on S and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of g. This is obtained by studying radial solutions of the equation Δu − λu + λup =0onSm and the number of solutions in terms of λ. 1. Introduction Any closed manifold admits metrics of constant scalar curvature. Given any Riemannian metric g on M n we consider its conformal class [g] and define the Yamabe constant of [g] as the minimum of the (normalized) total scalar curvature functional restricted to [g]: M sh dvolh Y (M,[g]) = inf − , ∈ n 2 h [g] Vol(M,h) n where sh and dvolh are the scalar curvature and volume element of h. It is elementary that the functional restricted to [g] is bounded below, and the fact that the infimum is actually achieved is a fundamental result, obtained in a series of steps by Hidehiko Yamabe [17], Thierry Aubin [2], Neil Trudinger [16] and Richard Schoen [14]. Since the critical points of the functional (restricted to [g]) are the metrics of constant scalar curvature in [g], it follows that minimizers are metrics of constant scalar curvature. -
Geometric Relativity
GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee 10.1090/gsm/201 Geometric Relativity GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201 Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Gradu- ate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identifiers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Differ- ential equations, Partial. | AMS: Differential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Differential geometry – Global differential geometry – Global Riemannian geometry, including pinching. msc | Differential geometry – Global differential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Differential geometry – Global differential geometry – Rigidity results. msc — Differential geometry – Global differential geometry – Spin and Spin. msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow, -
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. -
PCMI Lectures Final.Pdf
Contents Critical Metrics for Riemannian Curvature Functionals Jeff A. Viaclovsky 1 Critical Metrics for Riemannian Curvature Functionals 3 Introduction 3 Lecture 1. The Einstein-Hilbert functional 5 1. Notation and conventions 5 2. First variation 6 3. Normalized functional 7 4. Second variation 8 5. Transverse-traceless variations 9 Lecture 2. Conformal geometry 11 1. Conformal variations 11 2. Global conformal minimization 12 3. Green's function metric and mass 13 4. The Yamabe Problem 14 5. Generalizations of the Yamabe Problem 15 Lecture 3. Diffeomorphisms and gauging 17 1. Splitting 17 2. Second variation as a bilinear form 18 3. Ebin-Palais slice theorem (infinitesimal version) 19 4. Saddle point structure and the smooth Yamabe invariant. 21 Lecture 4. The moduli space of Einstein metrics 23 1. Moduli space of Einstein metrics 23 2. The nonlinear map 24 3. Structure of nonlinear terms 25 4. Existence of the Kuranishi map 26 5. Rigidity of Einstein metrics 27 Lecture 5. Quadratic curvature functionals 31 1. Quadratic curvature functionals 31 2. Curvature in dimension four 33 3. Einstein metrics in dimension four 34 4. Optimal metrics 35 5. Anti-self-dual or self-dual metrics 36 Lecture 6. Anti-self-dual metrics 39 i ii 1. Deformation theory of anti-self-dual metrics 39 2. Weitzenb¨ock formulas 40 3. Calabi-Yau metric on K3 surface 41 4. Twistor methods 43 5. Gluing theorems for anti-self-dual metrics 43 Lecture 7. Rigidity and stability for quadratic functionals 45 1. Strict local minimization 45 2. Local description of the moduli space 47 3. -
SOMMAIRE DU No 121
SOMMAIRE DU No 121 SMF Rapport moral ....................................... ............................... 3 MATHEMATIQUES´ Cantor et les infinis, P. Dehornoy ................................................... 29 Les immeubles, une th´eorie de Jacques Tits, prix Abel 2008, G. Rousseau .......... 47 EN HOMMAGE A` HENRI CARTAN (SUITE) La vie et l’œuvre scientifique de Henri Cartan, J-P. Serre ............................ 65 EN HOMMAGE A` THIERRY AUBIN Thierry Aubin (1942 – 2009), Bahoura, Ben Abdesselem, Delano¨e, Gil–Medrano, Holcman et al ................................................... ................... 71 Professor Aubin in our Memory, S.-Y. A. Chang, P. Yang ............................ 72 A tribute to Thierry Aubin, R. Schoen .............................................. 73 Un cours m´emorable (Paris VI, DEA 1976-77), Ph. Delano¨e ........................ 73 Thierry Aubin’s Work on Prescribing Scalar Curvature, J.L. Kazdan ................ 75 Aubin’s Contribution to Yamabe Problem, O. Gil-Medrano .......................... 77 Sur les travaux de Thierry Aubin en g´eom´etrie k¨ahl´erienne, A. Ben Abdesselem ...... 79 Grand salut au maˆıtre et `al’ami, A. Bahri .......................................... 81 C’´etait mon patron, D. Holcman ................................................... 82 T. Aubin, un math´ematicien curieux en m´ecanique des fluides, F. Coulouvrat ........ 83 In Memory of Thierry Aubin, M. A. Pinsky .......................................... 85 ENSEIGNEMENT De nouveaux programmes de math´ematiques en seconde.. -
The Contact Yamabe Flow
The Contact Yamabe Flow Von der FakultÄatfÄurMathematik und Physik der UniversitÄatHannover zur Erlangung des Grades eines Doktors der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von M. Sc. Yongbing Zhang geboren am 19.11.1978 in Anhui, China 2006 Referent: Prof. Dr. K. Smoczyk Koreferent: Prof. Dr. E. Schrohe Tag der Promotion: 09.02.2006 i Acknowledgements I am greatly indebted to my supervisor, Prof. Knut Smoczyk who has given me many help and support during the past years from fall of 2002. His helpful suggestions made it possible that my dissertation appears in the present form. I am also grateful to Prof. Jiayu Li and Prof. JÄurgenJost for their help and support. I am grateful to Prof. Guofang Wang, Prof. Xinan Ma and Prof. Xiux- iong Chen for their conversations in mathematics during my preparation for the dissertation. I have to give many thanks to Dr. Xianqing Li, Dr. Wei Li, Dr. Qingyue Liu, Ye Li, Xiaoli Han, Liang Zhao, Prof. Chaofeng Zhu, Prof. Yihu Yang, Prof. Qun Chen, Prof. Huijun Fan, Prof. Chunqin Zhou, Dr. Kanghai Tan, Dr. Bo Su, Dr. Ursula Ludwig, Konrad Groh, Melanie Schunert and many other friends with them I had learned a lot in mathematics and enjoyed a pleasant period in China and Germany. I am also grateful to the referee, Prof. Schrohe, for the discussion. He carefully reads this thesis and gives helpful suggestions. I am not only grateful to my family and many other friends for their many years' favor. iii Zusammenfassung Das Yamabe-Problem ist eine klassische Fragestellung aus der Di®erential- geometrie.