Geometry and Nonlinear Partial Differentia l Equations

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AMS/IP Studies in Advanced Mathematics Volume 29

Geometry and Nonlinear Partial Differentia l Equations

Dedicated to Professor Buqing Su in Honor of His 100t h Birthday

Proceedings o f the Conferenc e held July 30-31 , 2001 , at Zhejiang University, China

Shuxing Chen and S.-T. Yau, Editors

American Mathematical Society • International Press

https://doi.org/10.1090/amsip/029

Shing-Tung Yau , Genera l Edito r

2000 Mathematics Subject Classification. Primar y 34Axx , 47Jxx , 76Bxx , 70Kxx .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Geometry an d nonlinea r partia l differentia l equation s : dedicate d t o Professo r Buqin g S u i n hono r of his 100t h birthda y : proceeding s o f the conferenc e hel d Jul y 30-31 , 2001 , at Zhejian g University , China / Shuxin g Che n an d S.-T . Yau , editors .

p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 29 ) Includes bibliographica l references . ISBN 0-8218-3294- 8 (alk . paper ) 1. Geometry , Differential—Congresses . 2 . Differentia l equations , Nonlinear—Congresses .

I. Su , Buqing , 1902 - II . Yau , Shing-Tung , 1949 - III . Chen , Shuxing . IV . Series .

QA641 .G4445 200 2 516.3'6—dc21 200202776 1

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Contents

Preface vii

Essay honorin g Professo r Buqin g S u vii i

Dedication fo r th e Conferenc e a t Zhejian g University , Chin a i x

The addres s on the celebratio n fo r th e 100t h birthday o f Professor S u Buqin g CHAOHAO G U 1

A property o f Kahler-Ricc i soliton s o n complet e comple x surface s BING-LONG CHE N AN D X I - P I N G ZH U 5

Propagation o f singularitie s i n compressibl e viscou s fluid s SHUXING CHE N AN D YA-GUAN G WAN G 1 3

The Landau-Lifshit z equatio n an d it s gaug e equivalen t structur e QING DIN G 2 7

General flattene d Jaff e model s fo r galaxie s ZHENGLU JIANG , DAOYUA N FANG , HONGXI A LIU , AN D DAVI D MOS S 3 1

Scattering matrice s an d geodesie s o f locall y symmetri c space s LIZHEN J I 3 9

A not e o n enumeratin g rationa l curve s i n a K 3 surfac e JUN L i 5 3

Multi-phase computation s o f th e semiclassica l limi t o f th e Schrodinge r equation SHI JI N AN D XIANTA O L I 6 3

Towards a mirro r principl e fo r highe r genu s BONG LIAN , KEFEN G LIU , AN D SHING-TUN G YA U 7 7

Vortices i n two-dimensiona l Bose-Einstei n condensate s FANGHUA LI N AN D TAI-CHI A LI N 8 7

Sample pat h propertie s o f Gaussia n processe s ZHENGYAN LI N 11 5

Wavelets o n th e Heisenber g grou p LIZHONG PEN G 12 3

vi C O N T E N T S

On complet e submanifold s wit h paralle l mea n curvatur e vecto r YI-BING SHE N 13 3

Triple cover s o n smoot h algebrai c varietie s SHENG-LI TA N 14 3

Osculating C R manifold s b y nilpoten t Li e group s i n th e theor y o f severa l complex variable s W E I WAN G 16 5

Dynamics o f commutin g holomorphi c map s YUEFEI WAN G 17 5

Different behaviou r fo r th e solution s o f 1-dimensiona l chemotaxi s mode l with exponentia l growt h YIN YANG , HU A CHEN , AN D WEIA N LI U 18 1

Some progress i n classica l genera l relativit y SHING TUN G YA U 19 1

Long shoc k fo r supersoni c flo w pas t a curve d con e HUICHENG YI N 20 7

A compactness theore m fo r Yang-Mill s connection s Xi ZHAN G 21 7

The positiv e mas s theore m i n genera l relativit y XIAO ZHAN G 22 7

Extension theorem s fo r specia l holomorphi c function s XIANG-YU ZHO U 23 5

Preface

On Jul y 30-31 , 2001 , a conferenc e wa s hel d dedicate d t o th e one-hundredt h birthday o f Professo r Buqin g Su . Professo r S u wa s on e o f th e founder s o f moder n differential geometr y i n China . H e was also one of the founder s o f the mathematic s department o f Zhejiang University . Thus , i t i s a good time to celebrate the openin g of a ne w mathematic s cente r a t Zhejian g University . Th e cente r wil l b e formall y inaugurated o n August 12 , 2002. Th e construction o f the building where the cente r is locate d an d th e center' s operatio n fe e wa s primaril y donate d b y Mr . Tang , t o whom w e are mos t grateful . Hi s contributio n t o th e cente r wil l b e full y describe d during th e forma l inauguratio n o f the cente r thi s summer .

On the following pages , I have included a poem fo r the birthday o f Professor S u and a n introductio n fo r thi s conference , bot h writte n i n Chinese . Th e conferenc e was largel y dependen t o n th e effort s o f J . C . Chen , S . L . Wang , an d S . X . Chen . We als o woul d lik e t o than k al l o f th e peopl e wh o contribute d t o th e organizatio n of the conferenc e a s wel l a s the editin g o f the proceedings .

Shing-Tung Ya u

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Titles i n Thi s Serie s

29 Shuxin g Che n an d S.-T . Yau , Editors , Geometr y an d Nonlinea r Partia l Differentia l Equations, 200 2

28 Valenti n Afraimovic h an d Sze-B i Hsu , Lecture s o n Chaoti c Dynamica l Systems , 200 2

27 M . R a m Murty , Introductio n t o p-adi c Analyti c Numbe r Theory , 200 2

26 Raymon d Chan , Yue-Kue n Kwok , Davi d Yao , an d Qian g Zhang , Editors ,

Applied Probability , 200 2

25 Dongga o Deng , Dare n Huang , Rong-Qin g Jia , We i Lin , an d Jia n Zhon g Wong ,

Editors, Wavele t Analysi s an d Applications , 200 2

24 Jan e Gilman , Wil l ia m W . Menasco , an d Xiao-Son g Lin , Editors , Knots , Braids ,

and Mappin g Clas s Groups—Paper s Dedicate d t o Joa n S . Birman , 200 1

23 Cumru n Vaf a an d S.-T . Yau , Editors , Winte r Schoo l o n Mirro r Symmetry , Vecto r

Bundles an d Lagrangia n Submanifolds , 200 1

22 Carlo s Berenste in , Der-Che n Chang , an d Jingzh i Tie , Laguerr e Calculu s an d It s

Applications o n th e Heisenber g Group , 200 1

21 Jiirge n Jost , Bosoni c Strings : A Mathematica l Treatment , 200 1

20 L o Yan g an d S.-T . Yau , Editors , Firs t Internationa l Congres s o f Chines e

Mathematicians, 200 1

19 So-Chi n Che n an d Mei-Ch i Shaw , Partia l Differentia l Equation s i n Severa l Comple x

Variables, 200 1

18 Fangyan g Zheng , Comple x Differentia l Geometry , 200 0

17 Le i Gu o an d Stephe n S.-T . Yau , Editors , Lecture s o n Systems , Control , an d

Information, 200 0

16 Rud i Weikar d an d Gilber t Weinstein , Editors , Differentia l Equation s an d

Mathematical Physics , 200 0

15 Lin g Hsia o an d Zhoupin g Xin , Editors , Som e Curren t Topic s o n Nonlinea r

Conservation Laws , 200 0

14 Jun-ich i Igusa , A n Introductio n t o th e Theor y o f Loca l Zet a Functions , 200 0

13 Vasilio s Alexiade s an d Georg e Siopsis , Editors , Trend s i n Mathematica l Physics ,

1999

12 Shen g Gong , Th e Bieberbac h Conjecture , 199 9

11 Shinich i Mochizuki , Foundation s o f p-adic Teichmulle r Theory , 199 9

10 Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , Editors , Mirro r Symmetr y III ,

1999

9 Shing-Tun g Yau , Editor , Mirro r Symmetr y I , 199 8

8 Jiirge n Jost , Wilfri d Kendall , U m b e r t o Mosco , Michae l Rockner ,

and Karl-Theodo r Sturm , Ne w Direction s i n Dirichle t Forms , 199 8

7 D . A . Buel l an d J . T . Teitelbaum , Editors , Computationa l Perspective s o n Numbe r

Theory, 199 8

6 Harol d Levine , Partia l Differentia l Equations , 199 7

5 Qi-ken g Lu , Stephe n S.-T . Yau , an d Anatol y Libgober , Editors , Singularitie s an d

Complex Geometry , 199 7

4 Vyjayanth i Char i an d Iva n B . Penkov , Editors , Modula r Interfaces : Modula r Li e

Algebras, Quantu m Groups , an d Li e Superalgebras , 199 7

3 Xia-X i Din g an d Tai-Pin g Liu , Editors , Nonlinea r Evolutionar y Partia l Differentia l

Equations, 199 7

2.2 Wil l ia m H . Kazez , Editor , Geometri c Topology , 199 7

2.1 Wil l ia m H . Kazez , Editor , Geometri c Topology , 199 7

1 B . Green e an d S.-T . Yau , Editors , Mirro r Symmetr y II , 199 7