JOSAI MATHEMATICAL MONOGRAPHS

JMM 3. Differential Geometry

Proceedings of the First International Symposium on Differential Geometry held at Josai University, February 22 – 24, 2001

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii List of the Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Construction of flat tori in the 3-sphere and its applications

By Yoshihisa Kitagawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Some Liouville type inequalities and its applications to geometric problems

By Young Jin Suh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 On diffeomorphisms of a hyperbolic manifold

By Midori Goto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 Schrödinger flow and its applications in integrable systems

By Qing Ding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 On surfaces with holomorphic Gauss map in a certain symmetric space

By Masatoshi Kokubu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 On the Goldberg conjecture

By Kouei Sekigawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Acute triangulations of sphere and icosahedron

By Jinichi Itoh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 Some geometric properties of p-harmonic maps

By Shigeo Kawai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Maximum principles and its applications to submanifolds

By Katsuhiro Nonaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Comparison theorems for Riemannian manifolds with radial curvature

bounded below By Katsuhiro Shiohama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

On Anand conjecture By YuXin Dong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Notes on the relative Yamabe invariant By Kazuo Akutagawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105

Complete submanifolds in spheres By Qing-Ming Cheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Submanifolds with degenerate Gauss mapping By Goo Ishikawa, Makoto Kimura and Reiko Miyaoka . . . . . . . . . . . . . . . .125

Total curvature for open submanifolds of Euclidean spaces By Franki Dillen and Wolfgang Kühnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Existence of a family of complete minimal surfaces of genus one with one end and finite total curvature By Katsuhiro Moriya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Differential Geometry

JOSAI MATHEMATICAL MONOGRAPHS

3

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 1 – 1, sakado SAITAMA, JAPAN 350 – 0290

The First International Symposium on

Differential Geometry

JOSAI UNIVERSITY, SAKADO, SAITAMA, JAPAN

Programme

February 22, Thursday

13:05 – 13:10 Opening
13:10 – 13:55 Yoshihisa Kitagawa (Utsunomiya University)

Construction of flat tori in the 3-sphere and its applications

14:05 – 14:50 Young Jin Suh (Kyungpook National University)

Some Liouville type inequalities and its applications to geometric problems

15:10 – 15:30 Midori Goto (Fukuoka Institute of Technology)

On diffeomorphisms of a hyperbolic manifold

15:40 – 16:25 Qing Ding (Fudan Unversity)

Schrödinger flow and its applications in integrable systems

16:35 – 17:20 Masatoshi Kokubu (Tokyo Denki University)

On surfaces with holomorphic Gauss map in a certain symmetric space

February 23, Friday

9:00 – 9:45 Kouei Sekigawa (Niigata University)

On the Goldberg conjecture

9:55 – 10:40 Jinichi Itoh (Kumamoto University)

Acute triangulations of sphere and icosahedron

11:00 – 11:45 Shigeo Kawai (Saga University)

Some geometric properties of p-harmonic maps

11:55 – 12:15 Katsuhiro Nonaka (Josai University)

Maximum principles and its applications to submanifolds

13:40 – 14:25 Katsuhiro Shiohama (Saga University)

Comparison theorems for Riemannian manifolds with radial curvature bounded below

14:35– 15:20 YuXin Dong (Fudan University)

On Anand conjecture

15:40 – 16:25 Kazuo Akutagawa (Shizuoka University)

Notes on the relative Yamabe invariant

16:35 – 17:20 Qing-Ming Cheng (Josai University)

Complete submanifolds in spheres

February 24, Saturday

9:00 – 9:45 Reiko Miyaoka (Sophia University)

Submanifolds with degenerate Gauss mapping

9:55 – 10:40 Franki Dillen (Katholieke University Leuven)

Total curvature for open submanifolds of Euclidean spaces

11:00 – 11:45 Katsuhiro Moriya (University of Tsukuba)

Existence of a family of complete minimal surfaces of genus one with one end and finite total curvature

11:45 – 11:55 Closing

The schedule of bus (free): 17.30 (from university),

8.20 (from Sakado Station)

8.31 (on Saturday from Sakado Station)