Table Of ContentGeometry
and Topology
of Submanifolds X
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Geometry
and Topology
of Submanifolds X
Differential Geometry in honor of
Prof. S. S. Chem
Peking University, China 29 Aug - 3 Sept 1999
TU Berlin, Germany 26 - 28 Nov 1999
Editors
W. H. Chen and C. P. Wang
Peking University, China
A.-M. Li
Sichuan University, China
U. Simon and M. Wiehe
Technische Universitdt Berlin, Germany
L. Verstraelen
Katholieke Universiteit Leuven and
Katholieke Universiteit Brussel, Belgium
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GEOMETRY AND TOPOLOGY OF SUBMANIFOLDS X
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Professor Shiing-Shen Chern
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vii
DEDICATION
The authors and editors of this proceedings volume dedicate their contributions to
Professor Shiing-shen Chern
on the occasion of his birthday on October 26, 2000.
Shiing-shen Chern is one of the great mathematicians of the twentieth century. He has made
fundamental contributions to different fields:
- differential geometry
- integral geometry
- web geometry
- complex manifolds
- characteristic classes.
Additionally, he has stimulated research in various mathematical fields through his famous
conjectures and through the formulation of problems which challenged the mathematical
community. His teaching at various universities and schools and his lectures at conferences
influenced development as much as his individual encouragement, guidance and support
of young mathematicians. Of the enormous number of his Ph.D. students, many are now
famous mathematicians in their own right. S.S. Chern founded two world-famous centers
of excellence in mathematics: the Mathematical Research Institutes at Berkeley and at
Tianjin. Both institutions stimulate the international cooperation in mathematics and thus
the development of the field.
In particular, the editorial board of this volume would like to thank S.S. Chern for support
ing the cooperation between geometers in China and Germany, where he got his D.Sc. at
Hamburg University in 1936. This proceedings volume reports on two meetings arising from
a cooperative Chinese-German project, which originated in his proposals for cooperation and
research visits in the 80's. We provide a short description in the introduction.
CURRICULUM VITAE OF SHIING-SHEN CHERN
I. Born October 26, 1911 in Kashing, Chekiang Province, China.
B. Sc, Nankai University, Tientsin, China, 1930.
M. Sc, Tsinghua University, Peiping, China, 1934.
D. Sc, University of Hamburg, Germany, 1936.
China Foundation Postdoctoral Fellow at the Sorbonne, Paris, France, 1936-37.
Professor of Mathematics. Tsinghua University and Southwest Associated University,
Kunming, China 1937-43.
Member, Institute for Advanced Study, Princeton, USA, 1943-45.
Acting Director, Institute of Mathematics, Academia Sinica, Nanking, China, 1946-48.
Professor of Mathematics. University of Chicago, 1949-60.
Professor of Mathematics, University of California at Berkeley, 1960-79; Professor
VIM
Emeritus, 1979 to present.
Director, Mathematical Sciences Research Institute, Berkeley, 1981-84; Director Emer
itus, 1984 to present.
Became US citizen 1961.
Director, Nankai Institute of Mathematics, Tianjin, China, 1984 to present.
II. Visiting Professor or Member: Harvard University 1952, Eidgenossische Technische
Hochschule, Zurich 1953, Massachusetts Institute of Technology 1957, Institute for Ad
vanced Study 1964, University of California at Los Angeles 1966, Institut des Hautes
Etudes Scientifiques, Paris 1967, Instituto de Matematica Pura e Aplicada, Rio de
Janeiro, Brazil 1970, University of Warwick, Coventry, England 1972, Rockefeller Uni
versity, New York 1973, Eidgenossische Technische Hochschule 1981, Max Planck In
stitut fur Mathematik, Bonn, Germany 1982.
III. Member, Academia Sinica, 1948 to present.
Honorary Member, Indian Mathematical Society, 1950 to present.
Vice President, American Mathematical Society, 1962-64.
Fellow, American Academy of Arts and Sciences, 1963 to present.
Corresponding Member, Brazilian Academy of Sciences, 1971 to present.
Associate Founding Fellow, Third World Academy of Sciences, 1983 to present.
Foreign Member, Royal Society of London, 1985 to present.
Honorary Member, London Mathematical Society, 1986 to present.
Corresponding Member, Academia Peloritana, Messina, Sicily, 1986 to present.
Honorary Life Member, New York Academy of Sciences, 1987 to present.
Foreign Member, Academia dei Lincei, Rome, 1988 to present.
IV. LL. D. (hon), The Chinese University of Hongkong, 1969.
D. Sc. (hon), University of Chicago, 1969.
D. Sc. (hon), University of Hamburg, 1971.
Dr. Math. Eidgenossische Technische Hochschule Zurich, 1982.
D. Sc. (hon), SUNY at Stony Brook, 1985.
Hon Doctor, Nankai University, 1985.
V. Honorary Professor: Beijing University 1978, Nankai University 1978, Institute of Sys
tems Science, Academy of Sciences 1980, Chinan University, Canton 1980, Gradu
ate School, Academy of Sciences 1984, Nanjing University 1985, East China Normal
University 1985, Chinese University of Science and Technology 1985, Beijing Normal
University 1985, Chekiang University 1985, Hangchow University 1986, Fudan Univer
sity 1986, Shanghai University of Technology 1986, Tianjin University 1987, Tohoku
University, Japan 1987.
VI. Chauvenet Prize, Mathematical Association of America 1970.
National Medal of Science 1975.
Alexander von Humboldt Award, Germany 1982.
Steele Prize, American Mathematical Society 1983.
Wolf Prize, Israel 1983-84.
ix
INTRODUCTION
Since January 1999, the National Science Foundation of China (NSFC) and the Deutsche
Forschungs Gemeinschaft (DFG) have been supporting a joint cooperation project among
geometry research groups from Peking University, Sichuan University at Chengdu, and TU-
Berlin. Through this project, we jointly organized two international meetings in 1999:
- A Conference on Differential Geometry, with 50 participants from 9 countries, at Peking
University from August 29 - September 3, 1999;
- a Workshop on PDEs and Submanifolds, with 17 participants from 7 countries, at
TU-Berlin from November 26 - 28, 1999.
The conference at Beijing received significant support from Peking University, and the TU
workshop was partially supported by TU-Berlin. Moreover, NSFC and DFG supported
mutual research visits - and thus the conference participation - of Chinese and German
geometers from the cooperating institutions.
Our Chinese-German cooperation has its origins in three proposals of S.S. Chern:
- he proposed research visits of An-Min Li at the TU Berlin; since 1986, An-Min Li has
been to TU-Berlin many times, first as a Humboldt fellow, and later with support from
the DFG, the GADGET-program of the EU, the TU-Berlin;
- he proposed a lecture series by Udo Simon at the Nankai Institute of Mathematics,
which was realized in spring 1987;
- he proposed a period of study and research for Changping Wang at the TU-Berlin;
Wang got a special TU grant for Chinese graduates from 1989-1992, and from 1992-
1997 he took a TU position as lecturer.
These Chinese-German contacts were extended, and this led to mutual research visits of
several people, including that of Huili Liu, who visited TU-Berlin from Shenyan during
1993-1997. Li, Liu and Wang made many mathematical friends in Europe through their
visits to Berlin, and, in particular, this facilitated a close cooperation with geometers at KU-
Leuven. This fact finally led to results presented in Beijing and in Berlin that were published
together as volume X in the proceedings series "Geometry and Topology of Submanifolds",
of conferences held in Europe so far.
Two surveys in this volume concern conjectures of S.S. Chern:
- A.M. Li reports on higher order PDE's in affine differential geometry, in particular on
two recent solutions of Chern's version of the affine Bernstein problem - the proofs,
given independently by Trudinger-Wang and Li-Jia, are significantly different;
- L. Verstraelen's short note concerns Chern's conjecture on minimal hypersurfaces in
spheres.
The participants at the Berlin-Workshop formulated a series of problems in Affine Differential
Geometry (ADG); this problem list reflects the particular interest in geometric problems
related to challenging higher order PDE's. This list is combined with a continuation of the
reference lists on ADG from earlier proceedings volumes in the same series "Geometry and
X
Topology of Submanifolds". The editorial board would like to thank NSFC, DFG, Peking
University and TU-Berlin for the financial support mentioned.
All contributions of this volume have been refereed. Besides the few original surveys on
current research and open problems, all contributions are original research papers; no paper
will be published elsewhere.
Beijing, Chengdu,
Berlin, Leuven
July 2000 The editors
