Exterior Monge-Amp`ere Solutions - Mathematics - University of

47 Pages·2006·0.37 MB·English

Exterior Monge-Amp`ere Solutions
D. Burns, N. Levenberg and S. Ma’u
Abstract
We discuss the Siciak-Zaharjuta extremal function of a real convex
body in Cn, a solution of the homogeneous complex Monge-Amp`ere
equation on the exterior of the convex body. We determine several
conditions under which a foliation by holomorphic curves can be
found in the complement of the convex body along which the ex-
tremalfunctionisharmonic.Westudyavariationalproblemforholo-
morphic disks in projective space passing through prescribed points
at infinity. The extremal curves are all complex quadric curves, and
thegeometryofsuchcurvesallowsforthedeterminationoftheleaves
of the foliation by simple geometric criteria. As a byproduct we en-
counter a new invariant of an exterior domain, the Robin indicatrix,
which is in some cases the dual of the Kobayashi indicatrix for a
bounded domain.
1. Introduction.
For a function u of class C2 on a domain Ω ⊂ Cn, the complex Monge-
Amp`ere operator applied to u is
(ddcu)n := i∂∂u∧···∧i∂∂u
∂2u Yn
= 4nn!det[ ] ·(i/2)n dz ∧dz .
j,k=1,...,n j j
∂z ∂z
j k
j=1
Thus if u is plurisubharmonic (psh) and satisfies the homogeneous complex
Monge-Amp`ere equation (ddcu)n = 0 in Ω, then at each point of Ω the
complex Hessian of u has a zero eigenvalue. For a psh function u which
2000 Mathematics Subject Classification 32U15
Keywords: foliation,Kelvintransform,Monge-Amp`ereequation,plurisubharmonic,Robin
indicatrix
First author supported in part by NSF grants DMS-0104047 and DMS-0514070; third
author supported by a New Zealand Science & Technology Post-Doctoral fellowship
D. Burns, N. Levenberg and S. Ma’u
is only locally bounded, (ddcu)n is well-defined as a positive measure (cf.,
[BT76]).
Given a bounded set E ⊂ Cn, the Siciak-Zaharjuta extremal function is
defined as
V (z) := sup{u(z) : u ∈ L(Cn), u 6 0 on E}
E
whereL(Cn)denotestheclassofpshfunctionsuonCn withu(z) 6 log+|z|+
c(u), for some real constant c(u). If E is compact, this coincides with
1
sup{ log|p(z)| : degp > 0, ||p|| := sup|p(z)| 6 1}.
E
degp
z∈E
If E is non-pluripolar, the function V∗(z) := limsup V (ζ) is a locally
E ζ→z E
bounded psh function which satisfies (ddcV∗)N = 0 outside of E. The rele-
E
vantsituationsinthispaperoccurforE = D,Dbounded,smoothlybounded
and strictly lineally convex in Cn, and for E = K ⊂ Rn a convex body,
that is, a compact, convex set with non-empty interior. In the former case,
Cn \ D is foliated by holomorphic curves f(C \ 4) (4 denoting the unit
disk) on which the Monge-Amp`ere solution V is harmonic [Mom96], and
D
V is smooth on all of Cn \ D. In the latter situation the function V is
D K
continuous but it is not smooth. For a symmetric convex body (K = −K)
it was shown ([Lun85], [Bar92]) that there exists a continuous foliation of
Cn\K by analytic disks on which V is harmonic.
K
If K is not necessarily symmetric, the existence, through each point in
Cn \ K, of an analytic disk on which V is harmonic was demonstrated
K
in [BLM05]. This was acheived by using a decreasing sequence of strictly
convex σ-invariant (σ(z) := z) open sets D such that ∩D = K, and a
j j
normal families argument on a sequence of foliation curves f (C \ 4) for
j
Cn\D .
j
In this paper we return to the matters discussed in [BLM05] to examine
the properties of the Monge-Amp`ere solution more closely. In particular,
our goal is to clarify the situation for exterior Monge-Amp`ere solutions V
K
associated to general real convex bodies. We use a variational description of
the limit disks introduced in [Bur84] to base the study of the curves on their
behavior at infinity in projective space CPn containing Cn. We give criteria
for these curves to give a foliation of Cn \K which include the case when
the variational problem has a unique extremal curve. When these extremal
curves are not unique, we can sometimes retrieve a continuous foliation of
CPn\K by extremals, and even make a canonical choice of this foliation.
We begin in sections 2 and 3 by verifying that for a bounded, smoothly
boundedandstrictlylineallyconvexdomainD containingtheorigin,andits
2
Exterior Monge-Ampe`re Solutions
dual,D0,theKelvintransformK : D0\{0} → Cn\DdescribedbyLempertin
[Lem81]andutilizedin[BLM05]extendstoadiffeomorphismKb : Db0 → CPn\
D,whereDb0 istheblow-upofD0 attheorigin.Thisallowsustoparametrize
theleavesofthefoliationofCn\D bythehyperplaneatinfinity,H ,andto
∞
set up a variational problem in section 4 for extremal disks f , [c] ∈ H , in
c ∞
thespiritof[Lem81],anddualinasensetotheKobayashi-Roydenfunctional
for the infinitesimal Kobayashi metric. In section 3, a key role is played by
the second variation of the Kobayashi-Royden functional, already examined
in [Bur84]. Using this technique, we note in passing that the Kobayashi
indicatrix I (D0) ⊂ T0(D0) of a strictly lineally convex D0 ⊂⊂ Cn is itself
0 0
strictly convex (Corollary 3.2). This indicatrix is, in some sense, the best
circledapproximationofD0(see[Lem88]).Indirectanalogytothis,insection
4 we note that the individual extremal disks in our exterior problem may be
packaged into an exponential map from an indicatrix, the Robin indicatrix,
defined in the holomorphic normal bundle to H , to the complement CPn\
∞
D.Insections5and6wepasstothelimit∩D = K,asin[BLM05],anduse
j
the special form of the analytic disks in this case – they are complex ellipses
– to give a simple but useful geometric interpretation to the extremals of
the variational problem. Section 7 concerns a geometric criterion on K for
uniqueness of extremal curves passing through a given [c] ∈ H . In section
∞
8, we verify that if extremal curves f are unique for a real convex body K,
c
then we obtain a continuous foliation as in the symmetric case. This occurs,
for example, if K is strictly convex, but the general condition is somewhat
weaker than this. This foliation property depends on the geometry of the
real ellipses in K given by the intersection of the complex extremal curves
with Rn. In section 9 we show that in R2 for an arbitrary convex body we
maygetmanyfoliationsofthecomplementofK byconstructingappropriate
approximations of K by convex bodies K with unique extremals. Then, in
j
section 10, we construct one such foliation, albeit by different means, which
is, in a natural way, a “canonical” foliation. In fact, the set of extremals
through a given [c] ∈ H is parametrized by a convex closed subset of K,
∞
and the canonical curve f is the one parametrized by the barycenter of this
c
set. In the case where K = −K, this foliation reduces to that of [Lun85] and
[Bar92]. Finally, in section 11 we relate the Robin indicatrix of an arbitrary
convex body in Rn to that of a naturally associated symmetric one.
The present work is limited to domains K ⊂ R2 in sections 9 and 10 due
toatopologicalargument(continuityofintersection)weusetoshowthatour
family of extremal disks is actually a foliation (cf., the proof of Proposition
9.1). It is not clear that this is an essential obstruction, and this should be
3
D. Burns, N. Levenberg and S. Ma’u
clarified.Wehopetodiscusstheregularityofthefoliationdescribedhereand
the corresponding regularity of the Monge-Amp`ere solution V in a future
K
paper. The regularity depends on a more detailed study of the behavior of
the real ellipses in K used in this paper. Moreover, we hope to define certain
invariants associated to a convex body in Rn and its Robin indicatrix `a la
Lempert [Lem88].
2. The Behavior of the Kelvin Transform at the origin in D0.
We are given a domain D, bounded, smoothly bounded and strictly lineally
convexinCn,containingtheorigin.Thismeansforeacha ∈ ∂D,thecomplex
tangent hyperplane
n
TC(∂D) := {ζ = (ζ ,...,ζ ) ∈ Cn : X(ζ −a ) ∂ρ (a) = 0},
a 1 n j j ∂z
j
j=1
where ρ ∈ C2(D) is a defining function for D, satisfies TC(∂D)∩D = {a},
a
C
and ρ restricted to T (∂D) has a non-degenerate minimum at a. We want
a
to consider the Monge-Amp`ere solution V on the exterior of D ⊂⊂ Cn.
D
According to Lempert [Lem85], there is a domain
n
X
D0 := {z = (z ,...,z ) ∈ Cn : z p 6= 1 for all p = (p ,...,p ) ∈ D}
1 n j j 1 n
j=1
containing the origin dual to D, so that D0 is smoothly bounded and strictly
lineally convex in Cn. By [Lem81], D0 admits a plurisubharmonic Green’s
function
G(z) = G (z)
D0
:= sup{u(z) : u psh in D0, u 6 0, u(z)−log|z| = 0(1), z → 0}
for z ∈ D0 which is smooth in D0\{0} and has a logarithmic singularity at
z = 0. The Kelvin transform is a diffeomorphism
K : D0\{0} → Cn\D,
given by the formula
(G (z),...,G (z))
K(z) = z1 zn , (2.1)
z G (z)+...+z G (z)
1 z1 n zn
where G (z) = ∂G(z), j = 1,...,n. Lempert’s formula for V is
zj ∂zj D
V (K(z)) = −G(z). (2.2)
D
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Exterior Monge-Ampe`re Solutions
Reference [Lem81] provides detailed information about the nature of the
singularityofG(z)atz = 0,andinordertobeabletousethistounderstand
V better at infinity, we will have to examine the behavior of the Kelvin
D
transformnearz = 0.ItisusefultoreplaceD0 byDb0,theblow-upofD0 atthe
origin. Let π be the projection π : Db0 → D0, and E = π−1(0) the exceptional
divisor. Similarly it is useful to add the divisor at infinity H = CPn\Cn to
∞
thedomainCn\D,replacingitbyCPn\D.LetKb : Db0\E → Cn\D ⊂ CPn\D
be the diffeomorphism given by Kb = K◦π. The exact statement we want is
the following.
Theorem 2.1. The map Kb extends to a smooth diffeomorphism, also de-
noted Kb,
Kb : Db0 → CPn\D.
Proof. We first show that Kb extends smoothly across E, sending E to H∞.
That the resulting map is a diffeomorphism is more delicate. The problem
will be set up in this section, but completed only in the next.
We first have to recall some standard facts about coordinates on Db0 and
on CPn. First of all, note that the exceptional divisor E is identified with
P(T0D0), the projective space of holomorphic tangent directions at 0 ∈ D0.
0
We can assume without loss of generality that we are trying to extend Kb
smoothly across the point in E corresponding to the direction ∂ ∈ T0D0.
∂z1 0
Then there exist canonical coordinates (z1,η2,...,ηn) locally on Db0, so that
[ ∂ ] = 0, and the mapping π is given by
∂z1
π∗z = z
1 1
π∗z = z ·η , j = 2,...,n.
j 1 j
It follows from [Lem81] that
(π∗G)(z ,η) = log|z |+H(z ,η), (2.3)
1 1 1
whereη = (η ,...,η ),andH(z ,η)extendssmoothlyacrossE.Tocalculate
2 n 1
Kb locally at 0, we note that
π∗dz = dz
1 1
π∗dz = η dz +z dη , j = 2,...,n,
j j 1 1 j
from which we conclude immediately
∂ = dπ ( ∂ − 1 ·Pn η ∂ )
∂z1 ∗ ∂z1 z1 j=2 j∂ηj
∂ = dπ ( 1 ∂ ), j = 2,...,n.
∂zj ∗ z1∂ηj
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D. Burns, N. Levenberg and S. Ma’u
In turn, we conclude that
∂G 1 ∂H 1 ∂H ∂H
π∗ = + − (η +...+η ),
2 n
∂z 2z ∂z z ∂η ∂η
1 1 1 1 2 n
and
∂G 1 ∂H
π∗ = , j = 2,...,n.
∂z z ∂η
j 1 j
As to the denominator of Kb, we obtain
∂G ∂G ∂H
π∗(z +...+z ) = 1/2+z .
1 n 1
∂z ∂z ∂z
1 n 1
Gathering terms, we get
1/2+z ∂H −Pn η ∂H
Kb∗z1 = z11( 11/∂2z1+z1∂∂zHj1=2 j∂ηj ),
and
∂H
1
Kb∗zj = z1(1/2+∂ηzj1∂∂zH1), j = 2,...,n.
To examine the limit behavior of Kb as z1 → 0, i.e., near E, we need affine
coordinates in a neighborhood of the hyperplane H . It will suffice to look
∞
at
1
w = ,
1
z
1
and
z
j
w = , j = 2,...,n.
j
z
1
In these coordinates, H is given by w = 0. Substituting these into the
∞ 1
formulas above gives us
1/2+z ∂H
Kb∗w1 = z11/2+z ∂H −P1∂zn1 η ∂H ,
1∂z1 j=2 j∂ηj
and
∂H
Kb∗wj = 1/2+z ∂H∂−ηjPn η ∂H , j = 2,...,n.
1∂z1 j=2 j∂ηj
From this it is clear that Kb extends smoothly across E near z1 = 0, η = 0,
sending E to H and sending specifically (z ,η) = (0,0) to the point w =
∞ 1
(0,2∂H,...,2∂H) in our coordinates (w ,...,w ). More precisely, we will
∂η2 ∂ηn 1 n
see in the next section that (z ,η) = (0,0) gets sent to the point (0,...,0).
1
6
Exterior Monge-Ampe`re Solutions
To complete the proof of Theorem 2.1, it will suffice to show that the
differentialdKb∗ isnon-singularat(z1,η) = (0,0).ThisisbecauseH∞ iscom-
pact and simply-connected, and so Kb would induce a covering and therefore
a diffeomorphism of E to H , and therefore of a neighborhood of E to a
∞
neighborhood of H . The diffeomorphism property away from E was al-
∞
ready shown in [Lem85].
Let us now calculate the (real) differential dKb∗ at (z1,η) = (0,0).
 
∂w1 ∂w1 ... ∂w1 ∂w1 ∂w1 ... ∂w1
∂z1 ∂η2 ∂ηn ∂z1 ∂η2 ∂ηn
 · · · · · · · · 
 
 ∂wn ∂wn ... ∂wn ∂wn ∂wn ... ∂wn 
 ∂z1 ∂η2 ∂ηn ∂z1 ∂η2 ∂ηn 
 
dKb∗(0,0) =  .
 
 
 ∂w1 ∂w1 ... ∂w1 ∂w1 ∂w1 ... ∂w1 
 ∂z1 ∂η2 ∂ηn ∂z1 ∂η2 ∂ηn 
 · · · · · · · · 
 
∂wn ∂wn ... ∂wn ∂wn ∂wn ... ∂wn
∂z1 ∂η2 ∂ηn ∂z1 ∂η2 ∂ηn
At (0,0),
∂w ∂w ∂w ∂w
1 1 1 1
= 1, = = = 0, j = 2,...,n.
∂z ∂z ∂η ∂η
1 1 j j
Using elementary row and column operations, it suffices, therefore, to show
that the matrix
 ∂wi ∂wi 
∂ηj ∂ηj
 
A =  , i,j = 2,...,n,
 
 
∂wi ∂wi
∂ηj ∂ηj
is non-singular at (0,0). Since
∂wi (cid:2) ∂2H ∂H ∂H(cid:3)
(0,0) = 2 +2 ,
∂η ∂η ∂η ∂η ∂η
j i j i j
and
∂w ∂2H
i
(0,0) = 2 ,
∂η ∂η ∂η
j i j
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D. Burns, N. Levenberg and S. Ma’u
and the function H is real-valued, we obtain
 2(cid:2) ∂2H +2∂H ∂H(cid:3) 2 ∂2H 
∂ηi∂ηj ∂ηi∂ηj ∂ηi∂ηj
 
A =  , i,j = 2,...,n.
 
 
2 ∂2H 2(cid:2) ∂2H +2∂H ∂H(cid:3)
∂ηj∂ηi ∂ηi∂ηj ∂ηi∂ηj
In the next section we will give a geometric interpretation of the matrix A,
and show that for the D0 under consideration here it is non-singular.
3. Stability and Convexity of the Kobayashi Indicatrix.
In this section we will recall the interpretation of the function H(0,η), and
showthenon-singularityofthematrixAisequivalenttothelinealconvexity
of the Kobayashi indicatrix at z = 0 ∈ D0, which is implied by the stability
of the Kobayashi-Royden functional at 0.
First of all, let f (ζ) := f(ζ;η) be the Kobayashi-Royden extremal disk
η
f : 4 → D0, with f (0) = 0 ∈ D0 and with
η η
f0(0) = λ ·(1,η) = λ(f )·(1,η)
η (1,η) η
where λ = λ(f ) is maximal over
(1,η) η
{λ > 0|there isfholomorphic, f : 4 → D0, f(0) = 0, f0(0) = λ(1,η)}.
(3.1)
Let fbη be the proper transform of fη, i.e., π◦fbη = fη, so that fbη(0) = (0,η),
and fbη0(0) = λ(1,η)∂∂z1 +vη, where vη is tangent to E at (0,η). It follows from
[Lem81] that
(f∗G)(ζ) = log|ζ| for all η, (3.2)
η
and that f(ζ;η) is smooth in (ζ,η). Lifting equation (3.2) to Db0, we get,
using (2.3),
log|ζ| = fbη∗◦π∗G = log|fbη,1|+fbη∗H,
and evaluating at ζ = 0 gives
H(0,η) = −logλ . (3.3)
(1,η)
But then, by (3.1) and the definition of the infinitesimal Kobayashi metric
K on T0(D0), we have K (1,η) = λ−1 so that
0 0 0 (1,η)
eH(0,η) = K (1,η).
0
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Exterior Monge-Ampe`re Solutions
We will show that η = 0 is a critical point for K restricted to this complex
0
hyperplane through (1,0), and that its real Hessian is positive definite there.
Theorem 3.1. a) For j = 2,...,n, ∂H(0,0) = 0.
∂ηj
b) For a = (a ,...,a ) ∈ Cn−1, a 6= 0, we have
2 n
d2 X ∂2H ∂2H
H(0,ta)| = 2Re{ (0,0)a a + (0,0)a a } > 0.
dt2 t=0 ∂η ∂η i j ∂η ∂η i j
26i,j6n i j i j
Corollary 3.1. The matrix A above is non-singular.
As a result of Theorem 3.1, we see that the hyperplane {(1,η)|η ∈ Cn−1}
is tangent to the level set {v : K (v) = K (1,0) = eH(0,0)}, and that K
0 0 0
restricted to that hyperplane is strictly convex near the point (1,0). Recall
that the Kobayashi indicatrix I (D0) := {v ∈ T0(D0)|K (v) < 1}. Thus we
0 0 0
see that I (D0) is strictly lineally convex. Moreover, since K (tv) = |t|K (v),
0 0 0
we have the following corollary (cf., [Lem88], Remark 16.2).
Corollary 3.2. The Kobayashi indicatrix I (D0) is strictly convex.
0
We turn to the proof of Theorem 3.1.
Proof. We write O(4) = O(4,D0) for the holomorphic maps f : 4 → D0.
Define first, for k an integer > 4,0 < α < 1, Hk,α (D0) ⊂ Ck,α(4)∩O(4)
(0,(1,η))
by
Hk,α (D0) = {f|f(∂4) ⊂ ∂D0, f(0) = 0, f0(0) = λ(1,η), λ = λ(f) > 0}.
(0,(1,η))
Following [Lem81], in a neighborhood of f (4) we may find holomorphic
0
coordinates w = (w ,...,w ) and a defining function r for D0 such that
1 n
f (ζ) = f(ζ;0) = (ζ,0,...,0), ζ ∈ 4, and for ζ ∈ ∂4 we have:
0
i) r (f (ζ)) = ζ, r (f (ζ)) = 0, j = 2,...,n
w1 0 wj 0
ii) r (f (ζ)) = 0, j = 1,...,n
w1,wj 0
iii) r (f (ζ)) = δ (3.4)
wi,wj 0 i,j
iv) fora = (a ,...,a ) ∈ Cn−1, a 6= 0,
2 n
Pn Re{r (f (ζ))a a +r (f (ζ))a a } > 0.
i,j=2 wi,wj 0 i j wi,wj 0 i j
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D. Burns, N. Levenberg and S. Ma’u
We can now write, for f = (f ,...,f ) ∈ Hk,α (D0),
1 n (0,(1,η))
1 Z f (ζ) 1 Z 2π f (ζ)
1 1
λ = λ(f) = dζ = dθ.
2πi ζ2 2π ζ
∂4 0
Since λ is real, we can write this
1 Z 2π f (ζ) 1 Z 2π f (ζ)
1 1
λ(f) = Re{ dθ} = Re{ }dθ.
2π ζ 2π ζ
0 0
Let f(ζ;η) be the Kobayashi-Royden extremal disk for η, and let
d
f˙= f˙(ζ;a) = f(ζ;ta)|
t=0
dt
bethecorrespondingvariationvector.Asin[Pol83],wedifferentiateatt = 0
to get
d 1 Z 2π f˙ (ζ;a)
1
λ(f(ζ;ta))| = Re{ }dθ.
t=0
dt 2π ζ
0
Todeterminef˙(ζ;a),wedifferentiater(f(ζ;ta)) ≡ 0,att = 0andallζ ∈ ∂4
and use i):
0 = Re{Xn r (f (ζ))·f˙ (ζ;a)} = Re ζf˙ (ζ;a) = Re f˙1(ζ;a),
wj 0 j 1 ζ
j=1
from which we conclude dλ(1,ta)| = 0. This says that
dt t=0
n
X ∂λ
(0)a = 0
j
∂η
j
j=2
for all a; taking successively a = (1,...,0), a = (0,1,...,0),..., and a =
(0,...,1), we conclude that ∂λ(0) = 0 for j = 2,...,n. Using H(0,η) =
∂ηj
−logλ , this proves Theorem 3.1, part a).
(1,η)
As to part b) of the theorem, we have
d2 1 Z 2π f¨(ζ;a)
1
λ | = Re{ }dθ.
dt2 (1,ta) t=0 2π ζ
0
Differentiatingr(f(ζ;ta)) ≡ 0twiceatt = 0andusingii)andiii),weobtain
n
0 = Re{ζf¨(ζ;a)+2 X(cid:2)r (f (ζ))·[f˙(ζ;a)f˙ (ζ;a)]
1 wi,wj 0 i j
i,j=2
+r (f (ζ))·[f˙(ζ;a)f˙ (ζ;a)](cid:3)}+|f˙ (ζ;a)|2,
wi,wj 0 i j 1
10