Contemporary Mathematics
Volume 205, 1997
ON EXTREME POINTS AND THE STRONG MAXIMUM
PRINCIPLE FOR CR FUNCTIONS
S. Berhanu
ABSTRACT.
We give an example of a hypersurface in which the restrictions of
nonconstant holomorphic functions never attain a weak local maximum, but
the same property is not valid for continuous CR functions. In particular,
the hypersurface has no extreme points in the sense of [5]. We also establish
some necessary conditions for the validity of the strong maximum principle for
continuous CR functions on a hypersurface.
1.
INTRODUCTION
Let
M
be a smooth hypersurface in
C'
and consider the tangential Cauchy-
Riemann operators on A1. There are two kinds of maximum principles that one
can consider for the continuous solutions of the homogeneous equations, ie. for the
continuous CR functions.
DEFINITION
1.1. We say M satisfies the strong maximum principle if given
any connected open set U in M and any continuous CR function h on U, lhl
cannot have a weak local maximum at any point of U unless h is constant on U.
DEFINITION
1.2. We say M satisfies the weak maxzmum principle if gwen any
connected open set U in M and any continuous CR function h on U, lhl cannot
have a strong local maximum at any point of U.
If M satisfies the strong maximum principle, it clearly satisfies the weak max-
imum principle. On the other hand, a Levi flat hypersurface satisfies the weak
maximum principle but not the strong maximum principle. For a non Levi flat
example, consider
M
=
{(z1, ... ,
Zn-l,Xn
+
J=l(lz1!
2
+ · · · +
lzql
2)):
Zj
E
C,
x
n
E lR, and 1
~
q n - 1}, n
3.
The Levi form for this hypersurface has no strictly definite points and so by well-
known results (see for example [8], p.490) A1 satisfies the weak maximum principle.
However, the CR function
h
=
exp(J=l(xn
+
J=l(lz1l 2
+ · · · +
lzql 2
)))
has the property that lhl attains a weak local maximum at all points in A1 where
z1
= · · · =
Zq
=
0, and so A1 violates the strong maximum principle.
1991 Mathematics Subject Classification. Primary: 32H10, 32A25.
©
1997 American r..Iathematical Society
http://dx.doi.org/10.1090/conm/205/02649
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