ISSN 1344-7777

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Differential

Geometry

Proceedings of the First

I nternational Symposium

on Differential Geometry

February 22-24, 2001

JOSAI UNIVERSITY

Edited by

Qing-Ming CHENO

JOSAI UNIVERSITY

JOSAI UNIVERSITY

ORADUATE SCHOOL OF SCIENCE

lList　of　Specia1Issues

SCIENCE　BULLETIN　OF　JOSAI　UNIVERSITY

No．1

No．2

No．3

No．4

No．5

Convexity，Chaos　an（1趾acta1s

Proceedings　of　the　Symposium　on　Non1inear　Ana1ysis（NLA96）ヨSpr三ng1996

Kiyoko　NISHIZAWA，editor

Surgery　and　Geometric　Topo1ogy

Proceedings　of　the　Conference，September1996

Andrew　RANICKI，Masayuki　YAMASAKIヨed－itors

Mathematics　App1ied　to　Ec㎝omics　and　Finance

Proceed－ings　of　the　Interna．tiona1Conference，February1997

Shigeo　KOHNO，Kiyoko　NISHIZAWA，Masako　HAYASHI，editors

Compex　DynamicaI　SystemsProceedings　of　the　Symposium　on　Non－Linear　Ana1ysis（NLA97），May1997

Kiyoko　NISHIZAWA，editor

Wave1et，Chaos　and　Non1inear　PDEs

Proceedings　of　the　Symposium　on　App1ie（i　Mathe肌atics，Septe㎜ber1997

Kiyoshi　MIZOHATA，ed－itor

』OSA1MATトlEMAT1CAL　MONOGRAPHS

』MM1．

」MM2．

」MM3．

CONV1三X　ANALYS1S　AND　CHAOS

Proceedings　of　the　Symposiu㎜on　Non1inear　Ana1ysis（NLA98），

J皿1y1998

Kiyoko　NISHIZAWA，editor

COMPUTER　ALGεBRAProceedings　of　the　Symposiu㎜on　Non1inear　Ana1ysis（NLA99），

September1999

Kiyoko　NISHIZAWA，Chair　and　Editor

Di肝甘ential　Geometry

Proceedings　of　the　First　Internationa．1Symposium

on　D舐erentia1Geometry，Febru趾y2001

Qing－Ming　CHENG，Editor

JOSAl MATHEMATICAL MONOGRAPhlS

2001

DifFerential Geometry

Proceedings of the First International Symposium on DifFerential Geometry February 22 - 24, 2001 JOSAI UNIVERSITY

Edited by

Qing-Ming CHENG Department of Mathematics Josai University

JOSAI UNIVERSITY GRADUATE SCHOOL OF SCIENCE keyakidai I - l, sakado

SAITAMA, JAPAN 350 - 0290

JOSAl MATHEMATICAL MONOGRAPHS

EDITOR - IN - CHIEF HIRo¥'L~1(1 ISHIBASHI

Dean of t<he Graduate school of Science

MANAGING EDITOR QIN'c'.-~,'11N*G CHENG

Editorial Oflice of Jh,'1~'1

Gra.duate School of Science

Josai Uni¥'ersit>y

Sakado, SAITA~'IA, JAPAN, 350-0295

e-ma.il: chen'*'@math.josa,i.ac.jp

URL: http://math.josai.ac.jp/~ chen'.~/

tel +81-492-71-7690(ofiice)

fax +81-492-71-7985, 7690

EDITORIAL BOARD QING-'¥.IING CHENG

H11~OYUI(1 ISHIBASHI (EDITOR n¥' CiiIEF)

K~vol(o NISHIZA¥~"A

JOSAI MATHEMATICAL MONOGRAPHS

JMM 3. DifFerential Geometry

Proceed?;ngs of the First International Symposium on D~ffer'ential Geometry

h,eld at Josai Univ(-'rsity, Fe.bruary 22 - 24, 2001

COntentS

F07"eword . . . . . . v P7-eface . . . . . . . . . . . . . . . . . . . . . . . vi

Programme . . . . . . . . . . . . . . . . . . . . vii

List of the Contributors . . . . . . . ix Co'r?,struction offlat tori in the 3-sphere and its applications

By Yoshihisa Kitabcrawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Some Lio~l,ville type inequalities a7?,d its applications to geometric problems

OTl, d~ffe_om,o'rphisms of a hyperbolic manifold

By Midori Goto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Schrddi77,ger flow and its applications in integrable systems

By Qingo' Dint; ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ 23

O'/~ stLrfa,ces with holomo'rphic Gauss map in a certain symrn,etT'?;c space

By lv'lasa,toshi Kokubu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . 37

()'n, the Goldberg conjectt/7'e

By Kouei Sekigrawa . . . . . . . . . . . 43 Ac?Lte tr'ia,n,gulo,tions of sph,ere arrd icosah,edron

B~.r Jinichi Itoh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

S0'r7~e geom,etric propertics o,f p-/?,ar~rn,onic maps

By Shigeo Kawai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A,fa,xirn,?1'yn, p7'i7?,ciplcs a,7?,d ?;ts applica,tioT?'s to subm,a,rLifold,b'

By Katsl_illiro Non'a,ka, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Compa7"Isort tJ7,eorems for Riema?1;77;ian man~folds with 'radial curva,t,u.re

bourrded below

B~., Ka.tsuhiro Shiohama

On AT7,a??,d conjecture

By YuXin Dong .

Notes on the relative )/amabe invariant

By Ka.zuo Akutagawa.

Com,plete stl,bm,amfolds in spheres

By Qing-N'Iingr Cheng

Sub'marz~J'olds with degenerate Gauss mapping

By Goo Ishikawa, Makoto Kimur'a and Reiko ~,Iiyaoka

Totai curvature for open submanufolds of Euclidean spaces

By Fra,nki Dillen and Wolfgang Ktihnel

Existence of a family of complete minimal surfaces of ger2,us one with one

end and finite total curvature

By Kat,suhiro Moriya

81

93

l 05

l 15

l 25

l 30

l 49

』OSAl　MATHEMATlCAL　MONOGRA1PHS」MM3．01FFENR圧三NT1ALGEOMETRY

Pub玉ished　by　Graduate　Schoo1of　Science，

Josai　Universityヨ

Sakado，Sa．itama，350＿0295JAPAN

te1＋81－492－71－7690（・舐ce）

fax＋81＿492＿7L7685．7690

EmT0RIAL　B0ARD

Q1NG－MlNG　cHENGHIR0YUK玉ISH工BASHI（EDIT0R　IN　CHIEF）

KIY0K0NISHIZAWA

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JOSAI MATHEMATICAL MONOGRAPHS JMM 3. Differential Geometry

proceedings ofthe First International symposium on Dlfferential Geometry hetd at Josai University, February 22 - 24, 2001

Contents Foreword

Preface

Programme List ofthe Contributors

Construction offlat tori in the 3-sphere and its applications

By Yoshihisa Kitagawa .

Some Liouville type inequalities and its applications to geometric problems

By Young Jin Suh

On dlffeomorphisms ofa hyperbolic mamfold By Midori Goto

Schrddingerflow and its applications in integrable systems

By Qing Ding

On surfaces with holomorphic Gauss map in a certain symmetric space

By Masatoshi Kokubu

On the Goldberg conjecture

By Kouei Sekigawa

Acute triangulations ofsphere and icosahedron

Some geometric properties ofp-harmonic maps

Maximum principles and its applications to submantfolds

By Katsuhiro Nonaka

Comparison theorems for Riemannian mamfolds with radial curvature bounded below By Katsuhiro Shiohama

On Anand conjecture By YuXin Dong

Notes on the relative Yamabe invariant

Complete submamfolds in spheres

Submamfolds with degenerate Gauss mapping By Goo Ishikawa, Makoto Kimura and Reiko Miyaoka ......................

Total curvature for open submamfolds ofEuclidean spaces By Franki Dillen and Wolfgang Ktihnel

Existence ofa family ofcomplete minimal surfaces ofgenus one with one end and finite total curvature

v vi

vi i

ix

3

11

21

23

37

43

53

63

71

81

93

105

115

125

139

149