Indiana University Mathematics Journal c
Article electronically published on June 10th, 2009
Constant Rank Theorem in Complex Variables
Q UN L I
A BSTRACT. We establish a constant rank theorem for elementary symmetric functions in terms of complex Hessian matrix in
complex domains and complex manifolds. We also give some
application and discussion on it.
1. I NTRODUCTION
In the study of complex analysis and complex geometry, the class of plurisubharmonic functions plays a crucial role. For example, while studying the Kähler-Ricci
flow, a typical question is whether under the flow, the plurisubharmonicity will be
preserved or not, this relates to complex Laplacian operators, which actually is the
first order elementary symmetric function σ1 of some complex Hessian matrix. In
Kähler geometry, the celebrated work of Yau [35] on the Calabi conjecture opened
a vast field for the study of complex Monge-Ampère equation, which relates to the
function σn in Hessian matrix. Those are two extreme cases of k-th elementary
symmetric functions. For the intermediate cases, i.e., the k-th elementary symmetric functions, for 2 ≤ k ≤ n − 1, there are also many important illustrations.
In [4], Bedford-Kalka consider some nonlinear partial differential equation
involving the complex Hessian matrix (ui̄ ) on a complex manifold,
¯ k+1 = ∂ ∂u
¯ ∧ · · · ∧ ∂ ∂u
¯ = 0.
(∂ ∂u)
¯ with the complex Hessian matrix, the above
By identifying the (1, 1)-form ∂ ∂u
equation is equivalent to
σk+1 (ui̄ ) = 0,
or in other words: rank(ui̄ ) ≤ k. Their result is the following, provided that
u ∈ C 3 and the complex Hessian of u is of constant rank k, then there is a
1228
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foliation F n−k by complex manifolds of dimension n − k which has the property:
u is harmonic and ∂u/∂zj is holomorphic on each leaf M of F .
The existence of a foliation has been useful in the study of homogeneous complex Monge-Ampère equations and in the characterization of parabolic manifolds
as well. So, this posts a natural question: under what condition the complex
Hessian matrix (ui̄ ) is of constant rank, for u satisfies certain class of partial
differential equations? For example, if u is the solution to
σk (ui̄ ) = f ,
where σk denotes the k-th elementary symmetric function. For a complex Hessian
matrix (ui̄ ) with u at least C 2 , write its eigenvalue as λ1 , . . . , λn , then the k-th
elementary symmetric function of (ui̄ ) is,
σk (ui̄ ) =
X
λi1 · · · λik ,
where the sum is taken over all strictly increasing sequences i1 , . . . , ik of the indices
from {1, . . . , n}. Notice that
σn (ui̄ ) = λ1 · · · λn = det(ui̄ ),
and
σ1 (ui̄ ) = tr(ui̄ ) = ∆u,
are in the C 2 case complex Monge-Ampère operator and complex Laplace operator
respectively. Those two operators play an important role in both the theory of
complex analysis and complex geometry from the differential equations point of
view.
There have been tremendous works regarding those two operators in complex
variables, especially the study of complex Monge-Ampère equations has been an
active field for past decades. The foundation for the definition of complex MongeAmpère operator on general functions was laid by Chern-Levine-Nirenberg [18].
Later Bedford and Taylor did pioneering work on the complex Monge-Ampère
operators in a series of papers, [5], [6], [7], [8], [9]. Caffarelli, Kohn, Nirenberg
and Spruck [16] established fundamental regularity results of solutions to complex
Monge-Ampère equations. The complex Monge-Ampère equation is also investigated in connection with Kähler geometry. Yau’s fantastic work on it solves the
famous Calabi conjecture. See [1] and [27] for more references.
As to the study on intermediate Hessian equations, there are also some important results. The existence of a unique classical solution for the Dirichlet problem
of symmetric function of the eigenvalues of real Hessian matrix of a function on a
domain in Rn was proved by Caffarelli, Nirenberg and Spruck in [17]. Guan-Ma
[25] considered Hessian equation on Sn and provided a convex solution which
in turn solved the famous Christoffel-Minkowski problem of prescribing a convex
hypersurface with the k-th symmetric function of the principle radii.
Constant Rank Theorem in Complex Variables
1229
In the direction of complex Hessian equations, Blocki [12] studied the weak
solutions to the complex Hessian equations, and S.Li [33] established the existence
of admissible solutions to the complex Dirichlet problem for the general k-th
elementary symmetric functions in terms of complex Hessian in complex domains.
In this paper, we obtain a constant rank theorem for the k-th elementary
symmetric functions in terms of complex Hessian matrices. √
Let z1 , . . . , zn be complex coordinates in Cn , zj = xj + −1yj . For any C ∞
function u, we define
1
∂u
uj =
=
∂zj
2
u̄ =
1
∂u
=
¯
∂ zj
2
uj k̄ =
∂2u
.
∂zj ∂zk̄
!
p
∂u
∂u
,
− −1
∂xj
∂yj
!
p
∂u
∂u
,
+ −1
∂xj
∂yj
A real-valued smooth function u is called plurisubharmonic if (ui̄ ) is nonnegative definite.
Our result is the following.
Theorem 1.1. For any u that is a C 3 plurisubharmonic solution of the following
equation in a domain Ω of Cn ,
(1.1)
σk (ui̄ (z)) = f (z),
∀ z ∈ Ω,
if − log f is plurisubharmonic, then the complex Hessian U = (ui̄ ) is of constant
rank in Ω.
Notice that the natural class of solutions to (1.1) is not the plurisubharmonic
functions, but the so-called admissible solutions.
Let H = {all n × n Hermitian matrices}, for 1 ≤ k ≤ n, let Hk be the
connected cone in H ,
Hk = {W ∈ H | σ1 (W ) > 0, . . . , σk (W ) > 0}.
On a compact Kähler manifold M , if u is a C 2 -solution to the following equation,
σk (ui̄ + gi̄ ) = f (z),
in M.
we say u is an admissible solution if W = (ui̄ + gi̄ ) lies in Hk on M .
In geometric problems the class of plurisubharmonic solutions is desirable in
most cases.
For instance, recently Fu-Yau [20] established a beautiful result on the study
of the superstring theory with flux on non-Kähler manifolds, the key step for them
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is to find a reasonable Hermitian metric which expresses the geometry. Their geometric setting translates in the PDE sense to the solving of the following equation,
σ2 (eu gi̄ + tαe−u ρi̄ + 4αui̄ ) = f (u, |∇u|),
under the elliptic condition
(gi0̄ ) = (eu gi̄ + tαe−u ρi̄ + 4αui̄ ) > 0.
In other words, they need the solution to be strictly plurisubharmonic. Fu-Yau
solved the problem in the two-dimensional case, where σ2 is just the determinant of the positive definite matrix. In higher dimensional case, the equation is
no longer in terms of the determinant of some positive definite matrix, but the
expression of some elementary symmetric function. Under this circumstance, the
obtained solution is an admissible one, our constant rank theorem can be applied
to produce it to be plurisubharmonic. We will illustrate how constructural conditions could help reduce the admissible solution to a plurisubharmonic one in the
last section.
Besides serving as a model of producing a plurisubharmonic solution to certain geometric problems, our constant rank result can also be applied to the study
of extremal metrics in Kähler geometry, which is an important and active topic
in Kähler geometry. Kähler-Einsein metrics, Kähler metrics with constant scalar
curvature and other forms of elementary symmetric functions associated to the
Kähler Ricci curvature tensor are examples of this type of metrics. In many cases,
the extremal metrics are being solutions to certain fully nonlinear partial differential equations, in a joint paper with P. Guan and X. Zhang [24], we set up a
uniqueness result for Kähler-Einstein metrics.
The organization of the paper is as following, we’ll first list some preliminary
knowledge in elementary symmetric function theory and Kähler geometry, following with the proof of the main theorem. The last section is devoted to the
presentation of the plurisubharmonic functions and further discussion and possible applications as well.
2. P RELIMINARIES
We recall the definition of k-symmetric functions: For 1 ≤ k ≤ n, and λ =
(λ1 , . . . , λn ) ∈ Cn ,
X
σk (λ) =
λi1 · · · λik ,
where the sum is taken over all strictly increasing sequences i1 , . . . , ik of the indices
from the set {1, . . . , n}. The definition can be extended to symmetric matrices by
letting σk (W ) = σk (λ(W )), where λ(W ) = (λ1 (W ), . . . , λn (W )) are the eigenvalues of the symmetric matrix W . We also set σ0 = 1 and σk = 0 for k > n.
Constant Rank Theorem in Complex Variables
1231
Proposition 2.1. If W = (Wij ) is an n × n symmetric matrix, let F (W ) =
σk (W ) for 1 ≤ k ≤ n. Then the following relations hold:
σk (W ) =
n
1 X
δ(i1 , . . . , ik ; j1 , . . . , jk )Wi1 j1 · · · Wik jk ,
k! i ,...,i =1
1
k
j1 ,...,jk =1
F αβ B
=
F ij,r s B
=
∂F
(W )
∂Wαβ
n
X
1
(k − 1)!
δ(α, i1 , . . . , ik − 1; β, j1 , . . . , jk − 1)Wi1 j1 · · · Wik−1 jk−1 ,
i1 ,...,ik−1 =1
j1 ,...,jk−1 =1
∂2F
(W )
∂Wij ∂Wr s
n
X
1
(k − 2)!
δ(i, r , i1 , . . . , ik − 2; j, s, j1 , . . . , jk − 2)Wi1 j1 · · · Wik−2 jk−2 ,
i1 ,...,ik −2=1
j1 ,...,jk −2=1
where the Kronecker symbol δ(I, J) for indices I = (i1 , . . . , im ) and J = (j1 , . . . , jm )
is defined as



1,



δ(I, J) = −1,






0,
if I is an even permutation of J ;
if I is an odd permutation of J ;
otherwise.
We will need two lemmas in the following sections.
Lemma 2.2. For 1 ≤ k ≤ `, G = (λ1 , . . . , λ` ), 1 ≤ i, j ≤ `, i ≠ j , we denote
by σk (G|i) the symmetric function with λi = 0 and σk (G|ij) the symmetric function
with λi = λj = 0. Then the following hold:
(2.1)
σk (G)σ`−1 (G|α) − σk−1 (G|α)σ` (G) = σk (G|α)σ`−1 (G|α).
If 1 ≤ k ≤ `, and α ≠ β,
(2.2)
σk−1 (G|α)σk−1 (G|β) − σk (G)σk−2 (G|αβ)
2
= σk−
1 (G|αβ) − σk (G|αβ)σk−2 (G|αβ).
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1232
Lemma 2.3. For 1 ≤ k ≤ `, G = (λ1 , . . . , λ` ) and with λi ≥ 0, for 1 ≤ i ≤ `,
∀ α ≠ β and for all real numbers γ1 , . . . , γ` ,
(2.3)
X
σk (G|α)σ`−1 (G|α)σk−1 (G|α)γα2
α∈G
≥ σ` (G)
X
2
(σk−
1 (G|αβ) − σk (G|αβ)σk−2 (G|αβ))γα γβ .
α≠β
The proofs of the above lemmas can be found in [25].
Next we list some preliminary knowledge on Kähler geometry. Let (M, g, J)
be a compact Kähler manifold with the Kähler metric g . In local coordinate, its
Kähler form
p
ω = −1gi̄ dzi ∧ dz̄j
is a closed real (1, 1)-form, this is the same as saying that
∂gk̄
∂gi̄
=
∂zk
∂zi
∂gik̄
∂gi̄
=
∂ z̄k
∂ z̄j
and
for all i, j , k. The Kähler class of ω is the cohomology class [ω] in H 2 (M, R).
By the Hodge theory, any other Kähler metrics in the same class is of the form
ωϕ = ω +
p
¯
−1∂ ∂ϕ
for some real-valued function ϕ on M . The Christoffel symbols of the metric gi̄
are given by
¯ ∂gi`¯
k
Γij
= g k`
∂zj
and
Γı̄k̄̄ = g `k̄
∂g`ı̄
∂ z̄j
where (g i̄ ) = (gi̄ )−1 . It is easy to see that Γijk is symmetric in i and j and Γı̄k̄̄ is
symmetric in ı̄ and ̄. The curvature tensor of the metric gi̄ is defined as
Ri̄k`¯ = gm̄ R m ¯ = −
ik`
∂ 2 gi̄
∂zk ∂ z̄`
+ gmn̄
∂gin̄ ∂gm̄
.
∂zk ∂ z̄`
We see that Ri̄k`¯ is symmetric in i and k, in ̄ and `¯ and in the pairs {i̄} and
¯ .
{k`}
We say (M, g, J) has positive holomorphic bisectional curvature if
¯
Ri̄k`¯v i v ̄ w k w ` > 0
for all nonzero vectors v and w in the holomorphic tangent bundle of M .
Constant Rank Theorem in Complex Variables
1233
Given any (1, 1) tensor Ti̄ , its covariant derivatives are defined as
(2.4)
Ti̄,k =
∂
`
Ti̄ − Γik
T`̄ ,
∂zk
(2.5)
Ti̄,k̄ =
¯
∂
Ti̄ − Γ̄`k̄ Ti`¯ .
∂ z̄k
We shall need the following commutation formulas for covariant derivatives
Ti̄,k` = Ti̄,`k ,
Ti̄,k̄`¯ = Ti̄,`¯k̄ ,
mn̄
Rin̄k`¯ Tm̄ − g mn̄ Rm̄k`¯Tin̄ .
Ti̄,k`¯ = Ti̄,`k
¯ +g
Definition. We say that, on a compact Kähler manifold, the orthogonal bisectional curvature is bounded from below by r means that Riı̄j ̄ ≥ r for any
i ≠ j.
3. P ROOF OF THE T HEOREM
In this section, we will present the proof of Theorem 1.1. The basic idea is to apply
the strong minimum principle to some auxiliary function of complex Hessian
matrix.
Write U = (ui̄ ), suppose U attains minimal rank ` at z0 ∈ Ω, pick an open
neighborhood O of z0 , set
ϕ(z) = σ`+1 (U).
(3.1)
Write F = σk (U), and let
F αβ̄ =
∂σk (U)
.
∂uαβ̄
We first prove the following deformation argument.
Proposition 3.1. Assume U attains minimal rank ` at z0 , and O is an open
neighborhood of z0 ; then there exist constants c1 , c2 , depending only on u, f , n, and
k, such that,
(3.2)
X
F αβ̄ ϕαβ̄ ≤ f (z)σ` (z)
X
(log f )iı̄ + c1 |∇ϕ| + c2 ϕ.
i
in O .
Once Proposition 3.1 is proved, the main theorem follows immediately. Actually, assuming (3.2) holds, by the assumption that − log f is plurisubharmonic,
we derive the following differential inequality,
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1234
X
F αβ̄ ϕαβ̄ ≤ c1 |∇ϕ| + c2 ϕ.
Since u is a plurisubharmonic solution to (1.1), (F αβ̄ ) is strictly positive definite, the strong minimum principle then implies that
ϕ(z) = σ`+1 (z) ≡ 0,
which means U = (ui̄ ) has constant rank `.
In the rest we only need to prove Proposition 3.1. The proof is technically complicated; while the main task is to calculate the contraction summation
P αβ̄
F ϕαβ̄ , thanks to some algebraic properties of elementary symmetric funcP
P
tions, we could compare the quantities F αβ̄ ϕαβ̄ and f σ` (log f )i̄ and obtain
the differential inequality (3.2).
This deformation idea was first studied by Caffarelli-Friedman [14], KorevaarLewis [28] and Singer-Wong-Yau-Yau [34] for semilinear equations in real domains. Later Guan-Ma [25] established a fully nonlinear version when they constructed the convex solution to Christoffel-Minkowski problem on Sn . Recently
Caffarelli-Guan-Ma [15] constructed a general convexity principle for solutions
to a certain class of nonlinear elliptic equations in terms of real Hessians. Our
argument is a complex correspondent to Guan-Ma’s result.
Consider the following equation in a real domain Ω ⊂ Rn ,
σk (uij (x)) = f (x);
by [25], the condition for (uij ) to have constant rank is −f −1/k is a concave
function. In other words, if f ∈ C 2 , {(f −1/k )ij } is non-negative definite.
The above condition on f is natural in the sense that it can be regarded as
the dual condition for certain class of elliptic operators to be concave, which is the
usual technical requirement in classical elliptic PDE theory.
While our condition is different from the one in real problems, the main
difference between them comes from the fact that when considering third order
derivatives of the solution, in the real case, uijk is symmetric with respect to the
lower indices {ijk}; while in complex variables, ui̄k is symmetric with respect to
indices {ik} only. In other words, we could only commute indices with or without
bar at the same time, which causes to “lose” half useful quantities as in real case.
To explain this “loss” from the analysis point of view: as we know, the plurisubharmonic functions are in many ways analogous to convex functions. All convex
functions are plurisubharmonic, so the latter class is a generalization of the former. Nevertheless there are also big differences between them. Lelong [29] proved
that a plurisubharmonic function which is independent of the imaginary parts
of the variables is a convex one, in other words, if f is plurisubharmonic, and
f (z) = f (x + iy) = g(x) for x in a real convex domain in Rn and for all y in
Rn , then f is convex.
Constant Rank Theorem in Complex Variables
1235
The rest of this section is devoted to the proof of Proposition 3.1. For any
z ∈ O , let λ1 ≤ λ2 ≤ · · · ≤ λn be the eigenvalues of U at z. There is a
positive constant C > 0 such that λn ≥ λn−1 ≥ · · · ≥ λn−`+1 ≥ C . Let G =
{n − ` + 1, n − ` + 2, . . . , n} and B = {1, 2, . . . , n − `} be the “good” and “bad”
sets of indices respectively. Let ΛG = (λn−`+1 , . . . , λn ) be the “good” eigenvalues
of U at z, for simplicity, we also write G = ΛG if there is no confusion.
For two functions defined in an open set O ⊂ M , z ∈ O , we say that h(z) Ü
k(z) provided there exist positive constants c1 and c2 such that
(h − k)(z) ≤ (c1 |∇ϕ| + c2 ϕ)(z).
We also write h(z) ∼ k(z) if h(z) Ü k(z) and k(z) Ü h(z). Next, we write
h Ü k if the above inequality holds in O , with the constants c1 and c2 depending
only on n and C (independent of z and O ). Finally, h ∼ k if h Ü k and k Ü h.
For each fixed point z ∈ O , we choose a normal complex coordinate system
so that U is diagonal at z, and uiı̄ = λi , ∀ i = 1, . . . , n. We want to first calculate
ϕ and its derivatives up to second order.
Since U is diagonal at z, we have that
0 ∼ ϕ(z) ∼
X
X
uiı̄ σ` (G) ∼
uiı̄ ,
i∈B
i∈B
so
uiı̄ ∼ 0,
i ∈ B.
This relation yields that, for 1 ≤ m ≤ `,
σm (U) ∼ σm (G),


σm (G|j), if j ∈ G;
σm (U|j) ∼

σm (G),
if j ∈ B.


σm (G|ij), if i, j ∈ G;




if i ∈ B, j ∈ G;
σm (U|ij) ∼ σm (G|j),




σ (G),
if i, j ∈ B, i ≠ j.
m
Also
0 ∼ ϕα ∼ σ` (G)
X
uiı̄α ∼
i∈B
0 ∼ ϕᾱ ∼ σ` (G)
X
i∈B
X
uiı̄α ,
i∈B
uiı̄ᾱ ∼
X
i∈B
uiı̄ᾱ .
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By Proposition 2.1 in last section,


σ` (G),
i = j ∈ B;
∂ϕ
∼

∂ui̄

otherwise.
0,


if i = j, r = s, i ≠ r ;

σ`−1 (U|ir ),


2
∂ ϕ
=
= −σ`−1 (U|ij), if i = s, r = j, i ≠ j ;

∂ui̄ ∂ur s̄



0
otherwise.
S i̄ =
S i̄,r s̄
By the relations above, we calculate the second order derivatives of ϕ,
ϕαᾱ =
X
S i̄,r s̄ ui̄α ur s̄ ᾱ +
i,j,r ,s
=
X
X
S i̄ ui̄αᾱ
i,j
σ`−1 (U|ij)uiı̄α uj ̄ᾱ −
i,j ; i≠j
=
X
σ`−1 (U|ij)ui̄α uj ı̄ᾱ +
i,j, i≠j
X
+
i∈G
j∈B
−
X
+
i∈B
j∈G
X
+
i∈G
j∈B
X
+
X
+
i,j∈G
i≠j
X
+
i∈B
j∈G
X i,j∈B
i≠j
X
i,j∈G
i≠j
+
X
S i̄ ui̄αᾱ
i,j
σ`−1 (U|ij)uiı̄α uj ̄ᾱ
X σ`−1 (U|ij)ui̄α uj ı̄ᾱ
i,j∈B
i≠j
S i̄ ui̄αᾱ .
i,j
We want to simplify the expression above; notice that
X
σ`−1 (U|ij)uiı̄α uj ̄ᾱ =
X
σ`−1 (G|i)uiı̄α uj ̄ᾱ
i∈G, j∈B
i∈G, j∈B
=
X
σ`−1 (G|i)uiı̄α
i∈G
X
uj ̄ᾱ ∼ 0.
j∈B
P
σ`−1 (U|ij)uiı̄α uj ̄ᾱ ∼ 0.
P
For any i ∈ B fixed, and ∀ α, −uiı̄ᾱ ∼ j∈B, j≠i uj ̄ᾱ , so
Similarly,
X
i∈B, j∈G
σ`−1 (U|ij)uiı̄α uj ̄ᾱ ∼ −
i,j∈B, i≠j
X
=−
X
i∈B
So, we have
σ`−1 (G)uiı̄α uiı̄ᾱ
i∈B
σ`−1 (G)|uiı̄α |2 .
Constant Rank Theorem in Complex Variables
ϕαᾱ ∼ −
X
σ`−1 (G|i)|ui̄α |
i∈G, j∈B
i∈B
−
=
X
σ`−1 (G|j)|ui̄α |2 −
i∈B, j∈G
n
X
iı̄
S uiı̄αᾱ −
i=1
X
X
2
σ`−1 (G)|ui̄α |2 +
X
i,j∈B, i≠j
i
σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 ) −
i∈G, j∈B
X
∼
X
σ`−1 (G)|uiı̄α |2 −
σ` (G)uiı̄αᾱ −
i∈B
X
1237
S iı̄ uiı̄αᾱ
X
σ`−1 (G)|ui̄α |2
i,j∈B
2
2
σ`−1 (G|i)(|ui̄α | + |uj ı̄α | ) −
i∈G, j∈B
X
σ`−1 (G)|ui̄α |2 .
i,j∈B
Write F = σk (U), then
F αβ̄


σk−1 (G|α), if α = β ∈ G;





∂σk (U)
if α = β ∈ B ;
=
∼ σk−1 (G),

∂uαβ̄





0,
otherwise.


if i = j, r = s, i ≠ r ;
σk−2 (U|ir ),





F i̄,r s̄ = −σk−2 (U|ij), if i = s, j = r , i ≠ j ;






otherwise.
0,
As we calculated before,
ϕαᾱ ∼
n
X
S iı̄ uiı̄αᾱ −
i=1
X
σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 ) −
i∈G, j∈B
X
σ`−1 (G)|ui̄α |2 .
i,j∈B
Now, do the contraction,
(3.3)
n
X
F αᾱ ϕαᾱ ∼
α=1
−
F αᾱ S iı̄ uiı̄αᾱ
α=1 i=1
n X
X
α=1
n X
n
X
F αᾱ σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 ) −
n X
X
α=1
i∈G, j∈B
F αᾱ σ`−1 (G)|ui̄α |2 .
i,j∈B
We deal with the first term in (3.3), take the first order derivative to the equation
(1.1), we have that
n
X
F αβ̄ uαβ̄i = fi
α,β=1
and the second order derivatives,
and
n
X
F αβ̄ uαβ̄ı̄ = fı̄ ,
α,β=1
Q UN L I
1238
n
X
F αβ̄,γ s̄ uαβ̄i uγ s̄ ı̄ +
α,β,γ,s=1
n
X
F αβ̄ uαβ̄iı̄ = fiı̄ .
α,β=1
Remember that (F αβ̄ ) is diagonal, so
n
X
F αᾱ uαᾱiı̄ = fiı̄ −
α=1
σk−2 (U|αβ)uαᾱi uββ̄ı̄ +
α≠β
X
= fiı̄ −
+
α∈B
β∈G
+
X
X
X
+
α∈G
β∈B
X
+
α∈B
β∈G
∼ fiı̄ −
F αβ̄,r s̄ uαβ̄i ur ,s̄,ı̄
α,β,r ,s
X
= fiı̄ −
X
X
+
X +
σk−2 (U|αβ)uαᾱi uββ̄ı̄
α,β∈G α,β∈B
α≠β
α≠β
+
α∈G
β∈B
X
+
X σk−2 (U|αβ)|uαβ̄i |2
α,β∈G α,β∈B
α≠β
α≠β
σk−2 (G|αβ)uαᾱi uββ̄ı̄ +
X
X
σk−2 (G)|uαᾱi |2
α∈B
2
σk−2 (G|β)(|uαβ̄i | + |uβᾱi |2 )
α∈B, β∈G
+
σk−2 (U|αβ)uαβ̄i uβᾱı̄
α≠β
α,β∈G, α≠β
X
X
σk−2 (G|αβ)|uαβ̄i |2 +
α,β∈G, α≠β
X
σk−2 (G)|uαβ̄i |2 .
α,β∈B, α≠β
So, the first term in (3.3) becomes,
(3.4)
n X
n
X
F αᾱ S iı̄ uiı̄αᾱ
α=1 i=1
∼
X
X
X
σ` (G) fiı̄ −
σk−2 (G|αβ)uαᾱi uββ̄ı̄ +
σk−2 (G)|uαβ̄i |2
i∈B
α,β∈G, α≠β
+
X
α,β∈B
σk−2 (G|β)(|uaβ̄i |2 + |uβᾱi |2 ) +
α∈B, β∈G
X
σk−2 (G|αβ)|uαβ̄i |2 .
α,β∈G, α≠β
Next, rewrite the second term of (3.3),
(3.5)
n
X
X
F αᾱ σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 )
α=1 i∈G, j∈B
=
X
α∈B
+
X α∈G
X
F αᾱ σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 )
i∈G, j∈B
∼
Constant Rank Theorem in Complex Variables
∼
X
1239
σk−1 (G)σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 )
i∈G, α,j∈B
+
X
σk−1 (G|α)σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 ).
i,α∈G, ∈B
Finally, regroup the last term in (3.3),
n X
X
(3.6)
α=1
σ`−1 (G)F αᾱ |ui̄α |2
i,j∈B
=
X
+
α∈B
∼
X
X F αᾱ
α∈G
X
σ`−1 (G)|ui̄a |2
i,j∈B
σk−1 (G)σ`−1 (G)|ui̄α |2 +
i,j,α∈B
X
σk−1 (G|α)σ`−1 (G)ui̄α |2 .
i,j∈B, α∈G
Combining (3.4), (3.5), (3.6), we have
(3.7)
n
X
α=1
F αᾱ ϕαᾱ ∼
X
σ` (G)fiı̄
i∈B
−
X X
i∈B
+
X X
i∈B
+
σk−1 (G)σ`−1 (G|i)(|ui̄α |2 + |uj ı̄α |2 )
σk−1 (G)σ`−1 (G|i)(|ui̄α |
i∈G, j∈B
2
+ |uj ı̄α |2 )
σk−1 (G)σ`−1 (G)|uiı̄α |2
i,j∈B
X X
α∈G
2
i∈G, j∈B
X X
α∈B
−
σ` (G)σk−2 (G|αβ)|uαβ̄i |
α,β∈G, α≠β
X X
α∈G
−
α∈B, β∈G
X X
α∈B
−
σ` (G)σk−2 (G|β)(|uαβ̄i |2 + |uβᾱi |2 )
X X
i∈B
−
σ` (G)σk−2 (G)|uαβ̄i |2
α,β∈B
X X
i∈B
+
σ` (G)σk−2 (G|αβ)uαᾱi uββ̄ı̄
α,β∈G, α≠β
σk−1 (G|α)σ`−1 (G)|uiı̄α |2
i,j∈B
=
Q UN L I
1240
=
X
X X
σ` (G)fiı̄ −
i∈B
+
σ` (G)σk−2 (G|αβ)uαᾱi uββ̄ı̄
α,β∈G, α≠β
i∈B
X X
i∈B
|uαβ̄i |2 (σ` (G)σk−2 (G) − σk−1 (G)σ`−1 (G))
α,β∈B
X X
+
(|uαβ̄i |2 + |uβᾱi |2 )(σ` (G)σk−2 (G|αβ) − σk−1 (G)σ`−1 (G|β))
i∈B
+
α∈B, β∈G
X X
i∈B
−
α,β∈G, α≠β
X X
i∈G
−
σ` (G)σk−2 (G|αβ)|uαβ̄i |2
σk−1 (G|i)σ`−1 (G|β)(|uαβ̄i |2 + |uβᾱi |2 )
α∈B, β∈G
X X
i∈G
σk−1 (G|i)σ`−1 (G)|uαβ̄i |2 .
α,β∈G
By the Newton-MacLaurin inequality and the structural properties of the elementary symmetric functions, we have the following inequalities,
σ` (G)σk−2 (G) − σk−1 (G)σ`−1 (G) ≤ 0,
(3.8)
and
(3.9)
σ` (G)σk−2 (G|β) − σk−1 (G)σ`−1 (G|β)
= σ`−1 (G|β)uββ̄ σk−2 (G|β) − σk−1 (G)σ`−1 (G|β)
≤ 0.
Rewrite the sixth term in the right hand side of (3.7) (interchange i ↔ α),
(3.10)
X
i∈B
X
X
+
α,β∈G, α=β
=
α,β∈G, α≠β
X X
i∈B
σk−1 (G|α)σ`−1 (G|β)|uαβ̄i |2
σk−1 (G|α)σ`−1 (G|α)|uαᾱi |2
α∈G
+
X X
i∈B
σk−1 (G|α)σ`−1 (G|β)|uαβ̄i |2 .
α,β∈G, α≠β
Combining the second term on the right hand side of (3.10) and the fifth
term on the right hand side of (3.7), we have
(3.11)
X X
i∈B
|uaβ̄i |2 (σ` (G)σk−2 (G|αβ) − σ`−1 (G|β)σk−1 (G|α)) ≤ 0.
α,β∈G, α≠β
Constant Rank Theorem in Complex Variables
1241
Then, by (3.8), (3.9), (3.10), (3.11), we have the following,
X
(3.12)
∼
X
F αᾱ ϕαᾱ
σ` (G)fiı̄ −
i∈B
−
−
−
σk−1 (G|α)σ`−1 (G|α)|uαᾱi |
α∈G
X X
i∈G
σ` (G)σk−2 (G|αβ)uαᾱi uββ̄ı̄
α,β∈G, α≠β
i∈B
X X
i∈B
X X
2
σk−1 (G|i)σ`−1 (G|β)|uβᾱi |2
α∈B, β∈G
X X
σk−1 (G|i)σ`−1 (G)|uαβ̄i |2
i∈G
α,β∈B
i∈B
α∈G
!
X X
fi fı̄
−
=
σ` (G) fiı̄ −
σ` (G)σk−2 (G|αβ)uαᾱi uββ̄ı̄
f
i∈B
i∈B α,β∈G, α≠β
X X
−
σk−1 (G|α)σ`−1 (G|α)|uαᾱi |2
X
−
X X
i∈G
−
α∈B, β∈G
X X
i∈G
σk−1 (G|i)σ`−1 (G|β)|uβᾱi |2
σk−1 (G|i)σ`−1 (G)|uαβ̄i |2 +
α,β∈B
X
σ` (G)
i∈B
fi fı̄
.
f
By previous calculations,
fi =
n
X
F αᾱ uαᾱi =
X
α=1
fı̄ ∼
X
+
α∈B
X X
σk−1 (G|α)uαᾱi ,
F αᾱ uαᾱi ∼
α∈G
α∈G
σk−1 (G|β)uββ̄ı̄ .
β∈G
So we can rewrite the last term in (3.12) as the following:
(3.13)
1 X
f
σ` (G)fi fı̄
i∈B
∼
1
f
"
X X
i∈B
σ` (G)σk−1 (G|α)σk−1 (G|β)uαᾱi uββ̄ı̄
α,β∈G, α≠β
+
X X
i∈B
2
σ` (G)σk−1 (G|α)|uαᾱi |
α∈G
Next, we regroup the terms by ”Good” and ”Bad” indices. Let
#
2
.
Q UN L I
1242
1 X X
A=
f
σ` (G)σk−1 (G|α)σ`−1 (G|β)uαᾱi uββ̄ı̄
α,β∈G, α≠β
i∈B
−
X X
σ` (G)σk−2 (G|αβ)uαᾱi uββ̄ı̄ ,
α,β∈G, α≠β
i∈B
then
(3.14) f ·A =
X X
σ` (G)uαᾱi uββ̄ı̄ (σk−1 (G|α)σk−1 (G|β)
α,β∈G, α≠β
i∈B
=
X X
i∈B
− σk (G)σk−2 (G|αβ))
2
σ` (G)uαᾱi uββ̄ı̄ [σk−
1 (G|αβ) − σk (G|αβ)σk−2 (G|αβ)].
α,β∈G, α≠β
Let
B=
1 X X
f
2
2
σ` (G)σk−
1 (G|α)|uαᾱi | −
α∈G
i∈B
X X
σk−1 (G|α)σ`−1 (G|α)|uαᾱi |2
α∈G
i∈B
then,
(3.15) f · B =
X X
i∈B
=−
σk−1 (G|α)|uαᾱi |2 (σ` (G)σk−1 (G|α) − σk (G)σ`−1 (G|α))
α∈G
X X
σk (G|α)σ`−1 (G|α)σk−1 (G|α))|uαᾱi |2 .
α∈G
i∈B
The last equalities in (3.14) and (3.15) come from (2.1) and (2.2) in Lemma 2.2.
So,
f ·A+f ·B =
X
−
σk (G|α)σ`−1 (G|α)σk−1 (G|α)|uαᾱi |2
α∈G
i∈B
+ σ` (G)
X
X
2
[σk−
1 (G|αβ) − σk (G|αβ)σk−2 (G|αβ)]uαᾱi uββ̄ı̄
α,β∈G, α≠β
the last inequality follows from Lemma 2.3.
Finally we got
X
α
F αᾱ ϕαᾱ Ü
X
i∈B
σ` (G) fiı̄ −
fi fı̄
f
!
∼ f σ` (U)
so we are done with the proof of Proposition 3.1.
X
i
(log f )iı̄ ,
≤ 0;
Constant Rank Theorem in Complex Variables
1243
4. P LURISUBHARMONIC S OLUTIONS AND D ISCUSSIONS
Once we’ve got the constant rank theorem on flat domains, the next step is naturally to transplant it to complex manifolds. In the first part of this section, we’ll
give the statement of the constant rank result on complex manifolds as well as
a brief proof. Following this, we will illustrate a way to produce a plurisubharmonic function from an admissible solution to some Hessian equation. Finally
we’ll discuss an interesting observation which connects the constant rank theorem
to homogeneous complex Monge-Ampère equations.
The following theorem is an extended result of Theorem 1.1 in Kähler manifolds.
Theorem 4.1. Let (M, g) be a compact Kähler manifold with Kähler metric
(gi̄ ), and with the orthogonal bisectional curvature of g , Riı̄j ̄ being bounded from
below by r > 0. Suppose u is an admissible solution to the following equation, with
(ui̄ + gi̄ ) non-negative definite,
σk (ui̄ + gi̄ ) = f (z),
(4.1)
in M.
If the matrix ((log f )i̄ − r kgi̄ ) is non-positive definite for every z ∈ M , then
W = (ui̄ + gi̄ ) is of constant rank in M .
Proof. The proof goes the similar way as the proof of Theorem 1.1. Write
W = (wi̄ ) = (ui̄ + gi̄ ); we will work on a small neighborhood O of z0 ∈ M ,
where W (z0 ) attains the minimal rank `. Let ϕ(W ) = σ`+1 (W ); for any point
z ∈ O , we choose a normal coordinates at z so that W is diagonal at that point.
Also we might group the eigenvalues of W at z into G and B , the “good” and
“bad” sets of indices respectively, and |G| = `, |B| = n − `. Then doing the
similar contraction calculation as in (3.3), we get
(4.2)
n
X
α=1
F αᾱ ϕαᾱ ∼
n X
n
X
F αᾱ S iı̄ wiı̄αᾱ
α=1 i=1
n
X
X
α=1
n
X
X
−
−
F αᾱ σ`−1 (G|i)(|wi̄α |2 + |wj ı̄α |2 )
i∈G, j∈B
α=1
F αᾱ σ`−1 (G)|wi̄α |2 ,
i,j∈B
while by the computation of the covariant derivatives on the Kähler manifold, we
have that
wiı̄αᾱ = wαı̄iᾱ
= wαı̄ᾱi +
X
s
ws ı̄ Rαs̄iᾱ −
X
s
wαs̄ Rs ı̄iᾱ
= wαᾱiı̄ + wiı̄ Riı̄αᾱ − wαᾱ Rαᾱiı̄ .
Q UN L I
1244
Following the similar simplification steps as we did to (3.3), we have
X
F αᾱ ϕαᾱ Ü
X
fi fı̄
σ` (G) −
F αᾱ Rαᾱiı̄ wαᾱ + fiı̄ −
,
f
i∈B
α∈G, i∈B
X
Since σk is homogeneous of degree k and our assumption that the orthogonal
bisectional curvature is bounded from below by r in M ,
X
F αᾱ Rαᾱiı̄ wαᾱ ≥ r
X
F αᾱ wαᾱ = r kσk (W ) = r kf .
α
α∈G, i∈B
So we finally get
X
F
αᾱ
ϕαᾱ Ü σ` (G)
α
By our assumption
X
i∈B
"
fi fı̄
fiı̄ −
− r kf
f
(log fi̄ ) − r kgi̄
#
.
❐
is non-positive definite and the strong minimum principle, we can conclude the
rank of W is of constant ` everywhere.
One important application of the constant rank theorem is that one may produce a plurisubharmonic solution under certain geometric situations. To do this,
one may apply the classical continuity method by starting with a nicely picked
strictly plurisubharmonic function, then if the deformation process goes through,
one may produce a strictly plurisubharmonic solution to the original equation. If
this breaks down at some time t0 , at that time, by the continuity, the solution is
still plurisubharmonic, but not strictly. By our constant rank theorem, the rank of
the corresponding complex Hessian matrix must be a constant which is strictly less
than n. In a compact Kähler manifold with suitable geometric structure, certain
global conditions would then stand up to prevent this from happening.
Since last decades, there have been intensive works on the study of homogeneous complex Monge-Ampère equation; plurisubharmonic solutions to homogeneous complex Monge-Ampère equations are closely related to maximal functions
or Green functions. These functions appear in problems analogous to those of
classical potential theory, this field is usually called pluripotential theory. It has
been an active branch of mathematics in which crucial properties of plurisubharmonic functions are studied. See [1], [27] for more references.
The classical Green function in a domain in single variable is zero on the
boundary and has a logarithmic pole at one given point. Lempert [30] [31] introduced an analogous function in several complex variables, see also [19] and
[26].
Constant Rank Theorem in Complex Variables
1245
The pluri-complex Green function with logarithmic pole at a in a complex
domain Ω is defined as,
Ga (z) =
X
u(z) | u is plurisubharmonic in Ω,
u < 0, u(z) = log |z − a| + O(1) .
With the help of foliations, Lempert [31] showed that Ga (z) is smooth plurisubharmonic in Ω and moreover, it solves the homogeneous Monge-Ampère equation
in Ω \ {a}. This is a remarkable result since generally solutions to degenerate complex Monge-Ampère equations may not be in C 2 ([3], [2]).
Lempert’s construction was actually done for a smoothly bounded, strongly
convex domain. For general domains, B.Guan [21, 22] provided a C 1,α plurisubharmonic solution on a bounded smooth strongly pseudoconvex domain. The optimal C 1,1 regularity was obtained by Blocki [11]. Similar optimal C 1,1 regularity
was gained for a homogeneous Monge-Ampère equation in [23] by P.Guan when
he studied the extremal functions associated to the intrinsic norms introduced by
Chern-Levine-Nirenberg [18] and Bedford-Taylor
[7], in which case, the domain
S
is even more general, namely, M = Ω\( ni=1 Ωi ), where Ω, Ω1 , . . . , Ωn are smooth
bounded strongly pseudoconvex domains.
It is obvious to see that for any solution to a homogeneous complex MongeAmpère equation, the rank of the corresponding complex Hessian must be strictly
less than n. Conversely, if we could find a plurisubharmonic function whose
complex Hessian matrix has constant rank strictly less than n, then it of course
solves the homogeneous Monge-Ampère equation.
It is of great interest to note that in Lempert’s work [31], he furthermore
addressed that the complex Hessian matrix of the plurisubharmonic solution has
constant rank n − 1.
The above example indicates a very interesting direction through which we
might explore a way of incorporating the constant rank theorem into the study
of the homogeneous complex Monge-Ampère equations. It would be delightful
if we could dig out more connections between them and obtain further regularity
results for the solutions to homogeneous complex Monge-Ampère equations.
Acknowledgements This paper is part of the author’s Ph.D. thesis, she would
like to thank her advisor, Professor Pengfei Guan for bringing her attention to this
area and for his constant guidance and support.
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Department of Mathematics and Statistics
McGill University
Montreal, Quebec H3A 2K6, Canada
C URRENT ADDRESS :
Department of Mathematics and Statistics
Wright State University
3640 Colonel Glenn Hwy.
Dayton, OH 45435
E- MAIL: [email protected]
1248
Q UN L I
K EY WORDS AND PHRASES: plurisubharmonic; constant rank; complex Hessian equations.
2000 M ATHEMATICS S UBJECT C LASSIFICATION: 35; 53.
Received : February 8th, 2008; revised: June 10th, 2008.