˖ڍመڙጲ
http://www.paper.edu.cn
Here Q is the LYH quantity dened in Theorem 0.1 , which is a (p − 1, p − 1)-form valued
(Hermitian) quadratic form of V , Ric(ϕ) is a (p − 1, p − 1)-form dened by
Ric(ϕ)
I
p−1
,J
p−1
+ g
i
¯
l
g
k
¯
j
R
i
¯
j
ϕ
kI
p−1
,
¯
lJ
p−1
.
Here is how we organize the paper. In section 1, we prove that under the condition C
kp
the
r-positivity of the (p, p)-forms is preserved under the Hodge Laplacian heat equation. Section
2 and 3 are devoted to the proof of Theorem 0.1 and the proof of Theorem 0.2 respectively.
1 Maximum principle for r-positive (p, p)-forms
Let (M
m
, g) be a complete Kähler manifold of complex dimension m. Recall that a (p, p)-
form ϕ is called strongly positive if for any x ∈ M, ϕ can be written as a convex combination
ϕ(x) =
1
(p!)
2
γ
s
iα
s,1
∧ ¯α
s,1
∧ ··· ∧ iα
s,p
∧ ¯α
s,p
,
where γ
s
≥ 0 and α
s,µ
∈ Λ
1,0
T
∗
x
M for any s and 1 ≤ µ ≤ p. It is easy to check that ϕ is
strongly positive if and only if for any u ∈ Λ
p,0
T
1,0
x
M, (−
√
−1)
p
2
ϕ(u, ¯u) ≥ 0, where ϕ can be
locally written as
ϕ(x) =
(
√
−1)
p
2
(p!)
2
ϕ
I
p
,
¯
J
p
dz
I
p
∧ dz
¯
J
p
.
Here (z
1
, ··· , z
m
) are the local coordinates, I
p
= (i
1
, ··· , i
p
), J
p
= (j
1
, ··· , j
p
), dz
I
p
= dz
i
1
∧
··· ∧ dz
i
p
and dz
¯
J
p
= dz
¯
j
1
∧ ··· ∧ dz
¯
j
p
.
Recall that in [1], a (p, p)-form ϕ is said to be positive if for any x ∈ M and any vectors
v
1
, ··· , v
p
∈ T
1,0
x
M, ⟨ϕ,
1
√
−1
v
1
∧ ¯v
1
∧
1
√
−1
v
2
∧ ¯v
2
∧···∧
1
√
−1
v
p
∧ ¯v
p
⟩ ≥ 0. We simply write it as
ϕ ≥ 0 if ϕ is positive. Then ϕ ≥ 0 if and only if for any v
1
, ··· , v
p
∈ T
1,0
x
M,
ϕ
v
1
···v
p
,¯v
1
···¯v
p
+ (−
√
−1)
p
2
ϕ(v
1
, ··· , v
p
, ¯v
1
, ··· , ¯v
p
) =
ϕ
I
p
,
¯
J
p
v
I
p
v
¯
J
p
≥ 0.
Here v
I
p
= v
i
1
v
i
2
···v
i
p
, v
¯
J
p
= v
j
1
v
j
2
···v
j
p
. It is easy to check that when p = 0, 1, m − 1, m,
a (p, p)-form ϕ is positive if and only if ϕ is strong positive. In this paper, we mainly consider
another family of positive (p, p)-forms:
Denition 1.1. Let ϕ(x) be a (p, p)-form on a Kähler manifold (M
m
, g). We say ϕ(x) is
r-positive at the point x ∈ M, if for any X
s
= X
s,1
∧ X
s,2
∧ ··· ∧ X
s,p
∈ Λ
p,0
T
x
M, 1 ≤ s ≤ r,
(−
√
−1)
p
2
ϕ(
r
s=1
X
s
,
r
s=1
X
s
) ≥ 0. (1.1)
If ϕ is r-positive at any point x ∈ M, then we say that ϕ is a r-positive (p, p)-form on M
and simply write is as ϕ ≥
r
0. Furthermore, if for any nonzero (p, 0)-vector α =
r
s=1
X
s
,
(−
√
−1)
p
2
ϕ(α, ¯α) > 0, we say that ϕ is strictly r-positive, denoted by ϕ >
r
0.
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