# Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

Authors: Ruiwei Xu; Linfen Cao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Letf
(
x
) be a smooth strictly convex solution of det
(
∂
2
f
/
∂
x
i
∂
x
j
)
=
exp
(
1
/
2
)
∑
i
=
1
n
x
i
(
∂
f
/
∂
x
i
)
-
f defined on a domain Ω
⊂
R
n; then the graph M
∇
f of ∇
f is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space R
n
2
n with the indefinite metric ∑
d
x
i
d
y
i. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph M
∇
f is complete in R
n
2
n and passes through the origin then it is flat.


---

## Body

## 1. Introduction

LetM be an n-dimensional submanifold immersed into the Euclidean space R
n
+
m. Mean curvature flow is a one-parameter family X
t
=
X
(
·
,
t
) of immersions X
t
:
M
→
R
n
+
m with corresponding images M
t
=
X
t
(
M
) such that
(1)
d
d
t
X
(
x
,
t
)
=
H
(
x
,
t
)
,
x
∈
M
,
X
(
x
,
0
)
=
X
(
x
)
is satisfied, where H
(
x
,
t
) is the mean curvature vector of M
t at X
(
x
,
t
) in R
n
+
m. Self-similar solutions to the mean curvature flow play an important role in understanding the behavior of the flow and the types of singularities. They satisfy a system of quasilinear elliptic PDE of the second order as follows:
(2)
H
=
-
X
⊥
2
,
where (
⋯
)
⊥ stands for the orthogonal projection into the normal bundle N
M.Self-shrinkers in the ambient Euclidean space have been studied by many authors; for example, see [1–6] and so forth. For recent progress and related results, see the introduction in [7]. When the ambient space is a pseudo-Euclidean space, there are many classification works about self-shrinkers; for example, see [8–13] and so forth. But very little is known when self-shrinkers are complete not compact with respect to induced metric from pseudo-Euclidean space. In this paper, we will characterize self-shrinkers for Lagrangian mean curvature flow in the pseudo-Euclidean space from this aspect.Let(
x
1
,
…
,
x
n
;
y
1
,
…
,
y
n
) be null coordinates in 2n-dimensional pseudo-Euclidean space R
n
2
n. Then, the indefinite metric (cf. [14]) is defined by d
s
2
=
∑
i
=
1
n
‍
d
x
i
d
y
i. Suppose f
(
x
) is a smooth strictly convex function defined on domain Ω
⊂
R
n. The graph M
∇
f of ∇
f can be written as (
x
1
,
…
,
x
n
;
∂
f
/
∂
x
1
,
…
,
∂
f
/
∂
x
n
). Then, the induced Riemannian metric on M
∇
f is given by
(3)
G
=
∑
i
=
1
n
‍
∂
2
f
∂
x
i
∂
x
j
d
x
i
d
x
j
.
In particular, if function f satisfies
(4)
det
⁡
(
∂
2
f
∂
x
i
∂
x
j
)
=
exp
⁡
{
1
2
∑
i
=
1
n
‍
x
i
∂
f
∂
x
i
-
f
}
,
then the graph M
∇
f of ∇
f is a space-like self-shrinking solution for mean curvature flow in R
n
2
n.Huang and Wang [12] and Chau et al. [8] have used different methods to investigate the entire solutions to the above equation and showed that an entire smooth strictly convex solution to (4) in R
n is the quadratic polynomial under the decay condition on Hessian of f. Later Ding and Xin in [10] improve the previous ones in [8, 12] by removing the additional assumption and prove the following.Theorem 1.
Any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow inR
n
2
n with the indefinite metric ∑
i
‍
d
x
i
d
y
i is flat.These rigidity results assume that the self-shrinker graphs are entire. Namely, they are Euclidean complete. Here, we will characterize the rigidity of self-shrinker graphs from another completeness and pose the following problem.If a graphic self-shrinker is complete with respect to induced metric from ambient spaceR
n
2
n, then is it flat?In this paper, we will useaffine technique (see [15–18]) to prove the following Bernstein theorem. As a corollary, it gives a partial affirmative answer to the above problem.Theorem 2.
Letf
(
x
) be a C
∞ strictly convex function defined on a convex domain Ω
⊆
R
n satisfying the PDE (4). If there is a positive constant α depending only on n such that the hypersurface M
=
{
(
x
,
f
(
x
)
)
} in R
n
+
1 is complete with respect to the metric
(5)
G
~
=
exp
⁡
{
α
∑
‍
x
i
∂
f
∂
x
i
}
∑
‍
∂
2
f
∂
x
i
∂
x
j
d
x
i
d
x
j
,
then f is the quadratic polynomial.Remark 3.
Iff
(
x
) is a strictly convex solution to (4), then the graph {
(
x
,
∇
f
/
2
n
α
)
} is a minimal manifold in R
n
2
n endowed with the conformal metric d
s
2
=
exp
⁡
{
-
α
x
·
y
}
d
x
·
d
y.As a direct application of Theorem2, we have the following.Corollary 4.
Letf be a strictly convex C
∞-function defined on a convex domain Ω
⊂
R
n. If the graph M
∇
f
=
{
(
x
,
∇
f
(
x
)
)
} in R
n
2
n is a complete space-like self-shrinker for mean curvature flow and the sum ∑
‍
x
i
(
∂
f
/
∂
x
i
) has a lower bound, then M
∇
f is flat.When the shrinker passes through the origin especially, we have the following corollary.Corollary 5.
If the graphM
∇
f
=
{
(
x
,
∇
f
(
x
)
)
} in R
n
2
n is a complete space-like self-shrinker for mean curvature flow and passes through the origin, then M
∇
f is flat.


## 2. Preliminaries

Letf
(
x
1
,
…
,
x
n
) be a strictly convex C
∞-function defined on a domain Ω
⊂
R
n. Consider the graph hypersurface
(6)
M
:
=
{
(
x
,
f
(
x
)
)
∣
x
n
+
1
=
f
(
x
1
,
…
,
x
n
)
,
(
x
1
,
…
,
x
n
)
∈
Ω
}
.
For M, we choose the canonical relative normalization Y
=
(
0,0
,
…
,
1
). Then, in terms of the language of the relative affine differential geometry, theCalabi metric
(7)
G
=
∑
‍
f
i
j
d
x
i
d
x
j
is the relative metric with respect to the normalization Y. For the position vector y
=
(
x
1
,
…
,
x
n
,
f
(
x
1
,
…
,
x
n
)
), we have
(8)
y
,
i
j
=
∑
‍
A
i
j
k
y
k
+
f
i
j
Y
,
where “,” denotes the covariant derivative with respect to the Calabi metric G. We recall some fundamental formulas for the graph M; for details, see [19]. The Levi-Civita connection with respect to the metric G has theChristoffel symbols
(9)
Γ
i
j
k
=
1
2
∑
‍
f
k
l
f
i
j
l
.
TheFubini-Pick tensor
A
i
j
k satisfies
(10)
A
i
j
k
=
-
1
2
f
i
j
k
.
Consequently, for therelative Pick invariant, we have
(11)
J
=
1
4
n
(
n
-
1
)
∑
‍
f
i
l
f
j
m
f
k
n
f
i
j
k
f
l
m
n
.
TheGauss integrability conditions and theCodazzi equations read
(12)
R
i
j
k
l
=
∑
‍
f
m
h
(
A
j
k
m
A
h
i
l
-
A
i
k
m
A
h
j
l
)
,
(13)
A
i
j
k
,
l
=
A
i
j
l
,
k
.
From (12), we get the Ricci tensor
(14)
R
i
k
=
∑
‍
f
m
h
f
l
j
(
A
i
m
l
A
h
j
k
-
A
i
m
k
A
h
l
j
)
.
Introduce the Legendre transformation of f
(15)
ξ
i
=
∂
f
∂
x
i
,
i
=
1,2
,
…
,
n
,
u
(
ξ
1
,
…
,
ξ
n
)
=
∑
i
=
1
n
‍
x
i
∂
f
∂
x
i
-
f
(
x
)
.
Define the functions
(16)
ρ
:
=
[
det
⁡
(
∂
2
f
∂
x
i
∂
x
j
)
]
-
1
/
(
n
+
2
)
=
[
det
⁡
(
∂
2
u
∂
ξ
i
∂
ξ
j
)
]
1
/
(
n
+
2
)
,
Φ
:
=
∑
‍
f
i
j
(
ln
⁡
ρ
)
i
(
ln
⁡
ρ
)
j
=
∥
∇
ρ
∥
2
ρ
2
,
here and later the norm ∥
·
∥ is defined with respect to the Calabi metric. From the PDE (4), we obtain
(17)
∂
ln
⁡
ρ
∂
x
i
=
1
2
(
n
+
2
)
(
f
-
u
)
i
=
1
2
(
n
+
2
)
{
f
i
-
x
k
f
k
i
}
.
That is,
(18)
x
i
=
∑
‍
f
i
k
(
f
k
-
2
(
n
+
2
)
(
ln
⁡
ρ
)
k
)
.
Using (17) and (18), we can get
(19)
ρ
i
j
=
ρ
i
ρ
j
ρ
+
f
k
l
f
i
j
k
ρ
l
-
f
k
l
f
i
j
k
f
l
ρ
2
(
n
+
2
)
.
Put τ
:
=
(
1
/
2
)
∑
‍
f
i
j
(
ρ
i
/
ρ
)
f
j. From (19), we have
(20)
Δ
ρ
=
-
n
2
∥
∇
ρ
∥
2
ρ
+
τ
ρ
.
By (17), we get
(21)
4
(
n
+
2
)
2
Φ
=
∥
∇
u
∥
2
+
∥
∇
f
∥
2
-
2
(
f
+
u
)
,
and then
(22)
∥
∇
(
f
+
u
)
∥
2
=
4
(
n
+
2
)
2
Φ
+
4
(
f
+
u
)
.
Using (17) yields
(23)
Δ
(
f
+
u
)
=
2
n
+
(
n
+
2
)
2
Φ
.
Define a conformal Riemannian metric G
~
:
=
exp
⁡
{
α
(
f
+
u
)
}
G, where α is a constant.Conformal Ricci Curvature. Denote by R
~
i
j the Ricci curvature with respect to the metric G
~; then
(24)
R
~
i
j
=
R
i
j
-
(
n
-
2
)
α
2
(
f
+
u
)
,
i
j
+
(
n
-
2
)
α
2
4
(
f
+
u
)
,
i
(
f
+
u
)
,
j
-
1
2
(
α
Δ
(
f
+
u
)
+
(
n
-
2
)
α
2
2
∥
∇
(
f
+
u
)
∥
2
)
G
i
j
,
where “,” again denotes the covariant derivation with respect to the Calabi metric.Using the above formulas, we can get the following crucial estimates.Proposition 6.
Letf
(
x
1
,
…
,
x
n
) be a C
∞ strictly convex function satisfying PDE (4). Then, the following estimate holds:
(25)
Δ
Φ
≥
A
1
〈
∇
Φ
,
∇
ln
⁡
ρ
〉
+
1
4
〈
∇
(
f
+
u
)
,
∇
Φ
〉
-
A
2
∥
∇
Φ
∥
2
Φ
+
A
3
Φ
2
+
Φ
,
where
(26)
A
1
=
6
n
2
-
n
+
16
(
n
-
1
)
(
3
n
+
4
)
,
A
2
=
3
n
2
+
32
n
8
(
n
-
1
)
(
3
n
+
4
)
,
A
3
=
64
n
3
-
72
n
2
-
46
n
-
72
5
n
(
n
-
1
)
(
3
n
+
4
)
.Because its calculation is standard as in [16], we will give its proof in the appendix.For affine hyperspheres, Calabi in [20] calculated the Laplacian of the Pick invariant J. Later, for a general convex function, Li and Xu proved the following lemma in [17].Lemma 7.
The Laplacian of the relative Pick invariantJ satisfies
(27)
Δ
J
≥
n
+
2
n
(
n
-
1
)
∑
‍
f
i
l
f
j
m
f
k
n
A
i
j
k
(
ln
⁡
ρ
)
,
l
m
n
+
2
n
(
n
-
1
)
∥
∇
A
∥
2
+
2
J
2
-
(
n
+
2
)
4
4
Φ
2
,
where “,” denotes the covariant derivative with respect to the Calabi metric.Using Lemma7, we get the following corollary. For the proof, see the appendix.Corollary 8.
Letf
(
x
1
,
…
,
x
n
) be a C
∞ strictly convex function satisfying PDE (4); then
(28)
Δ
J
≥
J
2
-
20
(
n
+
2
)
8
Φ
2
+
1
4
〈
∇
J
,
∇
(
f
+
u
)
〉
+
J
-
n
(
n
-
1
)
∥
∇
(
f
+
u
)
∥
J
3
/
2
.


## 3. Proof of Theorem2

It is our aim to proveΦ
≡
0; thus, from definition of ρ,
(29)
det
⁡
(
f
i
j
)
=
const
.
everywhere on M. As in [8], by Euler homogeneous theorem, we get Theorem 2.Denote bys
(
p
0
,
p
) the geodesic distance function from p
0
∈
M with respect to the metric G
~. For any positive number a, let B
a
(
p
0
,
G
~
)
:
=
{
p
∈
M
∣
s
(
p
0
,
p
)
≤
a
}. Denote
(30)
A
:
=
max
⁡
B
a
(
p
0
,
G
~
)
{
(
a
2
-
s
2
)
2
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
}
,
B
:
=
max
⁡
B
a
(
p
0
,
G
~
)
{
(
a
2
-
s
2
)
2
exp
⁡
{
-
α
(
f
+
u
)
}
J
}
.Lemma 9.
Letf be a strictly convex C
∞-function satisfying the PDE (4). Then, there exist positive constants α and C, depending only on n, such that
(31)
A
≤
C
(
a
2
+
a
3
)
.Proof.
Step
1. We will prove that there exists a constant C depending only on n such that
(32)
A
≤
C
(
B
1
/
2
a
+
a
2
+
a
3
)
.
To this end, consider the function(33)
F
:
=
(
a
2
-
s
2
)
2
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
defined on B
a
(
p
0
,
G
~
), where α is a positive constant to be determined later. Obviously, F attains its supremum at some interior point p
*. We may assume that s
2 is a C
2-function in a neighborhood of p
*. Choose an orthonormal frame field on M around p
* with respect to the Calabi metric G. Then, at p
*,
(34)
Φ
,
i
Φ
-
α
(
f
+
u
)
,
i
-
4
s
s
,
i
a
2
-
s
2
=
0
,
(35)
Δ
Φ
Φ
-
∑
‍
(
Φ
,
i
)
2
Φ
2
-
α
Δ
(
f
+
u
)
-
12
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
4
s
Δ
s
a
2
-
s
2
≤
0
,
where “,” denotes the covariant derivative with respect to the Calabi metric G as before, and we used the fact ∥
∇
s
∥
G
2
=
exp
⁡
{
α
(
f
+
u
)
}. Inserting Proposition 6 into (35), we get
(36)
-
(
1
+
A
2
)
∑
‍
(
Φ
,
i
)
2
Φ
2
+
A
3
Φ
+
1
4
(
f
+
u
)
,
i
Φ
,
i
Φ
+
A
1
Φ
,
i
Φ
ρ
,
i
ρ
+
1
-
α
(
2
n
+
(
n
+
2
)
2
Φ
)
-
12
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
4
s
Δ
s
a
2
-
s
2
≤
0
.
Combining (34) with (36) and using the Schwarz inequality, we have
(37)
1
4
∑
‍
(
f
+
u
)
,
i
Φ
,
i
Φ
≥
1
8
α
∑
‍
[
(
f
+
u
)
,
i
]
2
-
2
α
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
,
A
1
∑
‍
Φ
,
i
Φ
ρ
,
i
ρ
≥
-
A
3
4
Φ
-
A
1
2
A
3
Φ
,
i
2
Φ
2
,
∑
‍
(
Φ
,
i
)
2
Φ
2
≤
2
(
α
2
∑
‍
[
(
f
+
u
)
,
i
]
2
+
16
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
)
.
Choose α small enough such that
(38)
2
(
1
+
A
2
+
A
1
2
A
3
)
α
≤
1
16
,
α
(
n
+
2
)
2
≤
A
3
4
,
100
n
α
≤
1
.
Then, by substituting the three estimates above, we get
(39)
A
3
2
Φ
+
1
16
α
(
f
+
u
)
,
i
2
-
exp
⁡
{
α
(
f
+
u
)
}
C
a
2
(
a
2
-
s
2
)
2
-
4
s
Δ
s
a
2
-
s
2
≤
0
,
here and later C denotes positive constant depending only on n.
Denotea
*
=
s
(
p
0
,
p
*
). If a
*
=
0, from (39), it is easy to complete the proof of the lemma. In the following, we assume that a
*
>
0. Now, we calculate the term 4
s
Δ
s
/
(
a
2
-
s
2
). Firstly, we will give a lower bound of the Ricci curvature Ric
(
M
,
G
~
). Assume that
(40)
max
⁡
B
a
*
(
p
0
,
G
~
)
{
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
}
=
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
(
p
~
)
,
max
⁡
B
a
*
(
p
0
,
G
~
)
{
exp
⁡
{
-
α
(
f
+
u
)
}
J
}
=
exp
⁡
{
-
α
(
f
+
u
)
}
J
(
q
~
)
.
For any p
∈
B
a
*
(
p
0
,
G
~
), by a coordinate transformation, f
i
j
(
p
)
=
δ
i
j and R
i
j
(
p
)
=
0  hold for i
≠
j. Then, at p,
(41)
R
i
i
≥
1
4
(
∑
m
‍
f
m
i
i
2
+
(
n
+
2
)
∑
m
‍
f
m
i
i
∂
∂
x
m
ln
⁡
ρ
)
≥
-
(
n
+
2
)
2
16
Φ
,
(
n
-
2
)
α
2
(
f
+
u
)
,
i
i
=
(
n
-
2
)
α
2
(
2
-
1
2
f
i
i
k
(
f
+
u
)
k
)
≤
(
n
-
2
)
α
+
(
n
-
2
)
α
2
4
∥
∇
(
f
+
u
)
∥
2
+
C
J
.
Then, using the Schwarz inequality and (22)–(24), we know that at the point p
(42)
Ric
(
M
,
G
~
)
≥
-
exp
⁡
{
-
α
(
f
+
u
)
}
×
{
C
Φ
+
C
J
+
α
[
3
(
n
-
2
)
α
(
f
+
u
)
+
2
(
n
-
1
)
]
}
G
~
.
If 3
(
n
-
2
)
α
(
f
+
u
)
+
2
(
n
-
1
)
≤
0, then
(43)
-
exp
⁡
{
-
α
(
f
+
u
)
}
α
[
3
(
n
-
2
)
α
(
f
+
u
)
+
2
(
n
-
1
)
]
≥
0
.
Otherwise,
(44)
exp
⁡
{
-
α
(
f
+
u
)
}
α
[
3
(
n
-
2
)
α
(
f
+
u
)
+
2
(
n
-
1
)
]
≤
C
.
Then, the Ricci curvature Ric
(
M
,
G
~
) on B
a
*
(
p
0
,
G
~
) is bounded from below by
(45)
Ric
(
M
,
G
~
)
≥
-
C
(
Φ
exp
⁡
{
α
(
f
+
u
)
}
(
p
~
)
+
J
exp
⁡
{
α
(
f
+
u
)
}
(
q
~
)
+
1
)
G
~
.
By the Laplacian comparison theorem, we get
(46)
s
Δ
s
a
2
-
s
2
=
exp
⁡
{
α
(
f
+
u
)
}
s
Δ
~
s
a
2
-
s
2
-
(
n
-
2
)
α
2
s
(
f
+
u
)
,
i
s
,
i
a
2
-
s
2
≤
C
3
exp
⁡
{
α
(
f
+
u
)
}
(
p
*
)
a
2
-
s
2
×
(
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
(
p
~
)
h
h
h
h
h
h
h
+
exp
⁡
{
-
α
(
f
+
u
)
}
J
(
q
~
)
+
1
)
s
+
α
16
(
f
+
u
)
,
i
2
+
C
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
,
where Δ
~ denotes the Laplacian with respect to the metric G
~.
Substituting (46) into (39) yields
(47)
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
≤
a
C
(
Φ
exp
⁡
{
α
(
f
+
u
)
}
(
p
~
)
h
h
h
h
h
h
h
h
+
J
exp
⁡
{
α
(
f
+
u
)
}
(
q
~
)
+
1
)
×
(
a
2
-
s
2
)
-
1
+
C
a
2
(
a
2
-
s
2
)
2
.
Note that
(48)
A
≥
[
(
a
2
-
s
2
)
2
exp
⁡
⁡
{
-
α
(
f
+
u
)
}
Φ
]
(
p
~
)
≥
(
a
2
-
s
2
)
2
(
p
*
)
exp
⁡
{
-
α
(
f
+
u
)
}
(
p
~
)
Φ
(
p
~
)
,
B
≥
[
(
a
2
-
s
2
)
2
exp
⁡
⁡
{
-
α
(
f
+
u
)
}
J
]
(
q
~
)
≥
(
a
2
-
s
2
)
2
(
p
*
)
exp
⁡
{
-
α
(
f
+
u
)
}
(
q
~
)
J
(
q
~
)
.
Multiplying by (
a
2
-
s
2
)
2
(
p
*
), at both sides of (47), yields
(49)
A
≤
C
a
(
A
1
/
2
+
B
1
/
2
)
+
C
(
a
2
+
a
3
)
.
Using the Schwarz inequality, we complete Step 1.
Step
2. We will prove that there is a constant C depending only on n such that
(50)
B
≤
C
(
A
+
a
2
+
a
4
)
.
Consider
(51)
H
=
(
a
2
-
s
2
)
2
exp
⁡
{
-
α
(
f
+
u
)
}
J
defined on B
a
(
p
0
,
G
~
), where α is the constant in (38). Obviously, H attains its supremum at some interior point q
*. Choose an orthonormal frame field on M around q
* with respect to the Calabi metric G. Then, at q
*,
(52)
J
,
i
J
-
α
(
f
+
u
)
,
i
-
4
s
s
,
i
a
2
-
s
2
=
0
,
(53)
Δ
J
J
-
∑
‍
(
J
,
i
)
2
J
2
-
α
Δ
(
f
+
u
)
-
12
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
4
s
Δ
s
a
2
-
s
2
≤
0
,
where “,” denotes the covariant derivative with respect to the Calabi metric G as before. Inserting Corollary 8 into (53), we get
(54)
J
-
20
(
n
+
2
)
8
Φ
2
J
+
1
4
∑
‍
J
,
i
J
(
f
+
u
)
,
i
+
1
-
∑
‍
(
J
,
i
)
2
J
2
-
α
(
2
n
+
(
n
+
2
)
2
Φ
)
-
n
(
n
-
1
)
∥
∇
(
f
+
u
)
∥
J
1
/
2
-
12
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
4
s
Δ
s
a
2
-
s
2
≤
0
.
Applying the Schwarz inequality, we have
(55)
1
4
∑
‍
J
,
i
J
(
f
+
u
)
,
i
≥
α
8
∑
‍
[
(
f
+
u
)
,
i
]
2
-
C
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
,
∑
(
J
,
i
)
2
J
2
≤
2
α
2
∑
‍
[
(
f
+
u
)
,
i
]
2
+
32
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
,
n
(
n
-
1
)
∥
∇
(
f
+
u
)
∥
J
1
/
2
≤
J
4
+
4
n
(
n
-
1
)
(
(
n
+
2
)
2
Φ
+
(
f
+
u
)
)
.
Inserting these estimates into (54) yields
(56)
3
4
J
-
20
(
n
+
2
)
8
Φ
2
J
-
C
Φ
+
α
16
∑
‍
(
f
+
u
)
,
i
2
-
C
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
C
(
f
+
u
)
-
4
s
Δ
s
a
2
-
s
2
≤
0
,
here and later C denotes different positive constants depending only on n.
We discuss two subcases.
Case
1. If
(57)
J
exp
⁡
{
α
(
f
+
u
)
}
(
q
*
)
≤
Φ
exp
⁡
{
α
(
f
+
u
)
}
(
q
*
)
,
then B
≤
A. In this case, Step 2 is complete.
Case
2. Now, assume that
(58)
J
exp
⁡
{
α
(
f
+
u
)
}
(
q
*
)
>
Φ
exp
⁡
{
α
(
f
+
u
)
}
(
q
*
)
.
Then, 1
>
(
Φ
/
J
)
(
q
*
). Thus,
(59)
3
4
J
-
C
Φ
+
α
16
∑
‍
[
(
f
+
u
)
,
i
]
2
-
C
a
2
exp
⁡
{
α
(
f
+
u
)
}
(
a
2
-
s
2
)
2
-
C
(
f
+
u
)
-
4
s
Δ
s
a
2
-
s
2
≤
0
.
The rest of the estimate is almost the same as in Step 1. The only difference is to deal with the term (
f
+
u
). If (
f
+
u
)
(
q
*
)
≤
0, then -
C
(
f
+
u
)
(
q
*
)
≥
0. We can drop this term.
Otherwise,exp
⁡
{
-
α
(
f
+
u
)
}
(
f
+
u
) has a uniform upper bound.
Using the same method as in Step 1, we can estimate the term4
s
Δ
s
/
(
a
2
-
s
2
) and finally get
(60)
B
≤
C
(
A
+
a
2
+
a
4
)
.
Then, combining the conclusion of Step 1, we get
(61)
A
≤
C
(
a
2
+
a
3
)
.
This completes the proof of Lemma 9.Proof of Theorem2.
For any pointq
∈
M, choose sufficient large constant R
0 such that q
∈
B
R
0
(
p
0
,
G
~
). Then, for all a
≥
R
0, q
∈
B
a
(
p
0
). Using Lemma 9, we know
(62)
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
(
q
)
≤
C
(
n
)
(
a
2
+
a
3
)
(
a
2
-
s
2
)
2
.
Now, let a
→
+
∞, and we have
(63)
0
≤
exp
⁡
{
-
α
(
f
+
u
)
}
Φ
(
q
)
≤
0
.
Consequently,
(64)
det
⁡
(
∂
2
f
∂
x
i
∂
x
j
)
(
q
)
=
const
.
This completes the proof of Theorem 2.


## 4. Appendix

Proof of Proposition6.
Letp
∈
M, and we choose a local orthonormal frame field of the metric G around p. Then,
(65)
Φ
=
∑
(
ρ
,
j
)
2
ρ
2
,
Φ
,
i
=
2
∑
‍
ρ
,
j
ρ
,
j
i
ρ
2
-
2
ρ
,
i
∑
(
ρ
,
j
)
2
ρ
3
,
Δ
Φ
=
2
∑
(
ρ
,
j
i
)
2
ρ
2
+
2
∑
‍
ρ
,
j
ρ
,
j
i
i
ρ
2
-
8
∑
‍
ρ
,
j
ρ
,
i
ρ
,
j
i
ρ
3
+
(
n
+
6
)
Φ
2
-
2
τ
Φ
,
where we used (20). In the case Φ
(
p
)
=
0, it is easy to get, at p,
(66)
Δ
Φ
≥
2
∑
(
ρ
,
i
j
)
2
ρ
2
.
Now, we assume that Φ
(
p
)
≠
0. Choose a local orthonormal frame field of the metric G around p such that ρ
,
1
(
p
)
=
∥
∇
ρ
∥
(
p
)
>
0, ρ
,
i
(
p
)
=
0, for all i
>
1. Then,
(67)
Δ
Φ
=
2
(
1
-
δ
+
δ
)
∑
‍
(
ρ
,
i
j
)
2
ρ
2
+
2
∑
‍
ρ
,
j
ρ
,
j
i
i
ρ
2
-
8
(
ρ
,
1
)
2
ρ
,
11
ρ
3
+
(
n
+
6
)
Φ
2
-
2
τ
Φ
,
where 1
>
δ
>
0 is a constant to be determined later. Applying (20), we obtain
(68)
2
∑
(
ρ
,
i
j
)
2
ρ
2
≥
2
n
n
-
1
(
ρ
,
11
)
2
ρ
2
+
4
∑
i
>
1
(
ρ
,
1
i
)
2
‍
ρ
2
+
2
n
n
-
1
(
ρ
,
1
)
2
ρ
,
11
ρ
3
+
n
2
2
(
n
-
1
)
Φ
2
-
4
n
-
1
ρ
,
11
ρ
τ
+
2
n
-
1
τ
2
-
2
n
n
-
1
Φ
τ
.
An application of the Ricci identity shows that
(69)
2
ρ
2
∑
ρ
,
j
ρ
,
j
i
i
=
-
2
n
(
ρ
,
1
)
2
ρ
,
11
ρ
3
+
n
(
ρ
,
1
)
4
ρ
4
+
2
R
11
(
ρ
,
1
)
2
ρ
2
+
2
ρ
,
1
ρ
2
(
ρ
τ
)
,
1
.
Substituting (68) and (69) into (67), we obtain
(70)
Δ
Φ
≥
2
δ
∑
‍
(
ρ
,
i
j
)
2
ρ
2
+
(
-
2
n
-
8
+
2
n
(
1
-
δ
)
n
-
1
)
×
(
ρ
,
1
)
2
ρ
,
11
ρ
3
+
2
R
11
(
ρ
,
1
)
2
ρ
2
+
(
n
2
(
1
-
δ
)
2
(
n
-
1
)
+
2
(
n
+
3
)
)
(
ρ
,
1
)
4
ρ
4
+
2
ρ
,
1
ρ
2
(
ρ
τ
)
,
1
-
4
n
-
2
-
2
n
δ
n
-
1
Φ
τ
+
(
1
-
δ
)
×
(
2
n
n
-
1
(
ρ
,
11
)
2
ρ
2
+
4
∑
i
>
1
(
ρ
,
1
i
)
2
‍
ρ
2
h
h
h
h
2
n
n
-
1
(
ρ
,
11
)
2
ρ
2
+
4
∑
i
>
1
(
ρ
,
1
i
)
2
‍
ρ
2
-
4
n
-
1
ρ
,
11
ρ
τ
+
2
n
-
1
τ
2
)
.
Note that
(71)
(
ρ
,
11
)
2
ρ
2
=
1
4
∑
‍
(
Φ
,
i
)
2
Φ
-
∑
i
>
1
(
ρ
,
1
i
)
2
‍
ρ
2
+
2
(
ρ
,
1
)
2
ρ
,
11
ρ
3
-
(
ρ
,
1
)
4
ρ
4
.
Then, (70) and (71) together give us
(72)
Δ
Φ
≥
2
δ
∑
‍
(
ρ
,
i
j
)
2
ρ
2
+
n
(
1
-
δ
)
2
(
n
-
1
)
∑
‍
(
Φ
,
i
)
2
Φ
+
(
6
n
(
1
-
δ
)
n
-
1
-
2
(
n
+
4
)
)
(
ρ
,
1
)
2
ρ
,
11
ρ
3
+
2
R
11
(
ρ
,
1
)
2
ρ
2
+
[
(
n
2
-
4
n
)
(
1
-
δ
)
2
(
n
-
1
)
+
2
(
n
+
3
)
]
(
ρ
,
1
)
4
ρ
4
+
1
-
δ
n
-
1
(
2
τ
2
-
4
ρ
,
11
ρ
τ
)
-
4
n
-
2
-
2
n
δ
n
-
1
Φ
τ
+
2
ρ
,
1
ρ
2
(
ρ
τ
)
,
1
.
Using the Schwarz inequality gives
(73)
2
ρ
,
11
ρ
τ
≤
7
3
∑
‍
(
ρ
,
i
j
)
2
ρ
2
+
3
7
τ
2
.
Using
(74)
(
ρ
,
1
)
2
ρ
,
11
ρ
3
=
1
2
Φ
,
i
ρ
,
i
ρ
+
Φ
2
,
and choosing δ
=
7
/
(
3
n
+
4
), we get
(75)
Δ
Φ
≥
n
(
1
-
δ
)
2
(
n
-
1
)
∑
‍
(
Φ
,
i
)
2
Φ
+
(
3
n
(
1
-
δ
)
n
-
1
-
(
n
+
4
)
)
×
∑
‍
Φ
,
i
ρ
,
i
ρ
+
2
R
11
(
ρ
,
1
)
2
ρ
2
+
[
(
n
2
+
8
n
)
(
1
-
δ
)
2
(
n
-
1
)
-
2
]
Φ
2
+
8
(
1
-
δ
)
7
(
n
-
1
)
τ
2
-
4
n
-
2
-
2
n
δ
n
-
1
Φ
τ
+
2
ρ
,
1
ρ
2
(
ρ
τ
)
,
1
.
In the following, we will calculate the termsR
11
(
(
ρ
,
1
)
2
/
ρ
2
) and (
ρ
,
1
/
ρ
2
)
(
ρ
τ
)
,
1. Note that (17) is invariant under an affine transformation of coordinates that preserved the origin. So, we can choose the coordinates x
1
,
x
2
,
…
,
x
n such that f
i
j
(
p
)
=
δ
i
j and ∂
ρ
/
∂
x
1
=
∥
grad
ρ
∥
(
p
)
>
0, (
∂
ρ
/
∂
x
i
)
(
p
)
=
0, for all i
>
1. From (19), we easily obtain
(76)
ρ
,
i
j
=
ρ
i
j
+
A
i
j
1
ρ
,
1
=
ρ
,
i
ρ
,
j
ρ
-
A
i
j
1
ρ
,
1
+
A
i
j
k
f
k
ρ
n
+
2
.
Thus, we get
(77)
Φ
,
i
=
2
ρ
,
1
ρ
,
1
i
ρ
2
-
2
ρ
,
i
(
ρ
,
1
)
2
ρ
3
=
-
2
A
i
11
(
ρ
,
1
)
2
ρ
2
+
2
ρ
,
1
f
k
A
k
i
1
(
n
+
2
)
ρ
,
(78)
∑
Φ
,
i
ρ
,
i
ρ
=
-
2
A
111
(
ρ
,
1
)
3
ρ
3
+
2
f
k
A
k
11
n
+
2
(
ρ
,
1
)
2
ρ
2
.
By the same method, as deriving (69), we have
(79)
∑
(
A
m
l
1
)
2
≥
n
n
-
1
∑
‍
(
A
i
11
)
2
-
2
n
-
1
A
111
∑
‍
A
i
i
1
+
1
n
-
1
(
∑
‍
A
i
i
1
)
2
.
Note that ∑
‍
A
i
i
1
=
(
(
n
+
2
)
/
2
)
(
ρ
1
/
ρ
). Therefore, by (14), (77), (78), and (79), we obtain
(80)
2
R
11
(
ρ
,
1
)
2
ρ
2
=
2
∑
‍
(
A
k
j
1
)
2
(
ρ
,
1
)
2
ρ
2
-
(
n
+
2
)
A
111
(
ρ
,
1
)
3
ρ
3
≥
n
2
(
n
-
1
)
∑
‍
(
Φ
,
i
-
2
(
ρ
,
1
f
k
A
k
i
1
/
(
n
+
2
)
ρ
)
)
2
Φ
+
(
n
+
2
)
(
n
+
1
)
2
(
n
-
1
)
∑
Φ
,
i
ρ
,
i
ρ
-
n
+
1
n
-
1
f
k
A
k
11
(
ρ
,
1
)
2
ρ
2
+
(
n
+
2
)
2
2
(
n
-
1
)
Φ
2
.
On the other hand, we have
(81)
2
ρ
,
1
ρ
2
(
ρ
τ
)
,
1
=
2
Φ
τ
+
1
n
+
2
∑
‍
A
1
i
k
f
k
f
i
ρ
,
1
ρ
+
Φ
.
Then, inserting (80) and (81) into (75), we get
(82)
Δ
Φ
≥
2
n
-
n
δ
2
(
n
-
1
)
∑
‍
(
Φ
,
i
)
2
Φ
-
(
n
+
2
)
(
n
-
5
)
+
6
n
δ
2
(
n
-
1
)
×
∑
‍
Φ
,
i
ρ
,
i
ρ
+
Φ
+
2
(
n
+
2
)
2
-
(
n
2
+
8
n
)
δ
2
(
n
-
1
)
Φ
2
+
8
(
1
-
δ
)
7
(
n
-
1
)
τ
2
-
2
n
(
1
-
δ
)
n
-
1
Φ
τ
+
1
n
+
2
×
∑
‍
A
1
i
k
f
k
f
i
ρ
,
1
ρ
-
n
+
1
n
-
1
(
ρ
,
1
)
2
ρ
2
f
k
A
k
11
-
2
n
(
n
-
1
)
(
n
+
2
)
∑
‍
Φ
,
i
f
k
A
k
i
1
Φ
+
2
n
(
n
-
1
)
(
n
+
2
)
2
×
∑
‍
(
f
k
A
k
i
1
)
2
.
Using (77), we have
(83)
1
n
+
2
∑
‍
A
1
i
k
f
k
f
i
ρ
,
1
ρ
-
n
+
1
n
-
1
(
ρ
,
1
)
2
ρ
2
f
k
A
k
11
=
1
2
f
i
Φ
,
i
-
2
n
-
1
(
ρ
,
1
)
2
ρ
2
f
k
A
k
11
.
One observes that the Schwarz inequality gives
(84)
2
n
(
n
-
1
)
(
n
+
2
)
∑
‍
Φ
,
i
f
k
A
k
i
1
Φ
≤
9
n
8
(
n
-
1
)
∑
‍
(
Φ
,
i
)
2
Φ
+
8
n
9
(
n
-
1
)
(
n
+
2
)
2
×
∑
‍
(
f
k
A
k
i
1
)
2
,
2
n
-
1
(
ρ
,
1
)
2
ρ
2
f
k
A
k
11
≤
9
(
n
+
2
)
2
10
n
(
n
-
1
)
Φ
2
+
10
n
9
(
n
-
1
)
(
n
+
2
)
2
×
∑
‍
(
f
k
A
k
i
1
)
2
,
2
n
Φ
τ
≤
τ
2
+
n
2
Φ
2
.
Note that by (17) we have
(85)
1
4
f
i
j
Φ
j
f
i
=
n
+
2
2
f
i
j
Φ
j
(
ln
⁡
ρ
)
i
+
1
4
Φ
j
x
j
=
n
+
2
2
f
i
j
Φ
j
(
ln
⁡
ρ
)
i
+
1
4
f
i
j
∂
Φ
∂
x
i
∂
u
∂
x
j
.
Then, inserting these estimates into (82) yields Proposition 6.Proof of Corollary8.
Now, we will calculate the term(
ln
⁡
ρ
)
,
i
j
k. In particular, if f satisfies PDE (4), choose the coordinate (
x
1
,
x
2
,
…
,
x
n
) such that f
i
j
(
p
)
=
δ
i
j; then we have
(86)
(
ln
⁡
ρ
)
,
i
j
k
=
1
n
+
2
(
A
i
j
k
+
A
i
j
k
,
p
f
,
p
)
-
(
ln
⁡
ρ
)
,
l
A
i
j
k
,
l
+
A
i
j
l
A
k
l
p
(
3
(
ln
⁡
ρ
)
,
p
-
2
n
+
2
f
,
p
)
.
Using (17), we have
(87)
3
(
ln
⁡
ρ
)
,
p
-
2
n
+
2
f
,
p
=
-
1
n
+
2
(
f
+
u
)
,
p
+
(
ln
⁡
ρ
)
,
p
.
By the Young inequality and the Schwarz inequality, we have
(88)
n
+
2
n
(
n
-
1
)
∑
‍
A
i
j
k
A
i
j
l
A
k
l
h
(
ln
⁡
ρ
)
,
h
≤
1
2
J
2
+
16
n
2
(
n
-
1
)
2
(
n
+
2
)
4
Φ
2
,
1
n
(
n
-
1
)
∑
‍
A
i
j
k
A
i
j
l
,
k
f
l
=
1
2
∑
‍
J
,
l
f
,
l
=
1
4
〈
∇
J
,
∇
(
f
+
u
)
〉
+
n
+
2
2
〈
∇
J
,
∇
ln
⁡
ρ
〉
,
n
+
2
n
(
n
-
1
)
∑
A
i
j
k
A
i
j
l
,
k
(
ln
⁡
ρ
)
,
l
=
n
+
2
2
∑
‍
J
,
i
(
ln
⁡
ρ
)
,
i
≤
1
n
(
n
-
1
)
∑
‍
(
A
i
j
k
,
l
)
2
+
(
n
+
2
)
2
4
J
Φ
≤
1
n
(
n
-
1
)
∑
‍
(
A
i
j
k
,
l
)
2
+
1
4
J
2
+
(
n
+
2
)
4
16
Φ
2
,
1
n
(
n
-
1
)
∑
‍
A
i
j
k
A
j
i
l
A
k
l
p
(
f
+
u
)
,
p
≤
n
(
n
-
1
)
∥
∇
(
f
+
u
)
∥
J
3
/
2
.
Thus, by inserting (88) into Lemma 7, we obtain Corollary 8.



---

*Source: 196751-2014-07-06.xml*