Function Theory on Manifolds Which Possess a Pole
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699
R. E. Greene H. Wu
Function Theory on Manifolds Which Possess a Pole
Springer-Verlag Berlin Heidelberg New York 1979
Authors
R. E. Greene Department of Mathematics University of California Los Angeles, CA 90024/USA H. Wu Department of Mathematics University of California Berkeley, CA 94?20/USA
AMS Subject Classifications (1970): Primary: 53C55, 32C10, 32F99, 35N15 Secondary: 53C20, 58G99, 32 F15, 32H20, 32 H10, 30A48 ISBN ISBN
3-540-09108-4 0-38?-09108-4
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the pub~sher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS
O.
Introduction
I.
Preliminaries
2.
The
Hessian
Theorem 3.
A
4.
5.
I
...........................................
5
Comparison
B
Theorem
Subharmonic
Functions
........
the Exponential Map and the Absence Functions .............................
43 43
of 56
...............................................
56
Theorem
D
...............................................
57
Criterion E
Appendix Bounded K~hler
of
Hyperbolicity
..........................
81
...............................................
83
................................................
94
Exhaustion Manifolds
Functions and a New .......................
Class of Hyperbolic ; ................
99
Theorem
F
...............................................
99
Theorem
G
...............................................
111
A
Digression
Appendix The
............................................
................................................
Bergman
Theorem 9.
for
19
C
Theorem
8.
19
Theorem
New
7.
..........................
...............................................
Quasi-isometry of Positive Harmonic
6.
Theorem
...............................................
Sub-mean-value Theorem
............................................
H
Manifolds Theorem
Bibliography
J
Metric
......................................
116 136 141
...............................................
144
Biholomorphic
180
to
Cn
...........................
...............................................
180
................................................
206
Index
.......................................................
211
Index
of
214
Special
Notations
..................................
0.
Introduction
This paper studies the function theory of Cartan-Hadamard manifolds, i.e., complete
simply-connected
tional curvature.
Rieman
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