Complex Analysis

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be sent to the Mathematics Editor at either Springer Basel AG Birkhäuser P.O. Box 133 CH-4010 Basel Switzerland or Birkhauser Boston 233 Spring Street New York, NY 10013 USA Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TeX is acceptable, but the entire collection of files must be in one particular dialect of TeX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. The total number of pages should not exceed 350. The first-mentioned author of each article will receive 25 free offprints. To the participants of the congress the book will be offered at a special rate. Complex Analysis Several Complex Variables and Connections with PDE Theory and Geometry Peter Ebenfelt Norbert Hungerbühler Joseph J. Kohn Ngaiming Mok Emil J. Straube Editors Birkhäuser Editors: Peter Ebenfelt Department of Mathematics University of California, San Diego (UCSD) 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112, USA e-mail: pebenfel@ucsd.edu Ngaiming Mok Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong SAR, China e-mail: nmok@hku.hk Norbert Hungerbühler Department of Mathematics University of Fribourg Chemin du musée 23 1700 Fribourg, Switzerland e-mail: norbert.hungerbuehler@unifr.ch Emil J. Straube Department of Mathematics Texas A&M University College Station, TX 77843, USA e-mail: straube@math.tamu.edu Joseph J. Kohn Department of Mathematics Fine Hall, Washington Road Princeton, NJ 08544-1000, USA kohn@math.princeton.edu 2000 Mathematics Subject Classification 32-06 Library of Congress Control Number: 2010926412 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-0346-0008-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover Design: Alexander Faust, Basel, Switzerland Printed in Germany ISBN 978-3-0346-0008-8 e-ISBN 978-3-0346-0009-5 987654321 www.birkhauser.ch Contents N. Hungerbühler Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Extended Curriculum Vitae of Linda Preiss Rothschild . . . . . . . . . . . . . . . . . . . . xi Publication List of Linda Preiss Rothschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv D. Barlet and H.-M. Maire  Oblique Polar Lines of X |f |2λ |g|2µ  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 S. Berhanu On Involutive Systems of First-order Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A. Bove, M. Mughetti and D.S. Tartakoff Gevrey Hypoellipticity for an Interesting Variant of Kohn’s Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 D.W. Catlin and J.P. D’Angelo Subelliptic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 J.P. D’Angelo Invariant CR Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 M. Derridj and B. Helffer On the Subellipticity of Some Hypoelliptic Quasihomogeneous Systems of Complex Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 F. Forstnerič Invariance of the Parametric Oka Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 S. Fu ¯ Positivity of the ∂-Neumann Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 K. Gansberger and F. Haslinger Compactness Estimates for the ∂-Neumann Problem in Weighted L2 -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 vi Contents P. Guan Remarks on the Homogeneous Complex Monge-Ampère Equation . . . . 175 J. Hounie A Radó Theorem for Locally Solvable Structures of Co-rank One . . . . 187 F. Lárusson Applications of a Parametric Oka Principle for Liftings . . . . . . . . . . . . . . 205 Ch. Laurent-Thiébaut Stability of the Vanishing of the ∂ b -cohomology Under Small Horizontal Perturbations of the CR Structure in Compact Abstract q-concave CR Manifolds . . . . . . . . . . . . . . . . . . . . . . . 213 L. Lempert Coherent Sheaves and Cohesive Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 G.A. Mendoza Characteristic Classes of the Boundary of a Complex b-manifold . . . . . 245 A. Meziani Solvability of Planar Complex Vector Fields with Applications to Deformation of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 263 N. Mok On the Zariski Closure of a Germ of Totally Geodesic Complex Submanifold on a Subvariety of a Complex Hyperbolic Space Form of Finite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 L. Ni The Large Time Asymptotics of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 301 M.-C. Shaw The Closed Range Property for ∂ on Domains with Pseudoconcave Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 D. Zaitsev New Normal Forms for Levi-nondegenerate Hypersurfaces . . . . . . . . . . . . 321 Complex Analysis Trends in Mathematics, vii–x c 2010 Springer Basel AG  Preface Norbert Hungerbühler The idea to organize a conference in honour of Linda Rothschild emerged in 2006. This idea began to substantiate in 2007 when the Swiss Mathematical Society assigned the traditional Spring Meeting to the University of Fribourg. An organizing committee was quickly formed: Organizing committee Norbert Hungerbühler Frank Kutzschebauch Bernhard Lamel Francine Meylan Nordine Mir University of Fribourg, Switzerland University of Berne, Switzerland University of Vienna, Austria University of Fribourg, Switzerland Université de Rouen, France In order to ensure a high-quality conference program, the search for a scientific committee began. Soon after, a distinguished group was found who started working right away: Scientific committee Peter Ebenfelt Franc Forstnerič Joseph J. Kohn Emil J. Straube University of California, San Diego, USA University of Ljubljana, Slovenia Princeton University, USA Texas A&M University, USA Spring Meeting of the Swiss Mathematical Society Conference on Complex Analysis 2008 Several Complex Variables and Connections with PDEs and Geometry In honour of Linda Rothschild, Fribourg, July 7–11 viii N. Hungerbühler Only a little while later it became clear that the subject and the top-class speakers who agreed to participate in the conference called for a proceedings volume to make the presented results available shortly after the conference. This project was carried out under the direction of the editorial board: Editorial board Peter Ebenfelt Norbert Hungerbühler Joseph J. Kohn Ngaiming Mok Emil J. Straube University of California, San Diego, USA University of Fribourg, Switzerland Princeton University, USA The University of Hong Kong Texas A&M University, USA Focus on youth The aim of the conference was to gather worldwide leading scientists, and to offer the occasion to PhD students and postdocs to come into contact with them. The committees explicitly encouraged young scientists, doctoral students and postdocs to initiate scientific contact and to aim at an academic career. The topic of the conference was apparently very attractive for young scientists, and the event an ideal platform to promote national and international doctoral students and postdocs. This aspect became manifest in a poster session where junior researchers presented their results. The conference was intended to have a strong component in instruction of PhD students: Three mini courses with introductory character were held by Pengfei Guan, Mei-Chi Shaw and Ngaiming Mok. These three mini courses have been very well received by a large audience and were framed by the series of plenary lectures presenting newest results and techniques. The participation of junior female researchers, PhD students and mathematicians from developing countries has been encouraged in addition by offering grants for traveling and accommodation. The subject The conference Complex Analysis 2008 has been devoted to the subject of Several Complex Variables and Connections with PDEs and Geometry. These three main subject areas of the conference have shown their deep relations, and how techniques from each of these fields can influence the others. The conference has stimulated further interaction between these areas. The conference was held in honor of Prof. Linda Rothschild who is one of the most influential contributors of the subject during the last decades. A particular aim was to encourage female students to pursue an academic career. In fact, female mathematicians have been well represented among the speakers, in the organizing committee and in the poster sessions. Preface ix Several Complex Variables is a beautiful example of a field requiring a wide range of techniques coming from diverse areas in Mathematics. In the last decades, many major breakthroughs depended in particular on methods coming from Partial Differential Equations and Differential and Algebraic Geometry. In turn, Several Complex Variables provided results and insights which have been of fundamental importance to these fields. This is in particular exemplified by the subject of Cauchy-Riemann geometry, which concerns itself both with the tangential CauchyRiemann equations and the unique mixture of real and complex geometry that real objects in a complex space enjoy. CR geometry blends techniques from algebraic geometry, contact geometry, complex analysis and PDEs; as a unique meeting point for some of these subjects, it shows evidence of the possible synergies of a fusion of the techniques from these fields. The interplay between PDE and Complex Analysis has its roots in Hans Lewy’s famous example of a locally non solvable PDE. More recent work on PDE has been similarly inspired by examples from CR geometry. The application of analytic techniques in algebraic geometry has a long history; especially in recent ¯ years, the analysis of the ∂-operator has been a crucial tool in this field. The ¯ ∂-operator remains one of the most important examples of a partial differential operator for which regularity of solutions under boundary constraints have been extensively studied. In that respect, CR geometry as well as algebraic geometry have helped to understand the subtle aspects of the problem, which is still at the heart of current research. Summarizing, our conference has brought together leading researchers at the intersection of these fields, and offered a platform to discuss the most recent developments and to encourage further interactions between these mathematicians. It was also a unique opportunity for younger people to get acquainted with the current research problems of these areas. Organization The conference was at the same time the 2008 Spring Meeting of the Swiss Mathematical Society. The event has profited from the organizational structures of the SMS and the embedding in the mathematical community of Switzerland. The University of Fribourg has proven to be the appropriate place for this international event because of its tradition in Complex Analysis, the central geographic location, and its adequate infrastructure. In turn, its reputation and that of the region has benefited from this conference. The conference has been announced internationally in the most important conference calendars and in several journals. Moreover, the event has been advertised by posters in numerous mathematics institutes worldwide, by e-mails and in the regular announcements of the Swiss Mathematical Society. x N. Hungerbühler Acknowledgment It becomes increasingly difficult to find sponsors for conferences of the given size, in particular in mathematics. We are all the more grateful to our sponsors who have generously supported the conference, and the proceedings volume in hand: List of Sponsors • • • • • • • • • • • • • • • • Ciba Roche Merck Serono Novartis Syngenta Swiss Mathematical Society Troisième cycle romand de mathématiques Swiss Academy of Sciences Stiftung zur Förderung der mathematischen Wissenschaften in der Schweiz Centre Interfacultaire Bernoulli CIB, EPFL Department of Mathematics, University of Fribourg Faculty of Sciences, University of Fribourg Rectorate, University of Fribourg Swiss National Science Foundation Walter Haefner Stiftung Swiss Doctoral Program in Mathematics In the name of the conference committees and of all participants, we would like to thank all sponsors – foundations, institutions and companies – very cordially for their contributions and the shown appreciation for our work as mathematicians: Thank you! We also thank the team of Dr. Thomas Hempfling of the Birkhäuser publishing company for their help and professional expertise during the production process of these proceedings. Finally, we would like to thank Elisabeth François and Claudia Kolly who assumed the secretariat of the conference. Fribourg, August 2009 Norbert Hungerbühler Complex Analysis Trends in Mathematics, xi–xx c 2010 Springer Basel AG  Extended Curriculum Vitae of Linda Preiss Rothschild Linda Rothschild was born February 28, 1945, in Philadelphia, PA. She received her undergraduate degree, magna cum laude, from the University of Pennsylvania in 1966 and her PhD in mathematics from MIT in 1970. Her PhD thesis was “On the Adjoint Action of a Real Semisimple Lie Group”. She held positions at Tufts University, Columbia University, the Institute for Advanced Study, and Princeton University before being appointed an associate professor of mathematics at the University of Wisconsin-Madison in 1976. She was promoted to full professor in 1979. Since 1983 she has been professor of mathematics at the University of California at San Diego, where she is now a Distinguished Professor. Rothschild has worked in the areas of Lie groups, partial differential equations and harmonic analysis, and the analytic and geometric aspects of several complex variables. She has published over 80 papers in these areas. Rothschild was awarded an Alfred P. Sloan Fellowship in 1976. In 2003 she won the Stefan Bergman Prize from the American Mathematical Society (jointly with Salah Baouendi). The citation read in part: “The Bergman Prize was awarded to Professors Salah Baouendi and Linda Rothschild for their joint and individual work in complex analysis. In addition to many important contributions to complex analysis they have also done first rate work in the theory of partial differential equations. Their recent work is centered on the study of CR manifolds to which they and their collaborators have made fundamental contributions. Rothschild, in a joint paper with E. Stein, introduced Lie group methods to prove Lp and Hölder estimates for the sum of squares operators as well as the boundary Kohn Laplacian for real hypersurfaces. In later joint work with L. Corwin and B. Helfer, she proved analytic hypoellipticity for a class of first-order systems. She also proved the existence of a family of weakly pseudoconvex hypersurfaces for which the boundary Kohn Laplacian is hypoelliptic but does not satisfy maximal L2 estimates.” xii Extended Curriculum Vitae of Linda Preiss Rothschild In 2005, Rothschild was elected a Fellow of the American Academy of Arts and Sciences, and in 2006 she was an invited speaker at the International Congress of Mathematics in Madrid. Rothschild served as President of the Association for Women in Mathematics from 1983 to 1985 and as Vice-President of the American Mathematical Society from 1985 to 1987. She served on the editorial committees of the Transactions of the AMS and Contemporary Mathematics. She is also an editorial board member of Communications in Partial Differential Equations and co-founder and co-editorin-chief of Mathematical Research Letters. She has served on many professional committees, including several AMS committees, NSF panels, and an organization committee for the Special Year in Several Complex Variables at the Mathematical Sciences Research Institute. She presented the 1997 Emmy Noether Lecture for the AWM. Rothschild has a keen interest in encouraging young women who want to study mathematics. A few years ago she helped establish a scholarship for unusually talented junior high school girls to accelerate their mathematical training by participating in a summer program. Educational Background B.A. Ph.D. University of Pennsylvania, 1966 in mathematics, Massachusetts Institute of Technology, 1970 Dissertation: On the Adjoint Action of a Real Semisimple Lie Group Advisor: Isadore Manual Singer Professional Employment 1982– 2001–05 1979–82 1981–82 1978 1976–77 1975–76 1974–75 1972–74 1970–72 1970–72 Professor, University of California, San Diego Vice Chair for Graduate Affairs, Mathematics Dept., UCSD Professor, University of Wisconsin Member, Institute for Advanced Study Member, Institute for Advanced Study Associate Professor, University of Wisconsin Visiting Assistant Professor, Princeton University Member, Institute for Advanced Study Ritt Assistant Professor, Columbia University Assistant Professor, Tufts University Research Staff, Artificial Intelligence Laboratory, M.I.T. Honors and Fellowships 2005 2003 1976–80 1966–70 Fellow, American Academy of Arts and Sciences Stefan Bergman Prize Alfred P. Sloan Foundation Fellow National Science Foundation Graduate Fellow Extended Curriculum Vitae of Linda Preiss Rothschild xiii Selected Invited Lectures • Invited address, International Congress of Mathematicians, Madrid, August 2006 • “Frontiers in Mathematics” Lecturer, Texas A&M University, September 1999 • Invited hour speaker, Sectional joint meeting of American Mathematical Society and Mathematical Association of America, Claremont, October 1997 • Emmy Noether Lecturer (Association for Women in Mathematics), Annual Joint Mathematics Meetings, San Diego January 1997 • Invited hour lecturer, Annual Joint Mathematics Meetings, Orlando, January 1996 • Invited hour speaker, Annual Summer meeting of American Mathematics Society, Pittsburgh, August1981 Students Mark Marson Joseph Nowak John Eggers Bernhard Lamel Slobodan Kojcinovic Robert Kowalski University University University University University University of of of of of of California, California, California, California, California, California, San San San San San San Diego, Diego, Diego, Diego, Diego, Diego, Selected National Committees and Offices National Science Foundation, Mathematics Division • Advisory Panel, 1984–87 and other panels 1997–99, 2004 American Mathematical Society (AMS) • Bocher Prize Committee 2001–04 • National Program Committee 1997–2000 Chair 1998–1999 • Nominating Committee, 1982–84, 1994–96 • Committee on Science Policy, 1979–82, 92–9 • AMS Vice President, 1985–87 • Committee on Committees, 1977–79, 1979–81 • Executive Committee, 1978–80 • Council of the AMS, 1977–80 1990 1994 1995 2000 2001 2002 xiv Extended Curriculum Vitae of Linda Preiss Rothschild Association for Women in Mathematics (AWM) • Noether Lecture Committee 1988–90, 1994–1997 Chair 1989–90 • Schafer Prize Committee 1993–94 • AWM President, 1983–85. Mathematical Association of America • Chauvenet Prize Committee, 1998–2000 Mathematical Sciences Research Institute • Board of Trustees, 1996–1999 • Budget Committee 1996–1998 California Science Museum • Jury to select California Scientist of the Year Award, 1995–1999 Institute for Pure and Applied Mathematics (IPAM) • Board of Trustees, 2002–2005 Editorial Positions • • • • • Co-Editor-in-Chief, Mathematical Research Letters, 1994– Editorial Board, Journal of Mathematical Analysis and Applications, 2001– Editorial Board, Communications in Partial Differential Equations, 1984– Editorial Board, Contemporary Mathematics, 1990–1994 Editor for complex and harmonic analysis, Transactions of the American Mathematical Society, 1983–1986 Publication List of Linda Preiss Rothschild [1] Peter Ebenfelt and Linda P. Rothschild. New invariants of CR manifolds and a criterion for finite mappings to be diffeomorphic. Complex Var. Elliptic Equ., 54(34):409–423, 2009. ISSN 1747-6933. [2] M.S. Baouendi, Peter Ebenfelt, and Linda P. Rothschild. Transversality of holomorphic mappings between real hypersurfaces in different dimensions. Comm. Anal. Geom., 15(3):589–611, 2007. ISSN 1019-8385. [3] Peter Ebenfelt and Linda P. Rothschild. Analyticity of smooth CR mappings of generic submanifolds. Asian J. Math., 11(2):305–318, 2007. ISSN 1093-6106. [4] Peter Ebenfelt and Linda P. Rothschild. Images of real submanifolds under finite holomorphic mappings. Comm. Anal. Geom., 15(3):491–507, 2007. ISSN 1019-8385. [5] M.S. Baouendi, Peter Ebenfelt, and Linda P. Rothschild. Projection on Segre varieties and determination of holomorphic mappings between real submanifolds. Sci. China Ser. A, 49(11):1611–1624, 2006. ISSN 1006-9283. [6] Peter Ebenfelt and Linda P. Rothschild. Transversality of CR mappings. Amer. J. Math., 128(5):1313–1343, 2006. ISSN 0002-9327. [7] Linda Preiss Rothschild. Iterated Segre mappings of real submanifolds in complex space and applications. In International Congress of Mathematicians. Vol. II, pages 1405–1419. Eur. Math. Soc., Zürich, 2006. [8] M. Salah Baouendi, Linda Preiss Rothschild, Jörg Winkelmann, and Dmitri Zaitsev. Lie group structures on groups of diffeomorphisms and applications to CR manifolds. Ann. Inst. Fourier (Grenoble), 54(5):1279–1303, xiv, xx, 2004. ISSN 0373-0956. [9] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. Dynamics of the Segre varieties of a real submanifold in complex space. J. Algebraic Geom., 12(1):81–106, 2003. ISSN 1056-3911. [10] Linda Preiss Rothschild. Mappings between real submanifolds in complex space. In Explorations in complex and Riemannian geometry, volume 332 of Contemp. Math., pages 253–266. Amer. Math. Soc., Providence, RI, 2003. [11] M.S. Baouendi, Nordine Mir, and Linda Preiss Rothschild. Reflection ideals and mappings between generic submanifolds in complex space. J. Geom. Anal., 12(4):543– 580, 2002. ISSN 1050-6926. [12] M.S. Baouendi, Linda Preiss Rothschild, and Dmitri Zaitsev. Equivalences of real submanifolds in complex space. J. Differential Geom., 59(2):301–351, 2001. ISSN 0022-040X. [13] M.S. Baouendi, Linda Preiss Rothschild, and Dmitri Zaitsev. Points in general position in real-analytic submanifolds in CN and applications. In Complex analysis and geometry (Columbus, OH, 1999), volume 9 of Ohio State Univ. Math. Res. Inst. Publ., pages 1–20. de Gruyter, Berlin, 2001. [14] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. Local geometric properties of real submanifolds in complex space. Bull. Amer. Math. Soc. (N.S.), 37(3):309–336 (electronic), 2000. ISSN 0273-0979. [15] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. Convergence and finite determination of formal CR mappings. J. Amer. Math. Soc., 13(4):697–723 (electronic), 2000. ISSN 0894-0347. xvi Extended Curriculum Vitae of Linda Preiss Rothschild [16] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. Rational dependence of smooth and analytic CR mappings on their jets. Math. Ann., 315(2):205–249, 1999. ISSN 0025-5831. [17] M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild. Real submanifolds in complex space and their mappings, volume 47 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1999. ISBN 0-691-00498-6. [18] M. Salah Baouendi and Linda Preiss Rothschild. Local holomorphic equivalence of real analytic submanifolds in CN . In Several complex variables (Berkeley, CA, 1995– 1996), volume 37 of Math. Sci. Res. Inst. Publ., pages 1–24. Cambridge Univ. Press, Cambridge, 1999. [19] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. CR automorphisms of real analytic manifolds in complex space. Comm. Anal. Geom., 6(2):291–315, 1998. ISSN 1019-8385. [20] M.S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild. Parametrization of local biholomorphisms of real analytic hypersurfaces. Asian J. Math., 1(1):1–16, 1997. ISSN 1093-6106. [21] Linda Preiss Rothschild. Holomorphically nondegenerate algebraic hypersurfaces and their mappings. In Multidimensional complex analysis and partial differential equations (São Carlos, 1995), volume 205 of Contemp. Math., pages 247–252. Amer. Math. Soc., Providence, RI, 1997. [22] M.S. Baouendi, Xiaojun Huang, and Linda Preiss Rothschild. Regularity of CR mappings between algebraic hypersurfaces. Invent. Math., 125(1):13–36, 1996. ISSN 0020-9910. [23] M.S. Baouendi and Linda Preiss Rothschild. Unique continuation of harmonic functions at boundary points and applications to problems in complex analysis. In Partial differential equations and functional analysis, volume 22 of Progr. Nonlinear Differential Equations Appl., pages 35–40. Birkhäuser Boston, Boston, MA, 1996. [24] M.S. Baouendi, Xiao Jun Huang, and Linda Preiss Rothschild. Nonvanishing of the differential of holomorphic mappings at boundary points. Math. Res. Lett., 2(6):737– 750, 1995. ISSN 1073-2780. [25] M.S. Baouendi and Linda Preiss Rothschild. Mappings of real algebraic hypersurfaces. J. Amer. Math. Soc., 8(4):997–1015, 1995. ISSN 0894-0347. [26] Serge Alinhac, M.S. Baouendi, and Linda Preiss Rothschild. Flat analytic discs attached to real hypersurfaces of finite type. Math. Res. Lett., 1(3):359–367, 1994. ISSN 1073-2780. [27] M.S. Baouendi and Linda Preiss Rothschild. Directions of analytic discs attached to generic manifolds of finite type. J. Funct. Anal., 125(1):149–171, 1994. ISSN 00221236. [28] M.S. Baouendi and Linda Preiss Rothschild. A generalized complex Hopf lemma and its applications to CR mappings. Invent. Math., 111(2):331–348, 1993. ISSN 0020-9910. [29] M.S. Baouendi and Linda Preiss Rothschild. A local Hopf lemma and unique continuation for harmonic functions. Internat. Math. Res. Notices, 8:245–251, 1993. ISSN 1073-7928. Publication List of Linda Preiss Rothschild xvii [30] M.S. Baouendi and Linda Preiss Rothschild. Unique continuation and a Schwarz reflection principle for analytic sets. Comm. Partial Differential Equations, 18(11): 1961–1970, 1993. ISSN 0360-5302. [31] M.S. Baouendi and Linda Preiss Rothschild. Images of real hypersurfaces under holomorphic mappings. J. Differential Geom., 36(1):75–88, 1992. ISSN 0022-040X. [32] M.S. Baouendi and Linda Preiss Rothschild. Remarks on the generic rank of a CR mapping. J. Geom. Anal., 2(1):1–9, 1992. ISSN 1050-6926. [33] M.S. Baouendi and Linda Preiss Rothschild. A general reflection principle in C2 . J. Funct. Anal., 99(2):409–442, 1991. ISSN 0022-1236. [34] M.S. Baouendi and Linda Preiss Rothschild. Holomorphic mappings of real analytic hypersurfaces. In Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), volume 52 of Proc. Sympos. Pure Math., pages 15–26. Amer. Math. Soc., Providence, RI, 1991. [35] M.S. Baouendi and Linda Preiss Rothschild. Minimality and the extension of functions from generic manifolds. In Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), volume 52 of Proc. Sympos. Pure Math., pages 1–13. Amer. Math. Soc., Providence, RI, 1991. [36] Serge Alinhac, M.S. Baouendi, and Linda Preiss Rothschild. Unique continuation and regularity at the boundary for holomorphic functions. Duke Math. J., 61(2):635–653, 1990. ISSN 0012-7094. [37] M.S. Baouendi and Linda Preiss Rothschild. Cauchy-Riemann functions on manifolds of higher codimension in complex space. Invent. Math., 101(1):45–56, 1990. ISSN 0020-9910. [38] M.S. Baouendi and Linda Preiss Rothschild. Geometric properties of mappings between hypersurfaces in complex space. J. Differential Geom., 31(2):473–499, 1990. ISSN 0022-040X. [39] M.S. Baouendi and Linda Preiss Rothschild. Transversal Lie group actions on abstract CR manifolds. Math. Ann., 287(1):19–33, 1990. ISSN 0025-5831. [40] M.S. Baouendi, S.R. Bell, and Linda Preiss Rothschild. Mappings of three-dimensional CR manifolds and their holomorphic extension. Duke Math. J., 56(3):503–530, 1988. ISSN 0012-7094. [41] M.S. Baouendi and Linda Preiss Rothschild. Extension of holomorphic functions in generic wedges and their wave front sets. Comm. Partial Differential Equations, 13(11):1441–1466, 1988. ISSN 0360-5302. [42] M.S. Baouendi and Linda Preiss Rothschild. Germs of CR maps between real analytic hypersurfaces. Invent. Math., 93(3):481–500, 1988. ISSN 0020-9910. [43] Alain Grigis and Linda Preiss Rothschild. L2 estimates for the boundary Laplacian operator on hypersurfaces. Amer. J. Math., 110(4):577–593, 1988. ISSN 0002-9327. [44] Gerardo A. Mendoza and Linda Preiss Rothschild. Analytic approximability of solutions of partial differential equations. Ark. Mat., 26(2):289–303, 1988. ISSN 00042080. [45] M.S. Baouendi, S.R. Bell, and Linda Preiss Rothschild. CR mappings of finite multiplicity and extension of proper holomorphic mappings. Bull. Amer. Math. Soc. (N.S.), 16(2):265–270, 1987. ISSN 0273-0979. xviii Extended Curriculum Vitae of Linda Preiss Rothschild [46] M.S. Baouendi and Linda Preiss Rothschild. CR mappings and their holomorphic extension. In Journées “Équations aux derivées partielles” (Saint Jean de Monts, 1987), pages Exp. No. XXIII, 6. École Polytech., Palaiseau, 1987. [47] M.S. Baouendi and Linda Preiss Rothschild. Normal forms for generic manifolds and holomorphic extension of CR functions. J. Differential Geom., 25(3):431–467, 1987. ISSN 0022-040X. [48] Lawrence Corwin and Linda Preiss Rothschild. Solvability of transversally elliptic differential operators on nilpotent Lie groups. Amer. J. Math., 108(3):589–613, 1986. ISSN 0002-9327. [49] M.S. Baouendi and Linda Preiss Rothschild. Analytic approximation for homogeneous solutions of invariant differential operators on Lie groups. Astérisque, 131:189– 199, 1985. ISSN 0303-1179. Colloquium in honor of Laurent Schwartz, Vol. 1 (Palaiseau, 1983). [50] M.S. Baouendi and Linda Preiss Rothschild. Semirigid CR structures and holomorphic extendability. In Proceedings of the conference on partial differential equations, Vol. 1, 2 (Saint Jean de Monts, 1985), pages Exp. No. 1, 4. Soc. Math. France, Paris, 1985. [51] M.S. Baouendi, Linda Preiss Rothschild, and F. Trèves. CR structures with group action and extendability of CR functions. Invent. Math., 82(2):359–396, 1985. ISSN 0020-9910. [52] Lawrence Corwin, Bernard Helffer, and Linda Preiss Rothschild. Smoothness and analyticity for solutions of first-order systems of partial differential equations on nilpotent Lie groups. Invent. Math., 81(2):205–216, 1985. ISSN 0020-9910. [53] Linda Preiss Rothschild. Integrability and holomorphic extendibility for rigid CR structures. In Pseudodifferential operators and applications (Notre Dame, Ind., 1984), volume 43 of Proc. Sympos. Pure Math., pages 237–240. Amer. Math. Soc., Providence, RI, 1985. [54] Linda Preiss Rothschild. Analyticity of solutions of partial differential equations on nilpotent Lie groups. In Lie group representations, III (College Park, Md., 1982/1983), volume 1077 of Lecture Notes in Math., pages 389–395. Springer, Berlin, 1984. [55] Alain Grigis and Linda Preiss Rothschild. A criterion for analytic hypoellipticity of a class of differential operators with polynomial coefficients. Ann. of Math. (2), 118(3):443–460, 1983. ISSN 0003-486X. [56] Linda Preiss Rothschild. A remark on hypoellipticity of homogeneous invariant differential operators on nilpotent Lie groups. Comm. Partial Differential Equations, 8(15):1679–1682, 1983. ISSN 0360-5302. [57] Linda Preiss Rothschild and David S. Tartakoff. Analyticity of relative fundamental solutions and projections for left invariant operators on the Heisenberg group. Ann. Sci. École Norm. Sup. (4), 15(3):419–440, 1982. ISSN 0012-9593. [58] Linda Preiss Rothschild. Local solvability of second-order differential operators on nilpotent Lie groups. Ark. Mat., 19(2):145–175, 1981. ISSN 0004-2080. [59] Linda Preiss Rothschild and David S. Tartakoff. Inversion of analytic matrices and local solvability of some invariant differential operators on nilpotent Lie groups. Comm. Partial Differential Equations, 6(6):625–650, 1981. ISSN 0360-5302. Publication List of Linda Preiss Rothschild xix [60] Linda Preiss Rothschild. Nonexistence of optimal L2 estimates for the boundary Laplacian operator on certain weakly pseudoconvex domains. Comm. Partial Differential Equations, 5(8):897–912, 1980. ISSN 0360-5302. [61] Linda Preiss Rothschild. A criterion for hypoellipticity of operators constructed from vector fields. Comm. Partial Differential Equations, 4(6):645–699, 1979. ISSN 03605302. [62] Linda Preiss Rothschild. Local solvability of left invariant differential operators on the Heisenberg group. Proc. Amer. Math. Soc., 74(2):383–388, 1979. ISSN 0002-9939. [63] Linda Preiss Rothschild. Smoothness of solutions of certain partial differential equations constructed from vector fields. In Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part, pages 227–230. Amer. Math. Soc., Providence, R.I., 1979. [64] Linda Preiss Rothschild and David Tartakoff. Parametrices with C ∞ error for cmb and operators of Hörmander type. In Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), volume 48 of Lecture Notes in Pure and Appl. Math., pages 255–271. Dekker, New York, 1979. [65] Linda Preiss Rothschild. Book Review: Estimates for the ∂-Neumann problem. Bull. Amer. Math. Soc., 84(2):266–270, 1978. ISSN 0002-9904. [66] Linda Preiss Rothschild. Parametrices for the boundary Laplacian and related hypoelliptic differential operators. In Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), pages 195– 203. Amer. Math. Soc., Providence, R. I., 1977. [67] Linda Preiss Rothschild and E.M. Stein. Hypoelliptic differential operators and nilpotent groups. Acta Math., 137(3-4):247–320, 1976. ISSN 0001-5962. [68] Linda Preiss Rothschild and Joseph A. Wolf. Eigendistribution expansions on Heisenberg groups. Indiana Univ. Math. J., 25(8):753–762, 1976. ISSN 0022-2518. [69] Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild. Compactness of certain homogeneous spaces of finite volume. Amer. J. Math, 97:248–259, 1975. ISSN 0002-9327. [70] Paul S. Wang and Linda Preiss Rothschild. Factoring multivariate polynomials over the integers. Math. Comput., 29:935–950, 1975. ISSN 0378-4754. [71] Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss-Rothschild. Automorphisms, orbits, and homogeneous spaces of non-connected Lie groups. Math. Ann., 212:145–155, 1974/75. ISSN 0025-5831. [72] Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild. Central idempotent measures on connected locally compact groups. J. Functional Analysis, 15:22–32, 1974. [73] Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild. Unbounded conjugacy classes in Lie groups and location of central measures. Acta Math., 132:225–243, 1974. ISSN 0001-5962. [74] Linda Preiss Rothschild. A distribution theoretic proof of Kirillov’s character formula for nilpotent Lie groups. Math. Z., 140:63–65, 1974. ISSN 0025-5874. xx Extended Curriculum Vitae of Linda Preiss Rothschild [75] Linda Preiss Rothschild and Joseph A. Wolf. Representations of semisimple groups associated to nilpotent orbits. Ann. Sci. École Norm. Sup. (4), 7:155–173 (1975), 1974. ISSN 0012-9593. [76] David L. Ragozin and Linda Preiss Rothschild. Central measures on semisimple Lie groups have essentially compact support. Proc. Amer. Math. Soc., 32:585–589, 1972. ISSN 0002-9939. [77] Linda Preiss Rothschild. On uniqueness of quasi-split real semisimple Lie algebras. Proc. Amer. Math. Soc., 24:6–8, 1970. ISSN 0002-9939.