Convex affine surfaces with constant affine mean curvature

CONVEX AFFINE SURFACES WITH CONSTANT AFFINE MEAN CURVATURE A. M a r t ~ n e z and F. M i l a n (I~ An i n t e r e s t i n g [SI]):

(open)

p r o b l e m in A f f i n e D i f f e r e n t i a l

the c l a s s i f i c a t i o n

strongly

convex

of all affine complete,

surfaces

M,

the u n i m o d u l a r real affine The ovaloid

compact

strongly

studied

constant

assertion

convex

surfaces

in A 3, (see [B] and The p r o b l e m

affine

mean

true

with

[P]),

x2>0,

to

this

problem

or

M

[LI]), and Gauss map, there

the

complete

some

are

surface

locally

affine m e a n curvature. .- L e t

A 3 with

M

be

constant

a

Gauss-Kronecker

of

affine

the

metric

3JK

+ 2HB

ellipsoid,

ii) iii) iv)

and

(II)

B 2 = H a - r.

an

elliptic

is,

H = 0 on M,

Then

been

affine

in

with

the

image

of

of

the

([L2]). which

(x I,x2,x3)~A3

affine

spheres

([LP],

characterize

I

with

Pick

the

x 1>0,

xlx2x3 =a>0,

invariant

[K]).

strongly

convex

surfaces

in A 3 w i t h

constant

strongly

mean

affine

H.

o f M,

intrinsic

the Pick for

convex,

curvature the

invariant, some

real

Denote

complete by

r,

Gaussian

respectively.

numbers

~

c and

d,

surface

and

of

the

following

the

surface

Q(a,2),

the

If 2 c > ~,

a 0, M is o n e

J

curvature

surfaces:

paraboloid,

image

M

([CI],[CY2],[J]

an hyperboloid, an

is

surface

obtained

sphere,

results

Q(a,2)={

curvature

J - c B 2 a d,

i) a n

locally curvature

we give a step in the c l a s s i f i c a t i o n of the

locally

(I)

where

mean

We obtain the f o l l o w i n g result,

affine

affine

have

conditions

known

hyperbolic

In this communication,

in

affine

affine-maximal

(affine

s a t i s f y i n g some a d d i t i o n a l assumptions,

THEOREM

"Every

ellipsoid".

complete,

that

complete,

([C3],

complete,

prove:

(see [Ch]) and states:

([C2]),

and

is a n

affine

constant

graph,

x3>0 } as

affine

for

surfaces,

affine

involving

H=constant

0

(I )

A3

with

and,

and

assumption

complete,

from below

surface

~

assumptions

following

convex,

~ + cr bounded

~ = J + H. T h e n

from

( II ),

in

is a c o r o l l a r y .

surface,

obtain

strongly

theorem:

convex,

a f fine

Problem

A 3, w i t h

affine

(II)

solution

in T h e o r e m

of

the

affine-maximal

a constant,

for

Affine

surface

some

real

in

number

paraboloid".

In p a r t i c u l a r : If

the

obtain 3.-

a f fine

In t h e

case

(B 2 v a n i s h e s some

Gauss-Kronecker

t h a t M is an e l l i p t i c

conditions

one

"Let M b e

can o b t a i n

a locally

H=constant