Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm

221 Pages·1993·13.771 MB·English

by  Professor Hirotaka Fujimoto (auth.)

Hirotaka Fujimoto
Value Distribution theory
of the Gauss Map
of Minimal Surfaces in Rm
Aspect~f
tvbthematic~
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Hirotaka Fujimoto
Value Distribution Theory
of the Gauss Map
of Mininlal Surfaces in Rm
Die Deutsche Bibliothek - CIp·E inheitsaufnahme
Fujimoto, Hirotaka: Value distribution of the
Gauss map of minimal surfaces in Rm I
Hirotaka Fujimoto. - Braunschweig; Wiesbaden:
Vieweg, 1993
- (Aspects of mathematics: E; Vol. 21)
ISBN·13: 978·3·322·80273·6 e·ISBN·13: 978·3·322·80271·2
DOl: 10.1007/978·3·322·80271·2
NE: Aspects of mathematics I E
Professor Hirotaka Fujimoto
Department of Mathematics Faculty of Science
Kanazawa University
Marunouchi, Kanazawa, 920
Japan
Mathematics Subject Classification: 53-02, 53AlO, 30·02, 30035
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Softcover reprint of the hardcover 1s t edition 1993
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Preface
These notes are based on lectures given at Instituto de Matematica e Esta
tistica in Universidade de Sao Paulo in the spring of 1990. The main sub
ject is function-theoretic, particularly, value-distribution-theoretic proper
ties of the Gauss maps of minimal surfaces in the euclidean space.
The classical Bernstein theorem asserts that there is no nonfiat min
imal surface in R3 which is described as the graph of a C2-function on
R2. On the other hand, the classical Liouville theorem asserts that there
is no bounded nonconstant holomorphic function on the complex plane.
The conclusions of these theorems have a strong resemblance. Bernstein's
theorem was improved by many reseachers in the field of differential ge
ometry, Heinz, Hopf, Nitsche, Osserman, Chern and others. On the other
hand, in the field of function theory, Liouville's theorem was improved
as the Casoratti-Weierstrass theorem, Picard's theorem and Nevanlinna
theory, which were generalized to the case of holomorphic curves in the
projective space by E. Borel, H. C art an , J. and H. Weyl and L. V. Ahlfors.
These results in two different fields are closely related. As for recent re
sults, in 1988 the author gave a Picard type theorem for the Gauss map
of minimal surfaces which asserts that the Gauss map of nonfiat complete
minimal surfaces in R3 can omit at most four values. Moreover, he ob
tained modified defect relations for the Gauss map of complete minimal
surfaces in R m, which have analogies to the defect relation in N evanlinna
theory. Moreover, several results related to these subjects were obtained
by X. Mo-R. Osserman, S. J. Kao, M. Ru and so on.
In this book, after developing the classical value distribution the
ory of holomorphic curves in the projective space, we will explain the
above-mentioned modified defect relations for the Gauss map of complete
minimal surfaces together with detailed proofs and related results.
As for prerequisites, it is assumed that the reader is acquainted with
basic notions in function theory and differential geometry. Although the
description is intended to be self-contained as far as possible, he will need
some basic definitions and results concerning exterior algebra, differen
tiable manifold, Riemann surface and projective space, all of which, except
for a few theorems, appear in standard texts for graduate students.
The author would like to express his thanks to Plinio Amarante Quirio
Simoes for inviting him to 1M E, for giving him the opportunity to lecture
vi Preface
and for his valuable comments. In addition, the author would like to
express his thanks to Antonio Carlos Aspert and Dr. Bennett Palmer for
reading the manuscript and pointing out some mistakes; and to T. Osawa
and K. Diederich for recommending to him that he writes this book.
Kanazawa, Japan
November 1992
Hirotaka FUJIMOTO
Table of Contents
Preface ............................................................. v
Introduction .......................................................
IX
Chapter 1
The Gauss map of minimal surfaces in R 3 .••••.••.......•...... 1
§1 .1 Minimal surfaces in R m ......................................... 1
§1.2 The Gauss map of minimal surfaces in Rm ..................... 10
§1.3 Enneper-Weierstrass representations of minimal surfaces in R3 .. 15
§1.4 Sum to product estimates for meromorphic functions ........... 22
§1.5 The big Picard theorem ........................................ 29
§1.6 An estimate for the Gaussian curvature of minimal surfaces .... 35
Chapter 2
The derived curves of a holomorphic curve .................... 46
§2.1 Holomorphic curves and their derived curves .................... 46
§2.2 Frenet frames .................................................. 54
§2.3 Contact functions .............................................. 61
§2.4 Nochka weights for hyperplanes in subgeneral position ... " ..... 67
§2.5 Sum to product estimates for holomorphic curves ............... 76
§2.6 Contracted curves ............................................. 82
Chapter 3
The classical defect relations for holomorphic curves ......... 90
§3.1 The first main theorem for holomorphic curves .................. 90
§3.2 The second main theorem for holomorphic curves ............... 97
§3.3 Defect relations for holomorphic curves ....................... 105
§3.4 Borel's theorem and its applications .......................... 112
§3.5 Some properties of Wronskians ................................ 120
§3.6 The second main theorem for derived curves .................. 130
viii Contents
Chapter 4
Modified defect relation for holomorphic curves .............. 140
§4.1 Some properties of currents on a Riemann surface ............. 140
§4.2 Metrics with negative curvature ............................... 146
§4.3 Modified defect relation for holomorphic curves ............... 152
§4.4 The proof of the modified defect relation ...................... 159
Chapter 5
The Gauss map of complete minimal surfaces in Rm ......... 166
§5.1 Complete minimal surfaces of finite total curvature ............ 166
§5.2 The Gauss maps of minimal surfaces of finite curvature ........ 175
§5.3 Modified defect relations for the Gauss map of minimal surfaces 182
§5.4 The Gauss map of complete minimal surfaces in R3 and R4 .... 186
§5.5 Examples .... , ................................................ 193
Bibliography ...................................................... 200
Index ............................................................. 205
Introduction
=
Let x (Xl, X2, X3) : M -+ R 3 be an oriented surface immersed in R 3.
By definition, the classical Gauss map g : M -+ 52 is the map which
maps each point p E M to the point in 52 corresponding to the unit
normal vector of M at p. On the other hand, 52 is canonically identified
with the extended complex plane C U {oo} or pI (C) by the stereographic
projection. We may consider the Gauss map g as a map of Minto PI(C).
Using systems of isothermal coordinates, we can consider M as a Riemann
surface. Our main concerns are minimal surfaces, namely, surfaces which
have minimal areas for all small perturbations. In 1915, S. Bernstein gave
the following theorem( [5]):
THEOREM 1. If a minimal surface M in R3 is given as the graph
of a C2-function f(XI, X2) on (Xl, x2)-plane, then M is necessarily a plane.
Thirty-seven years later, E. Heinz obtained the following improve
ments of this([45]) and E. Hopf and J. C. C. Nitche gave some related
results([48]' [56]).
,THEOREM 2. If a minimal surface M is given as the graph of a function
ofclassC2 on a disk ~R:= {(Xl,X2);Xr+X~ < R2} in the (XI,x2)-plane,
then there is a positive constant C, not depending on M, such that
IK(O)I ~ C/R2
holds, where K(O) is the Gaussian curvature of M at the origin.
Later, R. Osserman generalized these results to a minimal surface
which need not to be the graph of a function ([59], [60D. One of his
results is stated as follows:
THEOREM 3 ([59]). Let M be a simply-connected minimal surface im
mersed in R 3 and assume that there is some fixed nonzero vector no and
a number 0 > 0 such that all normals to M make angles at least 0 with
0 0
no. Then
IK(p)II/2 < _1_ 2 cos (00/2) (p EM),
- d(p) sin3(Oo/2)
x Introduction
where d(p) denotes the distance from p to the boundary of M.
Moreover, R. Osserman proved the following fact in the paper [61].
THEOREM 4. Let x : M -+ R3 be a nonffat complete minimal surface
immersed in R3. Then the complement of the image of the Gauss map is
of logarithmic capacity zero in Pl(C).
Theorem 4 is an improvement of Theorem 1. In fact, under the as
sumption of Theorem 1 the Gauss map of M omits the set which corre
sponds to the lower half of the unit sphere, which is not be of logarithmic
capacity zero. Therefore, M is necessarily fiat and consequently a plane.
In 1981, Theorem 4 was improved by F. Xavier in his paper [76] as
follows.
THEOREM 5. In the same situation as in Theorem 4, the Gauss map of
M can omit at most six points in Pl(C).
Afterwards, the author obtained the following theorem ([36]).
THEOREM 6. The number of exceptional values of the Gauss map of a
nonffat complete minimal surface immersed in R 3 is at most four.
Here, the number four is the best-possible. In fact, there are many
examples of nonfiat complete minimal surfaces immersed in R3 whose
Gauss maps omit exactly four values([63]), Among them, Scherk's surface
is the most famous.
Recently, the author gave also the following estimate of the Gaussian
curvature of a minimal surface in R3 related to Theorem 3([41]).
THEOREM 7. Let x = (Xl, X2, X3) : M -+ R 3 be a minimal surface
immersed in R 3 and let G : M -+ 52 be the Gauss map of M. Assume
that G omits five distinct unit vectors nl, ... ,ns E 82. Let ()ij be the
angle between ni and n j and set
) ;
L := min { sin (()~j 1 ~ i < j ~ 5 } .
Then, there exists some positive constant C, not depending on each min
imal surface, such that
1
log2 -
IK(p)ll/2 < ~ L (p EM).
- d(p) L3