Table Of ContentA LIOUVILLE TYPE THEOREM FOR SPECIAL
LAGRANGIAN EQUATIONS WITH CONSTRAINTS
8 MICAH WARRENAND YUYUAN
0
0
2 Abstract. We derive a Liouville type result for special Lagrangian
equationswithcertain“convexity”andrestrictedlineargrowthassump-
n
tions on the solutions.
a
J
8
] 1. Introduction
P
A
In this note, we show the following
.
h
t Theorem1.1. Letube asmooth solution to the special Lagrangian equation
a
m n
(1.1) arctanλ = c on Rn,
[ i
Xi=1
2
where λ s are the eigenvalues of the Hessian D2u(x). Suppose that
v i
9 (1.2) 3+(1 ε)λ2(x)+2λ (x)λ (x) 0
9 − i i j ≥
6 for all i, j, x and any small fixed ε > 0; and the gradient u(x) satisfies
0 ∇
. (1.3) u(x) δ(n) x
6 |∇ | ≤ | |
0 for large x and any fixed δ(n) < 1/√n 1. Then u must be a quadratic
7 | | −
polynomial.
0
:
v The special Lagrangian equation (1.1) arises in the calibrated geometry
i [HL]. A Lagrangian graph M = (x, u(x)) Cn = Rn Rn is called special
X ∇ ⊂ ×
when the calibrating n-form
r
a
Ω = Re(e √ 1cdz dz ... dz )
c − − 1 2 n
∧ ∧ ∧
is equal to the induced volume form along M; equivalently, u satisfies (1.1).
The equation (1.1) holds if and only if the gradient graph (x, u(x)) Cn
is a (volume minimizing) minimal surface in Rn Rn [HL, ∇Theorem⊂2.3,
×
Proposition 2.17].
By Fu’s classification result [F], any global solution to (1.1) on R2 is ei-
ther quadratic or harmonic; a harmonic function with any linear growth
condition on the gradient is certainly quadratic; see also [Y3] for a unique-
ness result for the global solutions to (1.1) with c > (n 2) π. In the case
| | − 2
n = 3, other Liouville-Bernstein type results hold true for (1.1) under the
Date: February 1, 2008.
Y.Y. is partially supported byan NSFgrant.
1
2 MICAHWARRENANDYUYUAN
following conditions respectively: λ K [Y2]; λ λ K [Y4]; or c = π
i i j
≥ − ≥ −
and the solution is strictly convex with quadratic growth [BCGJ]. While
boundedness of the Hessian alone is sufficient in dimension three, certain
boundedness and convexity are both needed for Liouville-Bernstein type re-
sults to be valid for (1.1) in the general dimension (n 4). The results hold
≥
with the assumptions that c = kπ and the solution is convex with linear
growth [B]; with the almost convex assumption λ ε(n) [Y2]; with the
i
≥ −
semi-convex assumption λ 1 + γ everywhere, or with the (“equiva-
i ≥ −√3
lent”) assumption λ √3 γ everywhere [Y4]; or with the assumption
i ′
| | ≤ −
λ λ 1 ε(n) [Y4]. (It is straightforward that any convex solution with
i j
≥ − −
aboundedHessian to (1.1) is aquadratic polynomial, by the well-known Cα
Hessian estimate of Krylov-Evans for now the convex elliptic equation (1.1);
see also [X, p. 217–218] for a different approach via the iteration argument
of [HJW].) A Liouville-Bernstein type result with the assumption λ K
i
| | ≤
and λ λ const > 3 was stated in [TW].
i j ≥ −2
The more general “convexity” condition (1.2) does not alone lead to any
Hessianboundforthesolutionsto(1.1),butdoesguaranteesthatthevolume
element V, which is a geometric combination of the eigenvalues, is subhar-
monic. Better yet, the Laplacian of V bounds its gradient; see Lemma 2.1,
which is a key piece in our proof of Lemma 2.2 on our Hessian estimates.
In fact, this paper grows out of our attempts towards deriving a Hessian
estimate in terms of the gradient, for solutions to the special Lagrangian
equation (1.1). The unpleasant technical assumption δ(n) < 1/√n 1 in
−
(1.3) reflects thelimitation of our currentarguments; theassumption is nec-
essary for us to push the Bernstein-Pogorelov-Korevaar technique to obtain
a Hessian estimate for special Lagrangian equations; see Lemma 2.2.
Once a Hessian bound for solutions to (1.1) is available, the “standard”
blow-down process from the geometric measure theory will show that the
global solution is a quadratic polynomial, provided certain convexity con-
ditions like (1.2) or others are available in the whole process (for n 4).
≥
(Unlike [JX], we could not generalize the iteration argument in [HJW] to
get a Liouville type result for now the larger image set (1.2) of the corre-
spondingharmonicGaussmaptotheLagrangianGrassmanian.) Thesimple
constraints λ K like λ 1 or λ √3 γ are easily shown to be
i i i
| | ≤ | | ≤ | | ≤ −
available in the blow-down process. An extra effort is needed to justify that
the nonlinear constraints (1.2) or others like λ λ const are preserved un-
i j
≥
der the C1,α convergence of the scaling process u (x) =u k2x /k2. Taking
k
advantage of the single elliptic equation (1.1), we apply the W2,δ estimates
(cid:0) (cid:1)
for solutions in terms of the supreme norm of the solution to extract a W2,δ
sub-convergent sequence, as in [Y1]. Then we extract another subsequence
withtheHessians converging almost everywhere. Thisjustifiesthatthecon-
straints (1.2) are preserved in the above blow-down process. Another route
of the justification is through Allard’s regularity result (cf. [S, Section 36]).
A LIOUVILLE TYPE THEOREM 3
Actually, Theorem 1.1 holds true for n = 3 without any growth condition
like (1.3). The condition (1.2) implies λ λ K, so as in [Y4] we can find
i j
≥ −
a bound on the Hessian (possibly for a new potential), and then draw the
conclusion. Note that the boundedness on the Hessian alone for n = 3 is
enoughforonetoruntheblow-downprocesstoobtainaLiouvilletyperesult;
see [F-C, Theorem 5.4] of Fischer-Colbrie. In general dimension n 4, we
≥
derive yet another Liouville-Bernstein type result for the solutions to (1.1)
with the bounded Hessian satisfying weaker constraints (3.1); see Theorem
3.1 in the appendix. One consequence of Theorem 3.1 coupled with the De
Giorgi-Allard ε-regularity theory is an improvement of the above mentioned
Liouville-Bernstein type result in [Y4], namely, any global solution to (1.1)
with λ 1 ε(n) everywhere or λ √3 + ε (n) everywhere is a
i ≥ −√3 − | i| ≤ ′
quadratic polynomial (for n 4). The argument is identical to the one in
≥
[Y2] with Proposition 2.1 there replaced by Proposition 3.1 here.
The desired Hessian estimate for special Lagrangian equations in the two
dimensional case follows from the gradient estimates in terms of the heights
of the two dimensional minimal graphs with any codimension by Gregori
[G], where some Jacobian estimates of Heinz were employed. For higher
dimensional and codimensional minimal graphs with the assumption that
the product of any two slopes is between 1 and 1, the gradient estimates
−
were obtained in [W], using an integral method developed for codimension
one minimal graphs. The gradient estimate for codimension one minimal
graphs is by now a classical result.
The general Hessian estimate for special Lagrangian equations is still a
puzzling issue to us.
Notation. ∂ = ∂ , ∂ = ∂2 , u = ∂ u, u = ∂ u, etc.
i ∂xi ij ∂xi∂xj i i ji ij
2. Proof Of Theorem 1.1
TakingthegradientofbothsidesofthespecialLagrangianequation(1.1),
we have
n
(2.1) gij∂ (x, u(x))= 0,
ij
∇
i,j
X
where gij is the inverse of the induced metric g = (g ) = I+D2uD2u on
ij
the surface (x, u(x)) Rn Rn. Simple geometric manipulation of (2.1)
(cid:0) (cid:1) ∇ ⊂ ×
yields the usual form of the minimal surface equation
(x, u(x)) = 0,
g
△ ∇
where the Laplace-Beltrami operator of the metric g is given by
n
1
= ∂ detggij∂ .
g i j
△ √detg
Xi,j (cid:16)p (cid:17)
4 MICAHWARRENANDYUYUAN
Because we are using harmonic coordinates x = 0, we see that also
g g
△ △
equals the linearized operator of the special Lagrangian equation (1.1) at u,
n
= gij∂ .
g ij
△
i,j
X
The gradient and inner product with respect to the metric g are
n n
v = g1kv , , gnkv
g k k
∇ ···
!
k=1 k=1
X X
n
v, w = gijv w , in particular v 2 = v, v .
h∇g ∇g ig i j |∇g |g h∇g ∇g ig
i,j=1
X
We begin by demonstrating a Jacobi inequality for the volume element
n
1
V = detg = (1+λ2)2.
i
i=1
p Y
Lemma 2.1. Suppose that u is a smooth solution to (1.1) satisfying (1.2).
Then
ε
lnV lnV 2
△g ≥ n|∇g |g
or equivalently
(2.2) △gV nε ≥ 2|∇gVVnεnε|2g.
Proof. By differentiating the minimal surface equation (2.1) again and per-
forming some long and tedious computation, one gets the standard formula
for lnV;seeforexample [Y2, Lemma2.1]. (Thegeneralformulafor min-
g
△
imalsubmanifoldsofanydimensionorcodimensionoriginates inSimons[Ss,
p. 90].) At any fixed point, we assume that D2u is diagonalized, then
n
lnV = (1+λ λ )h2 ,
△g i j ijk
i,j,k=1
X
where h = gii gjj gkku . Gathering all terms containing h2 =
ijk ijk ijj
h2 = h2 for a fixed i, we have
jij jji p p p
(1+λ2)h2 + (1+λ2)h2 + (1+λ λ )h2 + (1+λ λ )h2
i iii j jji i j ijj j i jij
j=i j=i j=i
X6 X6 X6
= (1+λ2)h2 + (3+λ2+2λ λ )h2 .
i iii j i j jji
j=i
X6
A LIOUVILLE TYPE THEOREM 5
Thus
n
(2.3) lnV = (1+λ2)h2 + (3+λ2+2λ λ )h2
△g i iii j i j jji
i=1 j=i
X X6
+2 (3+λ λ +λ λ +λ λ )h2 .
i j j k k i ijk
i<j<k
X
Condition (1.2) gives that
3+(1 ε)λ2+λ λ +λ λ +λ (λ +λ ) λ (λ +λ ) 0
− i i j i j k j i − k j i ≥
that is
S = 3+λ λ +λ λ +λ λ (λ λ )(λ +λ )+ελ2.
ijk i j j k k i ≥ k − i i j i
Switching λ and λ we also have
i j,
S = S (λ λ )(λ +λ )+ελ2.
ijk jik ≥ k − j j i j
By symmetry of S , we may assume
ijk
(2.4) λ λ λ ,
i k j
≥ ≥
then either (λ λ )(λ +λ ) or (λ λ )(λ +λ ) has to be non-negative,
k i i j k j j i
− −
thus
(2.5) S εmin λ2,λ2 .
ijk ≥ i j
We conclude that (cid:8) (cid:9)
n
(2.6) lnV (1+λ2)h2 + (3+λ2+2λ λ )h2 .
△g ≥ i iii j i j jji
i j=i
X X6
To bound the gradient, we compute, (still at the same fixed point with D2u
diagonalized,)
n
∂ lnV = gjjλ u ,
i j jji
j=1
X
then
2
n n
lnV 2 = gii gjjλ u
|∇g |g j jji
i=1 j=1
X X
2
n n n
(2.7) = λ h n λ 2h2 .
j jji ≤ j jji
i=1 j=1 i,j=1
X X X
6 MICAHWARRENANDYUYUAN
Combining (1.2) with (2.6) and (2.7) we have
ε
lnV lnV 2
g g
△ − n|∇ |
n
(2.8) ( 1+(1 ε)λ2 h2 + ( 3+(1 ε)λ2 +2λ λ h2 0.
≥ − i iii − j i j jji ≥
i=1 j=i
X (cid:2) (cid:3) X6 (cid:2) (cid:3)
The proof of Lemma 2.1 is complete. (cid:3)
Lemma 2.2. Suppose that u is a smooth solution to (1.1) on B (0) satis-
1
fying condition (1.2) and
1
u δ < .
|∇ |≤ √n 1
−
Then
D2u(0) C(n,δ,ε).
| | ≤
Proof. Set
1
v = u+α u(0),x or u+αx if u(0) = 0,
1
u(0) h∇ i ∇
|∇ |
where α = 1 δ /2. Now v satisfies in B the following
√n 1 − 1
−
(cid:16) (cid:17)
1
D2v = D2u, v(0) α, and v α+δ < .
|∇ |≥ |∇ | ≤ √n 1
−
ε
Set b = V n, and consider the function
+
w = ηb = v 2 (α+δ)2 x2 b 0.
|∇ | − | | ≥
h i
Apositive maximum forw willbeattained atapointpon theinterior, since
w(0) > 0 and w(x) vanishes on the boundary ∂B . At this point p,
1
η
(ηb) = 0 or η = b,
g g g
∇ ∇ −b∇
0 (ηb) = η b+2 η, b +b η
≥ △g △g h∇g ∇g ig △g
b 2
= η b 2|∇g |g +b η
g g
△ − b △
!
b η,
g
≥ △
A LIOUVILLE TYPE THEOREM 7
by the inequality (2.2) in Lemma 2.1. This last inequality implies a bound
on D2v(p) as the following. We have
| |
0 η = v 2 (α+δ)2 x2
g g
≥ △ △ |∇ | − | |
n hn i
= gij 2 (v v +v ∂ v ) (α+δ)2∂ x2
ki kj k ij k ij
− | |
" #
i,j=1 k=1
X X
n λ2 (α+δ)2 λ2 (α+δ)2
= 2 i − 2 1− (n 1)(α+δ)2 ,
1+λ2 ≥ 1+λ2 − −
i=1 i " 1 #
X
using the minimal surface equation (2.1) and assuming λ λ for all i.
1 i
| | ≥ | |
It follows that
1+(α+δ)2
1+λ2(p) .
1 ≤ 1 (n 1)(α+δ)2
− −
We get
ε
1+(α+δ)2 2
α2b(0) v(0) 2b(0) η(p)b(p) (α+δ)2 ,
≤ |∇ | ≤ ≤ "1 (n 1)(α+δ)2#
− −
then
δ 4εn 1+(α+δ)2 n
(2.9) 1+λ2(0) 1+ .
i ≤ α "1 (n 1)(α+δ)2#
(cid:18) (cid:19) − −
Therefore, we conclude the estimate D2u(0) C(n,δ,ε) in Lemma 2.2.
| | ≤ (cid:3)
Lemma 2.3. Let u C (Rn 0 ) be a solution to the special Lagrangian
∞
∈ \{ }
equation (1.1) and homogeneous of order 2; that is, u(x) = x 2u(x/ x ).
| | | |
Suppose that the eigenvalues λ of the Hessian D2u(x) satisfy (1.2). Then
i
u must be quadratic.
Proof. Lemma 2.3 follows from Proposition 3.1; nonetheless we give a direct
proof in the following. Considering (1.2), (2.4), and (2.5), we observe that
the coefficients of h2 in (2.3) are strictly positive. Accordingly,
ijk
n
(2.10) lnV c(λ) h2
△g ≥ ijk
i,j,k=1
X
with c(λ)> 0.
Since u is homogeneous of order 2, the homogeneous order 0 function
lnV attains its maximum along a ray. We infer from the strong maximum
principle that lnV const. It follows from (2.10) that D3u 0. Therefore,
u must be quadrati≡c, as claimed in Lemma 2.3. ≡ (cid:3)
Proof of Theorem 1.1. Now the Hessian bound is available by Lemma 2.2.
We run the “routine” blow-down procedure “in detail” to finish the proof of
Theorem 1.1, as in [Y2].
8 MICAHWARRENANDYUYUAN
Step 1. From the assumption that u(x) δ x for large x, we have on
the ball B (p) with any fixed p Rn|∇ | ≤ | |
R
∈
δ p
u(x) δ( p +R)= δ+ | | R.
|∇ | ≤ | | R
(cid:18) (cid:19)
A rescaled version of Lemma 2.2 with R going to then leads to a Hessian
bound, D2u(p) C(n,δ,ε) , K, which must ho∞ld at each point p Rn.
| | ≤ ∈
Step 2. Repeating verbatim the argument in [Y1, p.263–264], we show
that we can find a tangent cone of the special Lagrangian graph (x, u(x))
∇
at whosepotentialfunctionisC1,1,homogenousorder2,andstillsatisfies
∞
the “convexity” condition (1.2).
Without loss of generality, we assume u(0) = 0, ▽u(0) = 0. We “blow
down” u at .
∞
Set
u(kx)
u (x) = , k = 1,2,3, .
k k2 ···
We see that
u C(K,R),
k kkC1,1(BR) ≤
so there exists a subsequence, still denoted by u and a function u
k R
{ } ∈
C1,1(B ) such that u u in C1,α(B ) as k , and D2u K.
R k R R R
→ → ∞ ≤
By the fact that the family of viscosity solution is closed under C0 uniform
(cid:12) (cid:12)
limit, we know that u is also a viscosity solution of (cid:12) (cid:12)
R
n
F D2u = arctanλ = c on B .
i R
i=1
(cid:0) (cid:1) X
Applying the W2,δ estimate (cf. [CC] Proposition 7.4) to the difference
u u , we have
k R
−
D2u D2u C(K,R) u u 0as k .
k − R Lδ(BR/2) ≤ k k − RkL∞(BR) → → ∞
No(cid:13)(cid:13)te that D2uk ,(cid:13)(cid:13)D2uR K, so also
≤
(cid:12) (cid:12) D(cid:12) 2u (cid:12) D2u 0as k .
(cid:12) (cid:12) (cid:12) k −(cid:12) R Ln(BR/2) → → ∞
By a standard fac(cid:13)t from real ana(cid:13)lysis, there exists another subsequence and
(cid:13) (cid:13)
C1,1 function on B , still denoted by u and u such that D2u
R k R/2 k
{ } →
D2u almost everywhere as k . So D2u still satisfies (1.2) almost
R/2 R
→ ∞
everywhere on B .
R/2
The diagonalizing process yields yet another subsequence, again denoted
by u and v C1,1(Rn) such that u v in W2,n(Rn) as k ; v is
{ k} ∈ k → loc → ∞
a viscosity solution of (1.1) on Rn; D2v K; and D2v still satisfies (1.2)
almost everywhere on Rn. ≤
(cid:12) (cid:12)
The surfaces (x,▽u (x)) are mini(cid:12)mal (cid:12)in Rn Rn and their potentials u
k k
×
converge to v in W2,n(Rn), so by the monotonicity formula (cf. [S, p.84,
loc
Theorem 19.3] ), we conclude that M = (x,▽v(x)) is a cone.
v
A LIOUVILLE TYPE THEOREM 9
Step 3. We claim that M is smooth away from the vertex. Suppose
v
M is singular at P away from the vertex. We blow up M at P to get a
v v
tangent cone, which is a lower dimensional special Lagrangian cone crossing
a line; repeat the procedure if the resulting cone is still singular away from
the vertex. Finally we get a special Lagrangian cone which is smooth away
fromthevertex, andtheboundedeigenvalues of theHessian of thepotential
function satisfies (1.2), by a similar W2,δ argument as in Step 2. By Lemma
2.3, the cone is flat. This is a contradiction to Allard’s regularity result (cf.
[S, Theorem 24.2]).
Applying Lemma 2.3 to M , we see that M is flat.
v v
Step 4. Now with theflatness of M , afinalapplication of themonotonic-
v
ity formula yields that the original gradient graph (x, u(x)) is also a plane
(cf. [Y2, p.123]). Therefore, u is a quadratic polynom∇ial. (cid:3)
3. Appendix
We include here a uniqueness result for global solutions to the special La-
grangianequation(1.1)withboundedHessiansatisfyingcertain“convexity”
constraints (3.1). The constraints are only needed for n 4.
≥
Theorem3.1. Letube asmooth solution to the special Lagrangian equation
(1.1). Suppose that the eigenvalues λ of the Hessian D2u(x) are bounded
i
λ (x) K and satisfy
i
| |≤
(3.1) 3+λ2(x)+2λ (x)λ (x) 0
i i j ≥
for all i, j, and x. Then u must be a quadratic polynomial.
Proof. The proof is identical to the one of Theorem 1.1 with Lemma 2.3
replaced by the following proposition. (cid:3)
Proposition 3.1. Let u C (Rn 0 ) be a solution to the special La-
∞
∈ \{ }
grangianequation(1.1)andhomogeneous oforder2, thatisu(x) = x 2u(x/ x).
| | | |
Suppose that the eigenvalues λ of the Hessian D2u(x) satisfy (3.1) for all
i
i, j, and x= 0. Then u must be quadratic.
6
Proof. By (3.1), we certainly have (2.8) with ε= 0 in Lemma 2.1, that is
n
lnV 1+λ2 h2 + 3+λ2+2λ λ h2
△g ≥ i iii j i j jji
i=1 j=i
X(cid:0) (cid:1) X6 (cid:0) (cid:1)
n 3+λ2+2λ λ
1 j i j
(3.2) = u2 + u2 0.
i=1 1+λ2i 2 iii j=i (cid:16)1+λ2 2 1+λ(cid:17)2 jji ≥
X X6 j i
(cid:0) (cid:1) (cid:16) (cid:17)
Since u is homogeneous of order 2, the Hessian D(cid:0)2u(x)(cid:1)is homogeneous
of order 0, hence lnV must attain its maximum along a ray. The strong
maximum principle yields that lnV is constant, so in fact
(3.3) 0= lnV.
g
△
10 MICAHWARRENANDYUYUAN
We claim now that
(3.4) u = const
△
on Rn 0 . At any point p compute the derivative
\{ }
(3.5) ∂ ( u) = u
i jji
△
j
X
for all i. Still assumingthat D2uis diagonalized at p, an inspection of (3.2),
together with (3.3) shows that for all j with u = 0,
jji
6
(3.6) 3+λ2+2λ λ = 0.
j i j
From 3+λ2+2λ λ 0, we see that λ2 λ2. Solving (3.6) for λ we get
i i j ≥ i ≥ j j
λ = λ λ2 3, if λ < 0
j − i− i − i
q
λ = λ + λ2 3, if λ > 0 .
j − i i − i
The minimal surface equation (2.1q) at p then reads
p 1 1
0 = u = u = u .
△g i 1+λ2 jji 2 jji
Xj j 1+ −λi± λ2i −3 Xj
(cid:16) q (cid:17)
Hence ∂ ( u) = 0 and u is constant.
i
△ △
Differentiating (3.4), we see that each u satisfies
ij
u = 0.
ij
△
Applying the strong maximum principle once again to each (homogeneous
order 0) function u , we have immediately
ij
u = const;
ij
that is, u is quadratic. (cid:3)
References
[BCGJ] Bao, Jiguang; Chen, Jingyi; Guan, Bo; and Ji, Min, Liouville property and regu-
larity of a Hessian quotient equation. Amer.J. Math. 125 (2003), 301–316.
[B] Borisenko, Alexandr A., On a Liouville-type theorem for the equation of special
Lagrangiansubmanifolds.(Russian)Mat.Zametki52(1992),22–25;Englishtrans-
lation in Math. Notes 52 (1992), 1094–1096 (1993).
[CC] Caffarelli, Luis A. and Cabr´e, Xavier, Fully Nonlinear Elliptic Equations. Amer-
ican Mathematical Society Colloquium Publications, 43. American Mathematical
Society, Providence, RI,1995.
[F-C] Fischer-Colbrie, Doris, Some rigidity theorems for minimal submanifolds of the
sphere. Acta Math. 145 (1980), no. 1-2, 29–46.
[F] Fu,Lei,AnanalogueofBernstein’stheorem.HoustonJ.Math.24(1998),415–419.
[G] Gregori, Giovanni, Compactness and gradient bounds for solutions of the mean
curvature system in two independent variables. J. Geom. Anal. 4 (1994), no. 3,
327–360.
[HL] Harvey, Reese and Lawson, H. Blaine. Jr., Calibrated geometry. Acta Math. 148
(1982), 47–157.
