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

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