Math. Z. 230, 471–486 (1999)
c Springer-Verlag 1999
An extension of a theorem by K. Jörgens
and a maximum principle at infinity
for parabolic affine spheres?
L. Ferrer, A. Martı́nez, F. Milán
Departamento de Geometrı́a y Topologı́a, Universidad de Granada, E-18071 Granada, Spain
(e-mail: [email protected]; [email protected]; [email protected])
Received March 7, 1997; in final form September 5, 1997
1. Introduction
The aim of this paper is to study the following unimodular Hessian equation,
2 ∂ f
det
(1)
= 1 in Ω ,
∂xi ∂xj
where Ω is a planar domain and f is in the usual Hölder space C 2,α (Ω).
Without loss of generality we shall consider only locally convex solutions
of (1).
This equation arises in the context of an affine differential geometric
problem as the equation of a parabolic affine sphere (in short PA-sphere)
in the unimodular affine real 3-space (see [C1], [C2], [CY] and [LSZ]).
Contrary to the case of smooth bounded convex domains, little is known
about solutions of (1) when the domain is unbounded. Here, we recall a
famous result by K. Jörgens which asserts that any solution of (1) on Ω = R2
is a quadratic polynomial (see [J]) and we also mention a previous paper
(see [FMM]) where the authors study solutions of (1) on the exterior of a
planar domain that are regular at infinity.
Since the underlying almost-complex structure of (1) is integrable, one
expects PA-spheres (with their canonical conformal structure) to be conveniently described in terms of meromorphic functions. The reader will find in
Sect. 2 a complex representation of PA-spheres and, particularly, a complex
description for the solutions of (1).
?
Research partially supported by DGICYT Grant No. PB94-0796 and the GADGET III
program of the EU.
Mathematics Subject Classification (1991): 53A15
472
L. Ferrer et al.
Using this, in Sect. 3 we extend the results in [FMM] and [J] and obtain
how is the behaviour at infinity of any solution of (1) when Ω = Ωγ is the
outside of a plane Jordan curve γ. In particular, it is shown that any solution
behaves like a quadratic polynomial plus a logarithmic term. This allows to
define, for each one, its ellipse at infinity and its logarithmic growth rate. In
Sect. 4 we use these invariants to prove a Maximum Principle at infinity for
solutions of (1) on Ωγ , which improves some previous results by the authors
(see Theorems 6 and 7 in [FMM]). From here, we obtain the uniqueness of
solutions having the same behaviour at infinity for the Dirichlet Problem
associated to (1) on Ωγ .
Continuing the work [FMM] (where the authors studied properties of
symmetry in the compact case), Sect. 5 is devoted to give some applications
of the Maximum Principle to the study of symmetries of solutions of the
problem
2 ∂ f
det
= 1 in Ω 1 , f|∂Ω1 = 0 ,
(2)
∂xi ∂xj
where Ω 1 = {w ∈ C | |w| > 1}. In particular we prove that a solution f of
(2) is invariant under a reflection through a line by the origin if and only if
this line is an axis of the ellipse at infinity of f . In Appendix 1, we describe
a family of solutions of this problem with only one symmetry. Finally, in
Appendix 2, we give an alternative proof for the uniqueness of solutions
obtained in Sect. 4.
2. Parabolic affine spheres: a complex representation
We refer the reader to [C1], [C2], [CY], [FMM] and [LSZ] for more details
about affine surfaces.
Let X : M −→R3 be an oriented immersed locally strongly convex
surface in R3 with the suitable orientation so that the Euclidean second fundamental form σe is definite positive. We denote by Ke , Ne and dAe its
Euclidean Gauss curvature, its Gauss map and its element of area, respectively. Then M can be endowed with a metric ds2 given by
−1
ds2 = Ke 4 σe ,
which is invariant under unimodular affine transformations and it is called
the affine metric. Its element of 1area is known as the element of affine
area and it is given by dA = Ke4 dAe . Moreover, if we denote by ∆ the
Laplace-Beltrami operator associated with ds2 , one has the affine invariant
transversal vector field
1
ξ = ∆X ,
2
Parabolic affine spheres
473
which is called affine normal. Throughout we shall consider on M the
structure of Riemann surface induced by the affine metric ds2 .
One of the most interesting problems in affine differential geometry is the
study of the surfaces which are ”extremal” for the affinely invariant area under interior deformations. The Euler-Lagrange equation for this variational
problem is equivalent to the following system of differential equations
−1
∆(Ke 4 Ne ) = 0 .
−1
The immersion N = Ke 4 Ne : M −→R3 is called the affine conormal
map of X and consequently, X is extremal if and only if N is an harmonic
map on M . Since Calabi proved in [C3] that the second variation of the
affine area under interior deformations is always negative, these surfaces
are called affine maximal surfaces.
An important subset of affine maximal surfaces are the PA-spheres.
Definition 1. An oriented immersed locally strongly convex surface M in
R3 is a PA-sphere if and only if all the affine normal lines through each
point of M are mutually parallel, that is, the affine normal ξ is constant on
M or equivalently, M is affine maximal and its affine conormal map N lies
on a plane.
Now, let M be a smooth PA-sphere with a C 2,α -boundary (possibly empty).
We shall denote by (x1 , x2 , x3 ) a rectangular coordinate system in R3 . By
using an unimodular affine transformation if it is necessary, we can assume
that the affine normal vector of M is ξ = (0, 0, 1). Then the projection
on Π ≡ {x3 = 0} parallel to ξ, pξ : M −→Π, is an immersion and M
is, locally, the graph of a convex solution f : B−→R of (1) on a bounded
convex domain B in Π. That is, in a neighbourhood of each point, M is
given by
(3)
X (x1 , x2 ) = (X1 (x1 , x2 ), X2 (x1 , x2 ), X3 (x1 , x2 ))
= (x1 , x2 , f (x1 , x2 )) ,
where (x1 , x2 ) ∈ B. Moreover, it is easy to prove that the fundamental
unimodular affine invariants associated with M in this neighbourhood are
given by
(4)
ds2 =
2
X
fij dxi dxj ,
i,j=1
(5)
N = (N1 , N2 , N3 ) =
∂f
∂f
−
,−
,1
∂x1 ∂x2
,
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L. Ferrer et al.
2
f
where we denote by fij = ∂x∂i ∂x
, for i, j = 1, 2. Conversely, the graph
j
of a convex solution of (1) is a PA-sphere with affine normal vector field
ξ = (0, 0, 1) and affine metric given by (4).
Using standard notation of complex analysis (see [A]), one can define
the functions F, G : B−→C given by
∂f
∂f
G(x1 , x2 ) = x1 +
(6)
+ i x2 +
,
∂x1
∂x2
∂f
∂f
F (x1 , x2 ) = x1 −
(7)
+ i −x2 +
,
∂x1
∂x2
with (x1 , x2 ) in the former convex domain B. Taking into account (3), (5),
(6) and (7), the functions F and G can be written
G = (X1 − N1 ) + i (X2 − N2 ) ,
F = (X1 + N1 ) − i (X2 + N2 ) ,
(8)
(9)
hence, F and G are globally defined on M . From (1), (4), (6) and (7) one
can prove the following result (see the proof of Theorem 5 and Remark 6 in
[FMM]).
Lemma 1. Let M be a PA-sphere and F and G the functions above defined.
Then, one has
i) z = G is an holomorphic coordinate on M .
ii) ∂F
∂z
= 0, that is, F is an holomorphic function of z.
iii) ∂F < 1.
∂z
Moreover, if we denote w = x1 + ix2 , from (6) and (7) we have
1
1
∂f
w=
z + F (z) ,
= (z − F (z)) ,
(10)
2
∂w
4
and hence the function f can be rewritten as
Z z
1
1
1
1
(11) f (w) = |z|2 − |F (z)|2 + <(zF (z)) − <
F (ζ)dζ ,
8
8
4
2
z0
where < denotes the real part. On the other hand, since F is an holomorphic
function of z, from (10) one has,
f11 =
(1 − r1 )2 + r22
,
1 − r12 − r22
f22 =
(1 + r1 )2 + r22
,
1 − r12 − r22
f12 =
(12)
where r1 and r2 are the real and the imaginary part of
Thus, the affine metric is given by
!
∂F 2
1
2
|dz|2 .
1 − ds =
(13)
4
∂z 2r2
,
1 − r12 − r22
∂F
∂z ,
respectively.
Parabolic affine spheres
475
Analogously, from (5) and (10), one obtains
(14)
N=
!
−z + F (z)
,1
2
.
From the above considerations, it is not difficult to prove the following,
Theorem 1. (Complex representation)
i) Let M be a PA-sphere with affine normal ξ = (0, 0, 1) and the structure of
Riemann surface induced by its affine metric. Then there exist holomorphic
functions F and G on M such that dG does not vanish on M , |dF | < |dG|
and M can be represented, up a translation, by the immersion
Z
G+F 1 2 1 2 1
1
(15) X =
, |G| − |F | + <(GF ) − < F dG .
2
8
8
4
2
Moreover, the affine metric and the affine conormal map are given by
(16)
(17)
1
|dG|2 − |dF |2 ,
4
F −G
N=
,1 .
2
ds2 =
ii) Conversely, let M be a Riemann surface, F and G two holomorphic
functions on M such that dG does not vanish on M and |dF | < |dG|. Then
(15) defines a PA-sphere with affine normal (0, 0, 1) and with affine metric
and affine conormal map given by (16) and (17), respectively. Moreover, X
is singly-valued if and only if F dG does not have real periods.
Now and on, the pair (F, G) will be called a complex representation of
the PA-sphere M .
3. An extension of a theorem by K. Jörgens
As we mentioned in the introduction, K. Jörgens proved in [J] that all solutions of (1) on Ω = R2 are quadratic. In this section we use the complex
representation presented in Sect. 2 to describe the behaviour at infinity of
the solutions of (1) when Ω is either Ωγ , the exterior of a plane Jordan curve
γ, or the whole R2 .
Let f be a solution of (1) on Ω and let Mf = {(x1 , x2 , f (x1 , x2 )) |
(x1 , x2 ) ∈ Ω} be its graph. From Sect. 2, Mf is a PA-sphere with affine
normal (0, 0, 1) which can be represented as in (15), where the functions G
and F are given by (6) and (7). On the other hand, from (1) and (6) it is easy
to prove that
dx21 + dx22 ≤ |dG|2 .
(18)
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L. Ferrer et al.
Hence dG defines a metric on Mf which is complete at infinity, and then,
it is clear that Mf is conformally equivalent to either C, if Ω = R2 , or
Ω 1 = {τ ∈ C | |τ | > 1}, if Ω = Ωγ , where τ tends to infinity as x1 + ix2
tends to infinity (see [O]).
Moreover, a standard argument shows that G has at most a pole at infinity
dF
is bounded and thereby
of order m and, from Lemma 1, the function dG
holomorphic at infinity.
If Mf is conformally equivalent to C, we have that dG is a non vanishing
holomorphic one-form on C with at most a pole at infinity, and then, G as
well as F must be linear respect to τ with G non constant. Thus, Jörgens’
Theorem follows directly from (15).
On the contrary, if Mf is conformally equivalent to Ω 1 , we have that F
has at most a pole at infinity of order less or equal than m, and the function
G cannot be holomorphic at infinity, otherwise the function x1 + ix2 =
1
2 (G + F ) would be bounded. Furthermore, if m > 1, then the image of
the circle |τ | = k, for large k, by the map x1 + ix2 would wind around the
infinity more than once. This contradicts the fact that x1 + ix2 is one-to-one
on Ωγ , and then G has a pole at infinity of order one. Hence the function G
can be written as
∞
X
cn
,
G(τ ) = e
aτ + eb +
τn
n=1
a 6= 0 and τ ∈ Ω 1 . This allows us to assert that z = G
with e
a, eb, cn ∈ C, e
is another holomorphic coordinate on Mf defined on the exterior of a disk.
Thus, using (6), the transformation Lf given by
(19)
z = G(τ ) = Lf (w) = w + 2
∂f
∂w
b the outside of a plane Jordan curve in
is a global diffeomorphism from Ω,
Ωγ , onto the exterior of a disk Ω R of radius R. Moreover, from (7) and
b
Lemma 1, the function F : Ω R = Lf (Ω)−→C
given by
(20)
F (z) = w − 2
∂f
,
∂w
is holomorphic on Ω R and can be written as
(21)
F (z) = µz + ν +
∞
X
an
n=1
zn
,
on Ω R , where µ, ν, an ∈ C for n ≥ 2, a1 ∈ R and |µ| < 1.
From the above considerations and the complex representation (15) we
b as in (11).
have that f can be rewritten on Ω
Parabolic affine spheres
477
By using (11), (19), (20) and (21) we obtain
ν1
∂f
∂f
∂f
1
x1
x1 +
−
+ x2
f (w) =
2
∂x1
∂x2
4
∂x1
ν2
a1
∂f
+
x2 +
log(|z|2 ) + O(1) ,
−
(22)
4
∂x2
4
where ν = ν1 + iν2 and by O(1) we denote a term bounded in absolute
value by a constant. Since lim|z|→∞ (F (z) − µz) = ν, from (19) and (20)
one has,
∂f
1
=
{(1 + |µ|2 − 2µ1 )x1 + 2µ2 x2 } +
∂x1
1 − |µ|2
1
+
{a − (1 − µ1 )R1 + µ2 R2 },
1 − |µ|2
(23)
∂f
1
=
{(1 + |µ|2 + 2µ1 )x2 + 2µ2 x1 } +
∂x2
1 − |µ|2
1
+
{b + (1 + µ1 )R2 − µ2 R1 },
1 − |µ|2
(24)
where µ = µ1 + iµ2 , a = −ν1 + µ1 ν1 + µ2 ν2 , b = ν2 − µ2 ν1 + µ1 ν2 and R1
and R2 are the real and the imaginary part of F (z) − µz − ν, respectively.
Finally, with the above notations and using (22), (23) and (24) we obtain,
Theorem 2. Let f be a solution of the unimodular Hessian equation (1) on
Ω, where Ω is either C or Ωγ . Then f is quadratic or it is given, on the
exterior of some plane Jordan curve, by the expression
f (w) = E(f )(w) −
(25)
a1
log(|z|2 ) + O(1) ,
4
where
E(f )(w) =
(26)
1
1 + |µ|2 − 2µ1 x21
2
2(1 − |µ| )
+ 1 + |µ|2 + 2µ1 x22
1
ν2 b − ν 1 a
.
+
2µ2 x1 x2 + ax1 + bx2 +
1 − |µ|2
4
Definition 2. When k is a large positive number, from (25) and (26), the
ellipse
Ek ≡ E (f ) (w) = k ,
gives the shape of Mf at infinity. The ellipse Ek will be called the ellipse
at infinity associated with f . The center of this ellipse will be called the
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center of f and will be denoted by c(f ). From (26) we observe that the
ellipse at infinity is a circle if and only if µ = 0. In these cases the center of
the circle at infinity of f is given by c(f ) = ν1 − iν2 .
On the other hand, the real number a1 that appears in the expresion (25)
is known as the logarithmic growth rate of the function f and it will be
denoted by log(f ). One can observe that log(f ) indicates how much the
graph of f moves away from the elliptic paraboloid.
As Lf is known as the transformation of Lewy (see [S], pp. 167), the
function F given by (20) will be called the Lewy function of f .
Remark 1. From Theorem 2 we have that a solution of (1) when Ω is the
exterior of a bounded planar domain is regular at infinity in the sense of
[FMM].
4. A maximum principle at infinity
In [FMM] the authors proved the following Maximum Principle at infinity:
Theorem. Let f and g be convex solutions of (1) on ΩR with f = g in ∂ΩR .
Suppose that the graphs Mf and Mg of f and g, respectively, are regular
at infinity and f ≥ g on ΩS for some S > R. If there exists a sequence
{wn }n∈N in ΩR with limn→∞ |wn | = ∞ and
lim |f (wn ) − g(wn )| = 0,
n→∞
then f ≡ g.
We can observe that the assumption on the boundary in the former theorem
is very strong. The purpose of this section is, using the results of Sect. 3, to
prove the following Maximum Principle at infinity for any solution of (1)
on Ωγ without any assumption on the boundary.
Theorem 3. (Maximum Principle at infinity) Let f and g be solutions of
the problem (1) on Ωγ . If f ≥ g on Ωγ and there exists a sequence {wn }n∈N
in Ωγ with limn→∞ |wn | = ∞ and limn→∞ |f (wn ) − g(wn )| = 0, then
f ≡ g.
Proof. We can suppose that f > g. If not, there exists w ∈ Ωγ such that
f (w) = g(w) and since f and g are solutions of (1) we know that f − g
satisfies a linear elliptic operator (see [B], [GT]), thus, by the usual Maximum
Principle we have f ≡ g.
Let F1 and F2 be the Lewy functions of f and g, respectively. Using (21)
and the Lemma 1 of [FMM], F1 and F2 can be written as
(27) F1 (z) = µz + ν +
∞
X
an
n=1
zn
,
F2 (b
z ) = µb
z+ν+
∞
X
bn
,
zbn
n=1
Parabolic affine spheres
479
Fig. 1
where a1 , b1 ∈ R, ν, µ ∈ C, z = Lf (w) and zb = Lg (w). From (19), (20),
(25), (26) and (27) we obtain,
f (w) − g(w) =
b1 − a1
log(|z|2 ) + O(1) .
4
Hence by using that limn→∞ |f (wn ) − g(wn )| = 0, one gets that a1 = b1 .
−1
Let R > 0 be such that L−1
f and Lg are well defined on ΩR . For each
z ∈ ΩR we are going to denote by
−1
−1
(z),
f
(L
(28) x(z) = L−1
(z))
, y(z) = L−1
g (z), g(Lg (z)) ,
f
f
the corresponding points on Mf and Mg . From (14) the affine conormal
maps of Mf and Mg at x(z) and y(z) are given, respectively, by
!
!
−z + F1 (z)
−z + F2 (z)
(29) N1 (z) =
, 1 , N2 (z) =
,1 .
2
2
Let C be the curve which is the intersection of Mf with the vertical plane
through x(z) and y(z) and let P be the intersection point of the tangent
plane of Mf at x(z) and the vertical line passing through y(z) (see Fig. 1).
We can compute the diference f (w)−g(w) by computing λ1 = ||y(z)−P ||
and λ2 = ||x(e
z ) − P ||, where ze = Lf (L−1
g (z)). From the definition
of P we have that P − x(z) is tangent to Mf at x(z), and therefore
λ1 =< x(z) − y(z), N1 (z) >, where <, > denotes the usual inner product in R3 . Using (11), (27), (28) and (29) we find
1 −1
λ1 = < (F1 (z) − F2 (z)) (F1 (z) − z) + f (L−1
f (z)) − g(Lg (z)) =
4
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1
1 < (F1 (z) − F2 (z)) F1 (z) −
|F1 (z)|2 − |F2 (z)|2 +
4 Z
8
z
1
1
(F2 − F1 )(ζ)dζ = (F1 (z) − F2 (z)) F1 (z) − F2 (z) +
+ <
2
8
z
Z 0z
bq − aq
1
1
<
+ <
+ O(|z|−q ) ,
(F2 − F1 )(ζ)dζ =
q−1
2
2(1
−
q)
z
z0
=
where q = min{n ∈ N | an − bn 6= 0}. The expression O(|z|n ) will be
used to indicate a term which is bounded in absolute value by a constant
times |z|n for |z| large.
Next we shall show that λ2 is of order 2q. To see this we can write
λ2 = h(cos(θ))−1 where h is the distance between x(e
z ) and the tangent
line to C in x(z) and θ is the angle between the Euclidean normal vector to
C in x(z) and (0, 0, 1).
We know that in a neighbourhood of x(z) the order of h is the order
of d2 κ where d = ||x(z) − x(e
z )|| and κ is the curvature of C at x(z).
The distance d, close to x(z), is of the order of c(cos(θ))−1 , where c is the
horizontal distance between x(z) and y(z).
A straight computation using the expressions (10), (27) and (28) gives
that c = O(|z|−q ) and cos(θ) = O(|z|−1 ).
Finally we parametrize C by α(t) = (A1 t + B1 , A2 t + B2 , f (A1 t +
B1 , A2 t + B2 )) for some Ai , Bi ∈ R, i = 1, 2. Hence from (10), (12) and
(27), the curvature of C is given by
2
κ=
2
2
f
f
f
A21 ∂x∂1 ∂x
+ 2A1 A2 ∂x∂1 ∂x
+ A22 ∂x∂2 ∂x
1
2
2
3
∂f
∂f 2 2
(A21 + A22 + (A1 ∂x
+ A2 ∂x
) )
1
2
= O(|z|−3 ) ,
and we have that λ2 = O(|z|−2q ).
Thus, using the notation z = |z|eiγ we obtain
1
f (w) − g(w) =
< (bq − aq )eiγ(1−q) |z|(1−q) + O(|z|−q ) ,
2(1 − q)
which is contrary to f > g when |z| is large.
In an analogous way to the Proposition in [LR] we can prove the following
corollaries.
Corollary 1. Let f and g be solutions of (1) on Ωγ . Suppose that f = g on
∂Ωγ and f ≥ g on Ωγ . If log(f ) = log(g), then f ≡ g.
Corollary 2. Let f and g be solutions of (1) on Ωγ and let F1 and F2 be
the Lewy functions of f and g, respectively. Suppose that f = g on ∂Ωγ ,
log(f ) = log(g), lim|z|→∞ F1z(z) = lim|bz |→∞ F2zb(bz ) = µ, and
lim|z|→∞ (F1 (z) − µz) = lim|bz |→∞ (F2 (b
z ) − µb
z ), where z = Lf (w),
zb = Lg (w) and w ∈ Ωγ . Then f ≡ g.
Parabolic affine spheres
481
5. The exterior plateau problem
The aim of this section is to study the problem (2). Contrary to the case of the
minimal surface equation (see [LR]), in our case, as it will be shown later,
there exist different solutions of the problem (2) with the same logarithmic
growth rate. Then, we are going to restrict ourselves to solutions f of (2)
such that Mf satisfies some conditions of symmetry and we relate these
conditions with the ellipse at infinity of f .
Definition 3. A normal reflection is a reflection through a plane which
contains the line l ≡ {x1 = 0, x2 = 0}. Throughout we shall only consider
this kind of reflections. So we can give the following result.
Proposition 1. Let f be a solution of the problem (2). Then Mf is invariant
by a normal reflection through a plane Σ if and only if Σ ∩ C is an axis of
the ellipse at infinity of f .
Proof. If Mf is invariant by a normal reflection, then this reflection must be
an orthogonal reflection because of the condition on the boundary. Therefore,
there exists a rotation σ in the unimodular affine real 3-space given by
σ(x1 , x2 , x3 ) = (τ (x1 , x2 ), x3 ) ,
where τ (x1 , x2 ) = (cos(θ)x1 + sin(θ)x2 , − sin(θ)x1 + cos(θ)x2 ) and θ ∈
[0, π], such that fe = f ◦ τ −1 is a solution of (2) invariant by the reflection
through Σ1 ≡ {x2 = 0} parallel to (0, 1, 0).
Moreover, if z, ze are the transformations of Lewy associated with f and
fe, respectively and F , Fe are their Lewy functions, we have the following
relation between them,
(30)
ze = e−iθ z ,
a1
+ O(|e
z |−2 ) .
Fe (e
z ) = µe2iθ ze + νeiθ +
ze
From (30), it is easy to check that the ellipse at infinity associated with fe can
be obtained from the ellipse at infinity associated with f by the rotation τ .
This allows us to reduce the existence of one normal reflection to the study
of the reflection through Σ1 parallel to (0, 1, 0). The graph Mf is invariant
by this reflection if and only if f satisfies f (x1 , x2 ) = f (x1 , −x2 ) on Ω1 .
Let fb : Ω1 −→ R be the function given by fb(x1 , x2 ) = f (x1 , −x2 ), zb the
transformation of Lewy and Fb the Lewy function associated with fb. From
(21), F and Fb can be written as
(31)
F (z) = µz + ν +
∞
X
an
n=1
zn
,
Fb(b
z) = µ
bzb + νb +
∞
X
bn
.
zbn
n=1
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L. Ferrer et al.
From (19) and (20), zb and Fb satisfy
Fb (b
z (x1 , x2 )) = F (z(x1 , −x2 )) .
(32) zb(x1 , x2 ) = z(x1 , −x2 ) ,
If f ≡ fb, taking into account (32) we have that F (z) = F (z), and by
the expression of F given in (31) we obtain that µ and ν are real numbers.
Thus e ≡ {x2 = 0, x3 = 0} is an axis of the ellipse at infinity given by
E(f )(w) = k.
Conversely, if e is an axis of E(f )(w) = k, from (26), µ2 = b = 0 and
so µ and ν must be real numbers. Moreover, using expressions (31) and (32)
we get a1 =b1 and since µ and ν are real numbers we also obtain that µ = µ
b
and ν = νb. Thus f and fb satisfy the assumptions of Corollary 2 and we
conclude f ≡ fb.
From the above propositon we have the following consequences :
Corollary 3. Let f be a solution of (2). Then Mf is invariant by two different
normal reflections if and only if c(f ) = 0.
Corollary 4. Let f be a solution of the problem (2). Then Mf is invariant
by three normal reflections if and only if Mf is a revolution surface.
Proof. If Mf has more than two normal reflections, from the Proposition 1
the ellipse at infinity should be a circle. But if this happens, using again the
Proposition 1, we have that Mf is invariant by every normal reflection and
so it is a revolution surface.
Appendix 1: Examples with only one symmetry
Now we shall describe solutions of (2) which are invariant by only one
normal reflection.
Let ∆ be the region given by
∆ = {(a, b) ∈ R2 | a > −b > 0, 1 − a > −b} .
For each (a, b) ∈ ∆ and n a natural number different from zero we consider
the following complex functions:

(33) F(a,b,n) (z) = 2 
−a +
q
c2
b
+
2b
zn
n1

 −
−a +
q
c2
+
b
where c2 = a2 − b2 . Let α(a,b,n) be the curve in C given by
(34)
1
α(a,b,n) (t) = (a + b cos(nt))− n eit ,
t ∈ [0, 2π].
2b
zn
n2
 z,
Parabolic affine spheres
483
We shall denote by U(a,b,n) the unbounded region of C \ α(a,b,n) ([0, 2π]).
From (33) it is easy to see that at infinity F(a,b,n) can be written as
(35) F(a,b,n) (z) = µ(a, b, n)z + ν(a, b, n) +
a1 (a, b, n)
+ O(|z|−2 ) ,
z
where
µ(a, b, 1) = −
(36)
ν(a, b, 1) =
a1 (a, b, 1) =
a−c 2
b
2(a−c)(1−c)
bc
2c2 −a
c3
−a+c
b
µ(a, b, n) = −
ν(a, b, n) = 2
−a+c
b
2
n
1
n
,n ≥ 2
,n ≥ 2
a1 (a, b, 2) = − 1c
a1 (a, b, n) = 0
,n ≥ 3 .
From the expression (33) it is clear that F(a,b,n) is an holomorphic function on U(a,b,n) and it is given on α(a,b,n) by
1
F(a,b,n) (α(a,b,n) (t)) = 2 − (a + b cos(nt))− n e−it .
Furthermore, using (34), the derivative of F(a,b,n) verifies
2
∂F(a,b,n)
4λ(t)
=1−
(t))
<1,
(α
(a,b,n)
∂z
0
2
λ (t) + (1 + λ(t))2
1
where λ is a function given by λ(t) = (a+b cos(nt))− n −1, with t ∈ [0, 2π],
and thus F(a,b,n) satisfy
∂F(a,b,n) (37)
∂z < 1
on U(a,b,n) . Now we can consider, for each F(a,b,n) , the function TF(a,b,n) :
U(a,b,n) −→C given by
1
TF(a,b,n) (z) = (z + F(a,b,n) (z)) ,
2
which is clearly an immersion. Moreover, on the curve α(a,b,c) we have
γ(t) = TF(a,b,n) (α(a,b,n) (t)) = eit ,
t ∈ [0, 2π] .
Hence it is not difficult to prove, using a topological argument, that TF(a,b,n)
is an embedding on U(a,b,n) . Then, using (35), (37) and a similar argument as
in Theorem 4 of [FMM], one has that the function f(a,b,n) given from F(a,b,n)
484
L. Ferrer et al.
by the equation (11) is a solution of (1). Furthermore, from the expression
(10) we obtain that f(a,b,n) satisfy
(38)
∂f(a,b,n)
1
1
=
−1 + (a + b cos(nt))− n e−it ,
∂w
2
on the curve γ(t) = eit . Then from (38) it is easy to see that f(a,b,n) is constant
on the curve γ which means that f(a,b,n) is a solution of the problem (2).
Moreover, from (26) the ellipse at infinity is given by
1 1 − µ(a, b, n) 2 1 + µ(a, b, n) 2
E(f(a,b,n) )(w) =
x +
x
2 1 + µ(a, b, n) 1 1 − µ(a, b, n) 2
ν(a, b, n)
ν(a, b, n)2
−
x1 +
1 + µ(a, b, n)
4(1 + µ(a, b, n))
and then by using the Proposition 1 we obtain that Mf(a,b,n) is invariant by
the reflection through the plane Σ1 parallel to (0, 1, 0).
Remark 2. We observe that for n ≥ 2 we always have the relation ν(a, b, n)2
= −4 µ(a, b, n). Moreover, if we denote by Fn the set
Fn = {f(a,b,n) | (a, b) ∈ ∆} ,
using the expressions in (36) one can see that F1 , F2 and F3 are all disjoint
and Fn = F3 for n ≥ 4 .
Remark 3. Starting from one function of these families one can contruct an
uniparametric family of solutions of (2) which are invariant by one normal
reflection by means of the rotations given by the expressions (30).
Remark 4. We would like to notice that the only known solutions of the
problem (2) whose graphs are invariant by two normal reflections are surfaces of revolution. Then one wonders whether there exist solutions invariant
by only two normal reflections or not.
Appendix 2: An alternative proof of Corollary 2
Let f and g be solutions of the problem (1) satisfying the conditions of
Corollary 2. Then, from Theorem 2 we have that, up an unimodular affine
transformation, there exists a positive real number P such that for |w| > P
|w|2 a1
−
log(|w|2 ) + O(1) ,
2
4
|w|2 a1
g(w) =
−
log(|w|2 ) + O(1) .
2
4
f (w) =
(39)
Parabolic affine spheres
485
If u = f − g and v = f + g, we can define the functions
Φ = u1 v22 − u2 v12 ,
Ψ = −u1 v12 + u2 v11 ,
where by (·)i we denote ∂(·)
∂xi and by (·)ij we denote
A straight computation from (1) proves
∂ 2 (·)
∂xi ∂xj
for i, j = 1, 2.
∂Φ
∂Ψ
+
=0,
∂x1 ∂x2
and then, if R is a large number, using Stoke’s theorem we obtain,
Z
Z
(40)
u(Φdx2 − Ψ dx1 ) =
(u1 Φ + u2 Ψ )dx1 dx2 ,
∂AR
AR
where AR = {w = (x1 , x2 ) ∈ C | 1 ≤ x21 + x22 ≤ R2 } ∩ Ωγ . Since f and g
are convex functions, we have that the integrand on the right is nonnegative
and it vanishes only when both u1 and u2 vanish. Now we shall prove that the
integrand on the left tends to zero as R tends to infinity. Hence u1 = u2 = 0
and the proof will be concluded.
As u = 0 on ∂Ωγ , the left integrand reduces to the integral on CR =
{(x1 , x2 ) ∈ C | x21 + x22 = R2 }. From (39) we have u is bounded on CR
and ui = O(|w|−2 ). On the other hand, from (39), one can deduce that
vii = 2 + O(|w|−1 ) and vij = O(|w|−1 ) for i, j = 1, 2. Thus we have
Z
Z
u(Φdx
−
Ψ
dx
)
≤
K
sup
k(Φ,
Ψ
)k
ds = 2πK 0 R−1 ,
2
1 CR
CR
CR
with K, K 0 positive real numbers and the proof is finished. 2
References
[A]
[B]
A. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1979.
I. J. Bakelman, Convex analysis and nonlinear geometric elliptic equations,
Springer-Verlag Berlin Heidelberg, 1994.
[C1]
E. Calabi, Complete affine hyperspheres I, Ist. Naz. Alta Mat. Sym. Mat. Vol.
X,19-38 (1972).
[C2]
E. Calabi, Improper affine hyperspheres of convex type and a generalization of a
theorem by K. Jörgens, Mich. Math. J. 5, 108-126 (1958).
[C3]
E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104,
91-126 (1984).
[CY]
S.Y. Cheng, S.T. Yau, Complete affine hypersurfaces, Part.I. The completeness of
affine metrics, Comm. Pure Appl. Math. 39, 839-866 (1986).
[FMM] L. Ferrer, A. Martı́nez, F. Milán, Symmetry and Uniqueness of Parabolic Affine
Spheres, Math. Ann. 305, 311-327 (1996).
[GT]
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order,
2nd edit., Springer Verlag, 1983.
486
[J]
[LR]
[LSZ]
[NS]
[O]
[S]
L. Ferrer et al.
K. Jörgens, Über die Lösungen der Differentialgleichung rt − s2 = 1, Math. Ann.
127, 130-134 (1954).
R. Langevin, H. Rosenberg, A Maximum principle at infinity for minimal surfaces
and applications, Duke Math. J., Vol. 57, No.3, 819-828 (1988).
A.M. Li, U. Simon, Z. Zhao, Global affine differential geometry of hypersurfaces,
Walter de Gruyter, Berlin-New York, 1993.
K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge Univ. Press, 1994.
R. Osserman, A survey of minimal surfaces, Dover Publications, 2nd edition, New
York, 1986.
M. Spivak, Differential Geometry, Vol. IV, Publish or Perish, 1975.