On a class of fully nonlinear elliptic equations on Hermitian manifolds

0.22 MB·English

by  Bo Guan

ON A CLASS OF FULLY NONLINEAR ELLIPTIC EQUATIONS
ON HERMITIAN MANIFOLDS
3
BO GUAN AND WEI SUN
1
0
2 Abstract. We derive a priori C2 estimates for a class of complex Monge-Amp`ere

n
type equations on Hermitian manifolds. As an application we solve the Dirichlet
a
problemfortheseequationsunderthe assumptionofexistenceofasubsolution;the
J
existence result, as well as the second order boundary estimates, is new even for
4 Cn
2 bounded domains in .
Mathematical SubjectClassification (2010): 58J05,58J32,32W20,35J25,53C55.

]
P
A
.
h
t 1. Introduction
a
m
Let (Mn,ω) be a compact Hermitian manifold of dimension n 2 with smooth
[
≥
boundary∂M andχasmoothreal(1,1)formonM¯ := M ∂M. Defineforafunction
1
∪
v u C2(M),
3 ∈
6 √ 1
¯
8 χu = χ+ − ∂∂u
2
5
. and set
1
0
[χ] = χ : u C2(M) , [χ]+ = χ′ [χ] : χ′ > 0 .
3 u
∈ ∈ }
1
: In this paper we are(cid:8)concerned with(cid:9)the equatio(cid:8)n for 1 α n,
v ≤ ≤
i
X (1.1) χn =ψχn−α ωα in M, .
u u ∧
r
a
Werequireχ > 0sothatequation(1.1)iselliptic; wecallsuchfunctionsadmissible
u
orχ-plurisubharmonic. Consequently, weassumeψ > 0onM¯; equation(1.1)becomes
degenerate when ψ 0.
≥
When α = n this is the complex Monge-Amp`ere equation which plays extremely
important roles in complex geometry and analysis, especially in Ka¨hler geometry, and
has received extensive study since the fundamental work of Yau [34] (see also [1]) on
compact Ka¨hler manifolds and that of Caffarelli, Kohn, Nirenberg and Spruck [3]
for the Dirichlet problem in strongly pseudoconvex domains in Cn. For α = 1 equa-
tion (1.1) also arises naturally in geometric problems; it was posed by Donaldson [11]
1
2 BO GUAN ANDWEI SUN
in connection with moment maps and is closely related to the Mabuchi energy [5],
[33], [28].
Donaldson’s problem assumes M is closed, both ω, χ are Ka¨hler and ψ is constant.
It was studied by Chen [5], Weinkove [32], [33], Song and Weinkove [28] using para-
bolic methods. In [28] Song and Weinkove give a necessary and sufficient solvability
condition. Their result was extended by Fang, Lai and Ma [12] to all 1 α < n.
≤
In this paper we study the Dirichlet problem for equation (1.1) on Hermitian man-
ifolds. Given ψ C∞(M¯) and ϕ C∞(∂M), we wish to find a solution u C∞(M¯)
∈ ∈ ∈
of equation (1.1) satisfying the boundary condition
(1.2) u = ϕ on ∂M.
The Dirichlet problem for the complex Monge-Amp`ere equation in Cn was studied
by Caffarelli, Kohn, Nirenberg and Spruck [3] on strongly pseudoconvex domains.
Their result was extended to Hermitian manifolds by Cherrier and Hanani [8], [23],
andby the first author [14] to arbitrary bounded domains in Cn under the assumption
of existence of a subsolution. See also the more recent papers [16], [35], and related
work of Tosatti and Weinkove [29], [30] who completely extended the zero order
estimate of Yau [34] on closed Ka¨hler manifolds to the Herimatian case. In [25] Li
treated the Dirichlet problem for more general fully nonlinear elliptic equations in Cn
but needed to assume the existence of a strict subsolution. Li’s result does not cover
equation (1.1) as it fails to satisfy some of the key structure conditions in [25].
In this paper we prove the following existence result which is new even in the case
when M is a bounded domain in Cn and χ = 0; we assume 2 α n 2 as the
≤ ≤ −
cases α = 1 and α = n 1 were considered in [17] and [18], while for the complex
−
Monge-Amp`ere equation (α = n) it was proved in [16].
Theorem 1.1. Let ψ C∞(M¯), ψ > 0 and ϕ C∞(∂M). There exists a unique
∈ ∈
admissible solution u C∞(M¯) of the Dirichlet problem (1.1)-(1.2), provided that
∈
there exists an admissible subsolution u C2(M¯):
∈
χn ψχn−α ωα on M¯
(1.3) u ≥ u ∧
(u = ϕ on ∂M.
In order to solve the Dirichlet problem (1.1)-(1.2) one needs to derive a priori C2
estimates up to the boundary for admissible solutions. The most difficult step is
probably the second order estimates on the boundary.
FULLY NONLINEAR ELLIPTIC EQUATIONS 3
Theorem 1.2. Suppose ψ C1(M¯), ψ > 0 and ϕ C4(∂M) and u C2(M¯) is an
∈ ∈ ∈
admissible subsolution satisfying (1.3). Let u C3(M¯) be an admissible solution of
∈
the Dirichlet problem (1.1)-(1.2). Then
(1.4) max 2u C
∂M |∇ | ≤
where C depends on u , minψ−1, u and min c : c χ ω , as well as
| |C1(M¯) | |C2(M¯) { 1 1 u ≥ }
other known data.
This estimate is new for domains in Cn. Note that ∂M is assumed to be smooth
and compact in Theorem 1.2, but otherwise is completely arbitrary. In general, the
Dirichlet problem (1.1)-(1.2) is not always solvable in an arbitrary smooth bounded
domain in Cn without the subsolution assumption. In the theory of nonlinear elliptic
equations, many well known classical results assume certain geometric conditions on
the boundary of the underlying domain; see e.g. [27], [3], [2] and [4]. In [19], [13] and
[14], J.SpruckandthefirstauthorwereabletosolvetheDirichletproblemforrealand
complex Monge-Amp`ere equations on arbitrary smooth bounded domains assuming
theexistence of asubsolution. Their work was motivatedby applicationsto geometric
problems and had been found useful in some important problems such as the proof
by P.-F. Guan [20], [21] of the Chern-Levine-Nirenberg conjecture [6], and work on
the Donaldson conjectures [10] on geodesics in the space of Ka¨hler metrics; we refer
the reader to [26] for recent progress and further references on this fast-developing
subject.
On a closed Ka¨hler manifold (M,ω), Fang, Lai and Ma [12] proved second and
zero order estimates for equation (1.1) when χ is also Ka¨hler and ψ is constant. We
extend their second order estimates to Hermitian manifolds and for general χ and ψ.
Technically the major difficulty is to control extra third order terms which occur due
to the nontrivial torsion of the Hermitian metric. This was done in [17], [18] for α = 1
and α = n 1; the case 2 α n 2 is considerably more complicated. In order
− ≤ ≤ −
to solve the Dirichlet problem we also need global gradient estimates. Following [28]
and [12] let
(1.5) C (ω) = [χ] : χ′ [χ]+, nχ′n−1 > (n α)ψχ′n−α−1 ωα .
α
∃ ∈ − ∧
Theorem 1.3. Let u(cid:8) C4(M) C2(M¯) be an admissible solution of e(cid:9)quation (1.1)
∈ ∩
where ψ C2(M¯), ψ > 0. Suppose that χ C (ω). Then there are constants C ,C
α 1 2
∈ ∈
4 BO GUAN ANDWEI SUN
depending on u such that
| |C0(M¯)
(1.6) sup u C (1+sup u ),
1
|∇ | ≤ |∇ |
M ∂M
(1.7) sup∆u C (1+sup∆u).
2
≤
M ∂M
In particular, if M is closed (∂M = ) then u C and ∆u C on M.
1 2
∅ |∇ | ≤ | | ≤
The cone C (ω) was first introduced by Song and Weinkove [28] (α = 1) and Fang,
α
Lai and Ma [12] who derived the estimate (1.7) on a closed Ka¨hler manifold (M,ω)
when χ is also Ka¨hler and
χn
ψ = c := M ,
α
χn−α ωα
M R ∧
which is a Ka¨hler class invariant. As in [28], [12] the constant C in Theorem 1.3 is
R 2
independent of gradient bounds, i.e. C is independent of C .
2 1
The subsolution assumption (1.3) implies [χ] C (ω). On a closed manifold, a
α
∈
subsolution must be a solution or the equation has no solution. This is a consequence
of the maximum principle and a concavity property of equation (1.1).
The gradient estimate (1.6) is crucial to the proof of Theorem 1.1 and is also new
when ω and χ are Ka¨hler. Indeed, deriving gradient estimates for fully nonlinear
equations on complex manifolds turns out to be a rather challenging and mostly open
question. Only very recently were Dinew and Kolodziej [9] able to prove the gradient
estimateusing scaling techniques andLiouvilletypetheoremsforthecomplexHessian
equation
(1.8) ωn =ψχn−α ωα
u ∧
on closed Ka¨hler manifolds which is consequently solvable due to the earlier work of
Hou, Ma and Wu [24].
The proof of Theorem 1.3 is carried out in Sections 3 and 5 where we derive the
estimates for u and ∆u, the gradient and Laplacian of u, respectively. In Section 4
|∇ |
we establish the boundary estimates for second derivatives. These estimates allow us
to derive global estimates for all (real) second derivatives as in Section 5 in [16]
and apply the Evans-Krylov theorem since equation (1.1) becomes uniformly elliptic.
Theorem 1.1 may then be proved by the continuity method. These steps are all well
understood so we shall omit them. In section 2 we recall some formulas on Hermitian
manifolds.
FULLY NONLINEAR ELLIPTIC EQUATIONS 5
2. Preliminaries
Let g and denote the Riemannian metric and Chern connection of (M,ω). The
∇
torsion and curvature tensors of are defined by
∇
T(u,v) = v u [u,v],
u v
(2.1) ∇ −∇ −
R(u,v)w = w w w,
u v v u [u,v]
∇ ∇ −∇ ∇ −∇
respectively. Following the notations in [16], in local coordinates z = (z ,...,z ) we
1 n
have
∂ ∂
g = g , , gi¯j = g −1,
i¯j i¯j
∂z ∂z¯ { } { }
i j
 (cid:16) (cid:17)
(2.2)  Tk = Γk Γk = gk¯l ∂gj¯l ∂gi¯l ,
 ij ij − ji ∂z − ∂z
 i j

 (cid:16) (cid:17)
R = g ∂Γmik = ∂2gk¯l +gpq¯∂gkq¯∂gp¯l.
 i¯jk¯l − m¯l ∂z¯j −∂zi∂z¯j ∂zi ∂z¯j


Recall that for asmooth function v, v = v = ∂ ∂¯ v, v = ∂ v Γl v and
 i¯j ¯ji i j i¯jk k i¯j − ki l¯j
v = ∂¯v Γq v .
i¯jk¯l l i¯jk − lj iq¯k
We have (see e.g. [17]),
v v =Tl v ,
(2.3) i¯jk − k¯ji ik l¯j
(vi¯jk¯ −vik¯¯j =Tjlkvi¯l,
v v =gpq¯R v gpq¯R v ,
(2.4) i¯jk¯l − i¯j¯lk k¯liq¯ p¯j − k¯lp¯j iq¯
(vi¯jk¯l −vk¯li¯j =gpq¯(Rk¯liq¯vp¯j −Ri¯jkq¯vp¯l)+Tipkvp¯j¯l +Tjqlviq¯k −TipkTjqlvpq¯.
Let u C4(M) be an admissible solution of equation (1.1). As in [16] and [17], we
∈
denote g = χ +u , gi¯j = g −1 and w = trχ+∆u. Note that g is positive
i¯j i¯j i¯j i¯j i¯j
{ } { } { }
definite. Assume at a fixed point p M that g = δ and g is diagonal. Then
i¯j ij i¯j
∈
(2.5) u u =R u R u +2Re Tju TpTqu ,
i¯ikk¯ − kk¯i¯i kk¯ip¯ p¯i − i¯ikp¯ pk¯ { ik i¯jk}− ik ik pq¯
and therefore,
(2.6) g g =R g R g +2Re Tjg Tj 2g G
i¯ikk¯ − kk¯i¯i kk¯i¯i i¯i − i¯ikk¯ kk¯ { ik i¯jk}−| ik| j¯j − i¯ikk¯
where
(2.7) G =χ χ +R χ R χ +2Re Tjχ TpTqχ .
i¯ikk¯ kk¯i¯i − i¯ikk¯ kk¯ip¯ p¯i − i¯ikp¯ pk¯ { ik i¯jk}− ik ik pq¯
6 BO GUAN ANDWEI SUN
Let S (λ) denote the k-th elementary symmetric polynomial of λ Rn
k
∈
S (λ) = λ λ .
k i1··· ik
1≤i1<X···<ik≤n
In local coordinates we can write equation (1.1) in the form
(2.8) F(g ) := Sn(λ∗(gi¯j)) α1 = ψ α1
i¯j S (λ (g )) Cα
n−α ∗ i¯j n
(cid:16) (cid:17) (cid:16) (cid:17)
or equivalently,
(2.9) Cαψ−1 = S (λ∗(gi¯j))
n α
whereλ (A)andλ∗(A)denotetheeigenvaluesofaHermitianmatrixAwithrespectto
∗
g and to gi¯j , respectively. Unless otherwise indicated we shall use S to denote
i¯j α
{ } { }
S (λ∗(gi¯j)) when no possible confusion would occur. We shall also occasionally write
α
F(χ ) := F(g ) and F(χ ) := F(u +χ ), etc.
u i¯j u i¯j i¯j
Differentiating equation (2.9) twice at a point p where g = δ and g is diagonal,
i¯j ij i¯j
we obtain
(2.10) Cα∂ (ψ−1) = S (gi¯i)2g
n l − α−1;i i¯il
i
X
and
Cα∂¯∂ (ψ−1) = S (gi¯i)2g + S (gi¯i)2gj¯j(g g +g g )
n l l − α−1;i i¯il¯l α−1;i i¯jl j¯i¯l j¯il i¯j¯l
i i,j
(2.11) X X
+ S (gi¯i)2(gj¯j)2 g g g g
α−2;ij i¯il j¯j¯l j¯il i¯j¯l
−
i6=j
X (cid:0) (cid:1)
where for i , ,i 1, ,n ,
1 s
{ ··· } ⊆ { ··· }
S (λ) = S (λ ).
k;i1···is k |λi1=···=λis=0
We need the following inequality from [22]; see also Proposition 2.2 in [12],
n
S (λ) S (λ)S (λ)
α−1;i ¯ ¯ α−1;i α−1;j ¯
(2.12) ξ ξ + S (λ)ξ ξ ξ ξ 0
i i α−2;ij i j i j
λ ≥ S (λ) ≥
i α
i=1 i,j i,j
X X X
for λ = (λ ,...,λ ), λ > 0 and (ξ ,...,ξ ) Cn. Apply (2.12) to λ = gi¯i, ξ =
1 n i 1 n i i
∈
(gi¯i)2g and sum over l. We see that
i¯il
(2.13) S (gi¯i)3g g + S (gi¯i)2(gj¯j)2g g 0.
α−1;i i¯il i¯i¯l α−2;ij i¯il j¯j¯l
≥
i,l i6=j l
X XX
FULLY NONLINEAR ELLIPTIC EQUATIONS 7
Note also that
(S S gj¯j)(gi¯i)2gj¯jg g 0.
α−1;i α−2;ij j¯il i¯j¯l
− ≥
i6=j
X
We obtain from (2.11),
S (gi¯i)2g S (gi¯i)2gj¯jg g C.
(2.14) α−1;i i¯il¯l α−1;i i¯jl j¯i¯l
≥ −
i i,j
X X
Let u C2(M¯), χ > 0 such that
u
∈
(2.15) nχn−1 > (n α)ψχn−α−1 ωα.
u − u ∧
Thus there is ǫ > 0 such that
(2.16) ǫω χ ǫ−1ω.
u
≤ ≤
The key ingredient ofour estimates in thefollowing sections is the following lemma.
Lemma 2.1. There exist constants N,θ > 0 such that when w N at a point p
≥
where g = δ and g is diagonal,
i¯j ij i¯j
(2.17) S (gi¯i)2(u u ) θ S (gi¯i)2 +θ
α−1;i i¯i − i¯i ≥ α−1;i
i i
X X
and, equivalently,
(2.18) Fi¯j(u u ) θ Fi¯jg +θ.
i¯j − i¯j ≥ i¯j
i,j i,j
X X
Here and in the rest of this paper,
∂F
Fi¯j = (g ).
∂g i¯j
i¯j
It is well known that Fi¯j is positive definite.
{ }
An equivalent form of Lemma 2.1 and its proof are given in [12] (Theorem 2.8); see
also [15] where it is proved for more general fully nonlinear equations. So we shall
omit the proof here.
8 BO GUAN ANDWEI SUN
3. The gradient estimates
In this section we establish the a priori gradient estimates.
Proposition 3.1. Suppose χ C (ω) and let u C3(M) C1(M¯) be an admissible
α
∈ ∈ ∩
solution of (1.1). There is a uniform constant C > 0 such that
(3.1) sup u C(1+sup u ).
|∇ | ≤ |∇ |
M¯ ∂M
Proof. Let u C2(M¯), χ > 0 satisfy (2.15) and consider φ = Aeη where
u
∈
η = u u+sup(u u)
− −
M
and A is a constant to be determined. Suppose the function eφ u 2 attains its
|∇ |
maximal value at an interior point p M. Choose local coordinate around p such
∈
that g = δ and g is diagonal at p. At p we have
i¯j ij i¯j
∂ ( u 2) ∂¯( u 2)
i i ¯
(3.2) |∇ | +∂ φ = 0, |∇ | +∂ φ = 0
i i
u 2 u 2
|∇ | |∇ |
and
∂¯∂ ( u 2) ∂ ( u 2)∂¯( u 2)
i i i i ¯
(3.3) |∇ | |∇ | |∇ | +∂ ∂ φ 0.
i i
u 2 − u 4 ≤
|∇ | |∇ |
By direct computation,
(3.4) ∂ ( u 2) = (u u +u u ),
i k ik¯ ki k¯
|∇ |
k
X
∂¯∂ ( u 2) = (u u +u u +u u +u u )
i i k¯i k¯i ki k¯¯i ki¯i k¯ k k¯i¯i
|∇ |
k
X
= (u u +u u +u u +R u u )
(3.5) ki k¯¯i i¯ik k¯ i¯ik¯ k i¯ik¯l l k¯
k
X
2 2
+ u Tku Tku .
k¯i − il ¯l − il ¯l
Xk (cid:12) Xl (cid:12) Xk (cid:12)Xl (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Therefore, by (2.3) and (2.1(cid:12)0), (cid:12) (cid:12) (cid:12)
S (gi¯i)2∂¯∂ ( u 2) S (gi¯i)2 u 2
α−1;i i i α−1;i ki
|∇ | ≥ | |
i i,k
(3.6) X X
C u 2 C u 2 S (gi¯i)2.
α−1;i
− |∇ | − |∇ |
i
X
FULLY NONLINEAR ELLIPTIC EQUATIONS 9
From (3.2) and (3.4),
2 2
∂ ( u 2) 2 = u u 2 u 2 Re u u φ u u
i ki k¯ k ik¯ ¯i k ik¯
|∇ | − |∇ | −
(3.7) (cid:12) (cid:12) (cid:12)Xk (cid:12) Xk (cid:8) (cid:9) (cid:12)Xk (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) u 2 (cid:12)uki 2 2 u 2 Re ukuik¯φ¯i (cid:12) (cid:12)
≤|∇ | | | − |∇ |
k k
X X (cid:8) (cid:9)
by Schwarz inequality.
Combining (3.3), (3.6) and (3.7) we derive
2
S (gi¯i)2(φ C)+ S (gi¯i)2Re u u φ C.
(3.8) α−1;i i¯i − u 2 α−1;i k ik¯ ¯i ≤
i |∇ | i,k
X X (cid:8) (cid:9)
Next,
∂ φ = φ∂ η, ∂¯∂ φ = φ( ∂ η 2 +∂¯∂ η).
i i i i i i i
| |
Therefore,
1
(3.9) 2φ−1 Re ukuik¯φ¯i ≥2gi¯iRe uiη¯i − 2|∇u|2|ηi|2 −C
k
X (cid:8) (cid:9) (cid:8) (cid:9)
and
1
S (gi¯i)2η + S (gi¯i)2 η 2
α−1;i i¯i α−1;i i
2 | |
i i
X X
2 C
(3.10) ≤ − u 2 Sα−1;igi¯iRe uiη¯i + φ
|∇ | i
X (cid:8) (cid:9)
1 1
+C + S (gi¯i)2.
φ u 2 α−1;i
(cid:16) |∇ | (cid:17)Xi
For N > 0 sufficiently large so that Lemma 2.1 holds, we consider two cases: (a)
w > N and (b) w N. Without loss of generality we can assume that u > u
≤ |∇ | |∇ |
at p or otherwise we are done. Note that
2
(3.11) S gi¯iRe u η 4 S gi¯i = 4αS .
− u 2 α−1;i i ¯i ≤ α−1;i α
|∇ | i i
X (cid:8) (cid:9) X
In case (a) we have by Lemma 2.1
(3.12) S (gi¯i)2η θ +θ S (gi¯i)2.
α−1;i i¯i α−1;i
≥
i i
X X
So if S (gi¯i)2 K for some i and K sufficiently large we derive a bound u C
α−1;i
≥ |∇ | ≤
from (3.10) and (3.11) when A is sufficiently large.
10 BO GUAN ANDWEI SUN
Suppose that S (gi¯i)2 K for all i and assume g g . Note that
α−1;i 1¯1 nn¯
≤ ≤ ··· ≤
α
S 1
gi¯i α = .
≥ Cα ψ
i=1 n
Y
We have
g1¯1 α
(g1¯1)2 gi¯i S (g1¯1)2 K.
α−1;1
ψ ≤ ≤ ≤
i=2
Y
Therefore, for all 1 i n,
≤ ≤
S Cα−1(g1¯1)α−1 Cα−1(Kψ)α−1 K′.
α−1;i ≤ n ≤ n ≤
By Schwarz inequality,
1
2 S gi¯iRe u η 4 S + u 2 S (gi¯i)2 η 2
α−1;i i ¯i α−1;i α−1;i i
− ≤ 4|∇ | | |
i i i
(3.13) X (cid:8) (cid:9) X X
1
u 2 S (gi¯i)2 η 2 +C.
α−1;i i
≤ 4|∇ | | |
i
X
From (3.10), (3.12) and (3.13) we obtain
θ 1 1
0.
C − φ − u 2 ≤
|∇ |
This gives a bound for u when A is chosen sufficiently large.
|∇ |
In case (b) we have
η 2 η 2
(3.14) Sα−1;i(gi¯i)2|ηi|2 ≥ |∇η|2miinSα−1;i(gi¯i)2 ≥ |w∇α+|1 ≥ |N∇α+|1.
i
X
Substituting this into (3.10), we derive from (3.11) and (2.16),
η 2 C C C
(3.15) 2|N∇α+|1 ≤5αSα + φ + φ + u 2 −ǫ Sα−1;i(gi¯i)2.
(cid:16) |∇ | (cid:17)Xi
This gives a bound u C. (cid:3)
|∇ | ≤