A new way to Dirichlet problems for minimal surface systems in arbitrary dimensions and codimensions

0.09 MB·English

by  Jing Mao

A new way to Dirichlet problems for minimal surface
systems in arbitrary dimensions and codimensions
5
1
0 Jing Mao
2

n
a
DepartmentofMathematics,HarbinInstituteofTechnology(Weihai),Weihai,264209,China
J
3 [email protected],[email protected]
1
Abstract

]
G
In this paper, by considering a special case of the spacelike mean curvature flow investi-
D
gated byLiand Salavessa [6], wegetacondition for theexistence ofsmooth solutions of the
.
h Dirichlet problem for the minimal surface equation in arbitrary codimension. We also show
t that our condition issharper than Wang’s in [13, Theorem 1.1] provided the hyperbolic angle
a
m q oftheinitialspacelike submanifold M satisfiesmax coshq >√2.
0 M0
[

1
v
1 Introduction
9
4
9
Let W be a bounded C2 domain in the Euclidean n-space Rn and f : ¶ W Rm be a continuous
2
→
0 map from the boundary of W to Rm. The Dirichlet problem for the minimal surface system asks
1. whetherthereexistsaLipschitzmap f :W Rm suchthatthegraphof f isaminimalsubmanifold
50 in Rn+m and f ¶ W =f . Form=1 and any→mean convexdomainW , Jenkinsand Serrin [4]proved
|
1 theexistenceofthesolutionsforthisDirichletproblemandthesmoothnessofallofthesolutions.
:
v TheDirichletproblemiswellunderstoodowingtothepioneeringworksofJenkinsandSerrin[4],
Xi De Giorgi [3], and Moser [9]. However, they treated the Dirichlet problem for just hypersurfaces.
r For surfaces with higher codimension, very little is known. Is the high codimensional Dirichlet
a
problem solvable, or under what kind of assumptions could one obtain the existence of solutions
of this problem? Lawson and Osserman [5] gave some nice examples to show how important
the boundary data is for the solvability of high codimensional Dirichlet problems. If the minimal
submanifoldisadditionallyrequiredtobeLagrangian,theminimalsurfacesystembecomesafully
nonlinearscalarequation
Im det(I+√ 1D2f) =0,
−
(cid:16) (cid:17)
whereI istheidentitymatrixandD2f = ¶ 2f istheHessianmatrixof f. Caffarelli,Nirenberg,
¶ xi¶ xj
and Spruck [2] solved this Dirichlet prob(cid:16)lem wi(cid:17)th the prescribed boundary value of f. For n=2
and any convex planar domain, the existence of solutions of this problem was proved by Rado´ in
MS0C2010:53C21;53C44;35K55
KeyWords:Dirichletproblem;Spacelikesubmanifold;Meancurvatureflow;Maximumprinciple.
1
J.Mao 2
[11](see also [5]). ForC2,a smallDirichlet boundarydata offinitecodimension,Smale [12] used
themethodoflinearizationto provethesolvabilitysuccessfully.
Recently, under some assumption about the boundary data, Wang [13] proved the existence
of smooth solutionsof the Dirichlet problem for minimal surface systems in arbitrary dimensions
and codimensions by using his results in [14, 15] on high codimensional mean curvature flow of
submanifolds. Surprisingly, by considering a special case of the spacelike mean curvature flow
(MCFfor short)consideredin [6], wecan provethefollowingresult.
Theorem 1.1. Let W be a bounded and closedC2 convex domain in Rn (n 2) with diameter d .
If y :W Rm satisfies ≥
→
4nh 2d sup D2y +√2sup Dy <1, (1.1)
0
W ¶ W | |
(cid:12) (cid:12)
then the Dirichlet problem for the min(cid:12)imal s(cid:12)urface system is solvable for y ¶ W in smooth maps.
Hereh isaconstantdefinedby(3.2),dependingonlyonthespacelikegraph|ofW ,andforx W ,
0
∈
Dy (x):= sup Dy (x)(v)
| | | |
v=1
| |
and
D2y (x):= sup D2y (x)(v,v)
v=1
(cid:12) (cid:12) | | (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
arethenormandthesquarednormof thedifferentialDy (x):Rn Rm.
→
Thepaperisorganizedasfollows. WerecallsomeusefulfactsaboutspacelikeMCFin[6],and
establish the relation between the parametric and the non-parametric forms of the flow in Section
2. At the end of Section 2, as in [13], a boundary gradient estimate is derived by using the initial
mapas abarrier surface. Theorem 1.1willbeprovedinthelastsection.
2 Some useful facts
Assume the Riemannian manifold (S ,g ) to be closed and of dimension n 2, and the Rieman-
1 1
≥
nian manifold (S ,g ) to be complete, of dimension m 1. Let f : S S be a smooth map
2 2 1 2
from (S ,g ) to (S ,g ). Let M = S S be a pseud≥o-Riemannian m→anifold with the metric
1 1 2 2 1 2
×
g¯=g g . Let M beaspacelikegraph defined by
1 2
−
M =G = (p, f(p)) p S ,
f 1
{ | ∈ }
anddenotebygtheinducedmetriconM. Clearly,if f isaconstantmap,M isaslice. Ifwedenote
thisspacelikeimmersionbyF =id f,thenwesaythatthespacelikegraphM evolvesalongthe
g
×
MCFif
d F(x,t)=H(x,t), x M, t >0,
dt ∀ ∈ ∀
(2.1)

F( ,0)=F,

·

J.Mao 3
where H is the mean curvature vector of M =(M,F g¯)=F(M), and id is the identity map. The
t t∗ t
hyperbolicangleq can bedefined by(thisdefinitioncan also beseen in[1, 7])
1
coshq = , (2.2)
det(g f g )
1 ∗ 2
−
which is used to measure the deviation fropm a spacelike submanifold to a slice. Assume, in addi-
tion, that the Ricci curvature of S satisfies Ricci (p) 0, and the sectional curvatures of S and
1 1 1
S satisfyK (p) K (q),forany p S ,q S . Besid≥es,thecurvaturetensorR ofS andallits
2 1 2 1 2 2 2
≥ ∈ ∈
covariant derivatives are bounded. By Theorem 1.1, Propositions 5.1, 5.2 and 5.3 in [6], we have
thefollowing.
Theorem 2.1. Let f bea smoothmap fromS to S suchthatF :M M isa compact spacelike
1 2 0
→
graphof f. Then
(1) A unique smooth solution of (2.1) with initial condition F a spacelike graphic submanifold
0
existsin amaximaltimeinterval[0,T)for someT >0.
(2) coshq defined by (2.2) has a finite upper bound, and the evolving submanifold M remains a
t
spacelikegraphofa map f :S S whenever theflow(2.1)exists.
t 1 2
(3) B , H , (cid:209) kB , and (cid:209) kH→, forallk, areuniformlybounded.
k k k k k k k k
(4)ThespacelikeMCF (2.1)existsforall thetime.
Now, we would like to explain the connection between the spacelike MCF (2.1) and the high
dimensional Dirichlet problem. Let W Rn be a closed and bounded domain, and y : W Rm
⊂ →
beavector-valuedfunction. Then thegraph ofy can beseen as thespacelikeembeddingid y :
W Rn Rm = Rn+m with the pseudo-Riemannian metric g¯ = g g = ds2 ds2, wher×e ds2
and→ds2 a×re the standard Euclidean metrics of Rn and Rm, respect1iv−ely2. For t1h−e spa2celike MCF1
2
(2.1), choosing S 1 =W Rn and S 2 =Rm, if we require F ¶ W =(id y ) ¶ W , then the immersed
⊂ | × |
mapping F, with F =F, should be a smooth parametric solution to the Dirichlet problem of the
t 0
spacelikeMCF, thatis,
dF =H,
dt

 F ¶ W =id y ¶ W .
| × |
In alocal coordinatesystem x1,...,xn onS =W , thespacelikeMCFis thesolution
1
{ }
F =FA(x1,...,xn,t), A=1,2,...,n+m,
tothefollowingsystemofparabolicequations
¶ F = (cid:229) n gij ¶ 2F ⊥,
¶ t ¶ xi¶ xj
i,j=1 !
where gij = (g ) 1 is the inverse of the induced metric g = g ¶ F, ¶ F , and ( ) and ( )
ij − ij ¶ xi ¶ xj · ⊤ · ⊥
denotethetangentandthenormalpartsofavectorinRn+m,respect(cid:16)ively. Th(cid:17)eEinsteinsummation
convention that repeated indices are summed over is adopted in the rest of the paper. As in the
proofof[13, Lemma2.1], wecan easily provethefollowinglemma.
J.Mao 4
Lemma 2.2. Wehave
1 ¶ ¶ F ¶ 2F ⊥
D F = √Ggij = gij ,
√G¶ xi ¶ xj ¶ xi¶ xj
(cid:18) (cid:19) (cid:18) (cid:19)
where G=det(g ).
ij
Lemma2.2tellsusthatD F isalwaysinthenormaldirection,whichimpliesthatg ¶ F,D F =
¶ xi
0 for1 i n. (cid:16) (cid:17)
≤ ≤
Similar to [13, Proposition 2.2], we can also derive a relation between parametric and non-
parametricsolutionstothespacelikeMCF equation.
Proposition 2.3. Suppose that F is a solution to the Dirichlet problem for spacelike MCF (2.1)
and that each F(W ,t) can be written as a graph over W Rn. Then there exists a family of
diffeomorphismsr of W suchthatF =F r isof theform ⊂
t t t t
◦
F(x1,...,xn)=(x1,...,xn, f1,..., fm)
e
and f =(f1,..., fm):W [0,T) Rm satisfies
e
× →
¶ fa =gij ¶ 2fa , a =1,...,m,
¶ t ¶ xi¶ xj
(2.3)

 f ¶ W =y ¶ W ,
| |
where 
gij =(g ) 1 and g =d (cid:229)m ¶ fb ¶ fb . (2.4)
ij − ij ij− ¶ xi · ¶ xj
b =1
Conversely, if f =(f1,..., fm):W [0,T) Rm satisfies(2.3), thenF =I f is a solutionto
× → ×
¶
⊥ e
F(x,t) =H(x,t).
¶ t
(cid:18) (cid:19)
By applying the maximum principleefor scalar pearabolic equations (see, for instance, [10]) to
thesecond-orderparabolicequation(2.3), wehavethefollowing.
Proposition 2.4. Let f = (f1,..., fm) : W [0,T) Rm be a solution to equation (2.3). If
× →
sup Df isbounded,then
W [0,T)
| |
×
sup fa supy a ,
W [0,T) ≤ W
×
with y =(y 1,...,y m)theinitialmapgiven inequation(2.3).
By usingtheinitialdatay :W Rm as abarrier surface, wecan obtaintheboundarygradient
→
estimateas follows.
J.Mao 5
Theorem2.5. LetW beaboundedC2 convexdomaininRn withdiameterd . Supposethattheflow
(2.3)existssmoothlyonW [0,T). Then wehave
×
4nd
Df < sup D2y +√2sup Dy , on ¶ W [0,T),
| | 1 x W | | ¶ W | | ×
−
where x =sup Df 2.
W [0,T)
| |
×
Proof. We use a method similar to that of the proof of [13, Theorem 3.1]. Denote by P the sup-
porting (n 1)-dimensional hyperplane at a boundary point p, and d the distance function to P.
p
−
Let f =(f1,..., fm)beasolutionofequation(2.3). Considerthefunctiondefined by
S(x1,...,xn,t)=vlog(1+kd ) (fa y a )
p
− −
on Rn for each a =1,2,...,m, where k,v>0 are to be determined. The Laplace operator on the
graph (G ,g=F g¯) isgivenby D =gij ¶ 2 , withg satisfying(2.4). Clearly,
f t∗ ¶ xi¶ xj ij
f f
gij = d ij fifj −1 =d ij+ i j ,
− 1 Df 2
−| |
(cid:0) (cid:1)
where f f = (cid:229) m ¶ fb ¶ fb . Therefore, the eigenvalues of gij are between 1 and 1/(1 x ). By
i j b =1 ¶ xi ¶ xj −
direct computation,weknowthatS satisfies thefollowingevolutionequation
d vk vk2 ¶ d ¶ d
D S= ( D d )+ gij p p D y a . (2.5)
dt − 1+kd − p (1+kd )2 ¶ xi ¶ xj −
p p
(cid:18) (cid:19)
Sinced isa linearfunction,D d =0,then (2.5)isreduced to
p p
d vk2 ¶ d ¶ d
D S= gij p p D y a . (2.6)
dt − (1+kd )2 ¶ xi ¶ xj −
p
(cid:18) (cid:19)
Since Dd = 1, d (y) p y d for any y W , and the fact that the eigenvalues of gij are
p p
| | ≤ | − | ≤ ∈
between 1 and1/(1 x ),wehave
−
vk2 ¶ d ¶ d vk2 vk2
gij p p ,
(1+kd )2 ¶ xi ¶ xj ≥ (1+kd )2 ≥ (1+kd )2
p p
and
¶ 2y a n
D y a = gij D2y .
¶ xi¶ xj ≤ 1 x | |
(cid:12) (cid:12) −
(cid:12) (cid:12)
Hence, if (cid:12) (cid:12)
(cid:12) (cid:12)
vk2 n
sup D2y ,
(1+kd )2 ≥ 1 x W | |
−
then,togetherwith(2.6),wehave(d D )S 0on[0,T). Ontheonehand,byconvexity,wehave
dt − ≥
S>0 on theboundary ¶ W ofW except for S=0 at p. On theotherhand, S 0 on W att =0. By
≥
thestrongmaximumprincipleforsecond-orderparabolicpartial differentialequations,we have
S>0, on W (0,T).
×
J.Mao 6
The same conclusion can also be obtained for a new function S := vlog(1+kd )+(fa y a ).
′ p
−
Hence, wehavethatthenormalderivativessatisfy
¶ (fa y a ) fa y a
− (p) lim | − |
(cid:12) ¶ n (cid:12) ≤ dp(x)→0 dp(x)
(cid:12) (cid:12)
vlog[1+kd (x)]
(cid:12) (cid:12) p
(cid:12) (cid:12) < lim =vk.
dp(x) 0 dp(x)
→
So, by changing coordinates of Rm, we may assume ¶ fa /¶ n=0 for all a except for a =1 such
thattheinequality
¶ f ¶y
<vk+ (2.7)
¶ n ¶ n
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
holds. (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
TheDirichletboundaryconditionimplies
D¶ W f = D¶ W y , on ¶ W , (2.8)
(cid:12) (cid:12) (cid:12) (cid:12)
where D¶ W f is defined by D(cid:12)(cid:12)¶ W f :=(cid:12)(cid:12) su(cid:12)(cid:12)p Df((cid:12)(cid:12)x)w for x ¶ W and w being taken over all unit
w
| | | | | | ∈
vectorstangentto ¶ W . Combining(2.7)and(2.8), wehave
¶y 2
Df < vk+ + D¶ W y 2 √2 Dy +vk, on ¶ W .
| | s ¶ n | | ≤ | |
(cid:18) (cid:12) (cid:12)(cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Now, it is not difficult to find out that if we want to prove our assertion here, we actually need
(cid:12) (cid:12)
to minimize vk under the constraint vk2 n sup D2y . In fact, the minimum (vk) of
(1+kd )2 ≥ 1−x W | | min
the function vk is obtained when k=d 1, and (vk) =4nd (1 x ) 1sup D2y . The theorem
− min − W
− | |
follows.
3 Proof of the main theorem
Now,byapplyingtheconclusionswerecalledandderivedinSection2,wecanproveTheorem1.1
as follows.
ProofofTheorem1.1. WedividetheproofofTheorem1.1 intofivesteps.
Step 1. By the Schauder fixed-point theorem (see, for instance, Theorem 8.1 on p. 199 of [8]
for a detailed description and the proof of the Schauder fixed-point theorem), the solvability of
a
the parabolic system (2.3) can be reduced to the estimates of the solution (f ) of the uniformly
parabolicsystem
dfa =g˜ij ¶ 2fa , a =1,...,m,
dt ¶ xi¶ xj
(3.1)

 f ¶ W =y ¶ W ,
| |

J.Mao 7
withthecoefficients
(cid:229)m ¶ ub ¶ ub
g˜ij = d (cid:229)m ¶ ub ¶ ub −1 =d +b =1 ¶ xi ¶ xj
ij− ¶ xi ¶ xj ij 1 Du 2
b =1 ! −| |
for any u =(u1,...,um) with uniformC1,g bound. The property of being uniformly parabolic of
(3.1)isequivalentto Du <1foralltimet [0,T). Fortunately, Du <1for0 t <T essentially
| | ∈ | | ≤
corresponds to the fact that the evolving submanifold M is spacelike for all the time t [0,T),
t
∈
which can be obtained directly from Theorem 2.1 (2). Now, (3.1) is a decoupled system of linear
parabolicequations,whichisuniformlyparabolicandwhoserequiredestimatefollowsfromlinear
theory for scalar equations. Therefore, we know that the system (2.3) has the solution on a finite
timeinterval.
In fact, there is another way to show the short-time existence of the solution of (2.3). More
precisely, by Theorem 2.1 (1), we can also get the short-time existence, since in our case, as
explained before, we choose S to be a closed domain in Rn and S = Rm, which implies that
1 2
thesystem(2.3)is justaspecial case ofthespacelikeMCF (2.1)providedweadditionallyrequire
F ¶ W =(id y ) ¶ W , i.e., f ¶ W =y ¶ W .
| × | | |
Step 2. Denote the graph of f by M . We show that Df <1 holds under the assumption of
t t t
| |
Theorem1.1. Similarto(3.4)and(3.5)in[6],foreachpoint p S ,wecanchooseanorthonormal
1
∈
basisforthetangentspaceT M and forthenormalspaceN M givenas follows
p p
ei = 1 ai+(cid:229) l ib ab , i=1,...,n,
1−(cid:229) b l i2b b !
q
ea = 1 aa +(cid:229) l ja aj , a =n+1,...,n+m,
1−(cid:229) jl j2a j !
q
where ai i=1,...,n is a g1-orthonormal basis of TpS 1 of eigenvectors of f∗g2, aa a =n+1,...,n+m is
ag2-or{tho}normalbasisofTf(p)S 2, and df = l ia withl ia =d a ,n+il i. Herel{i,1} i n,are the
− ≤ ≤
eigenvalues of f g . So, the spacelike condition on M implies l 2 <1 for each 1 i n. We list
∗ 2 i
≤ ≤
them non-increasingly as l 2 l 2 ... l 2 0. By the classical Weyl’s perturbation theorem,
1 1 n
the ordering eigenvalues l 2 :≥S ≥[0,1)≥is a c≥ontinuous and locally Lipschitz function. For each
i 1
p S , denote by s=s(p)= 1,→2, ,n the rank of f at the point p, which implies l 2 >0 and
1 s
∈ { ··· }
l =l = =l =0. Therefore, wehaves min m,n . In fact,afterthissetting,wehave
s+1 s+2 n
l ia =d a ,n+i =·0·i·fi>s,ora >n+s. Undertheor≤thonor{malb}asis e1,...,en,...,en+m ,by(2.2)
thehyperbolicangleq satisfies { }
1 1
coshq = = .
det(g1 f∗g2) (cid:213) n (1 l 2)
− i=1 − i
p q
Let
h :=maxcoshq . (3.2)
t
Mt
J.Mao 8
By[6,Proposition4.3],weknowthat(inourcase,S =W Rn,S =Rm)theevolutionequation
1 2
ofcoshq hereshouldbe ⊂
d ln(coshq )= ln(coshq ) B 2 (cid:229) n l 2 hm+i 2 2 (cid:229) l l hm+jhm+i ,
dt △ − k k − i ik − i j ik jk
( k,i=1 k,i<j )
(cid:0) (cid:1)
where B 2 =(cid:229) n (cid:229) n+m (ha )2 is the squared norm of the second fundamental form. Hence,
k k i,j=1 a =n+1 ij
thereexistssomenonnegativeconstantℓ suchthat
d
ln(coshq ) D ln(coshq ) ℓ B 2 D ln(coshq ).
dt ≤ − k k ≤
Then bythemaximumprincipleforparabolicequations,wecan obtain
h h =maxcoshq
t 0
≤ M
0
for0<t T ¥ , whichimpliesl 2(t)<1 and
i
≤ ≤
1 l 2(t) (cid:213) n 1 l 2(t) 1 1
− i ≥ − i ≥ h 2 ≥ h 2
i=1 t 0
(cid:0) (cid:1)
forany0<t T ¥ . On theotherhand,ifweassume
≤ ≤
4nh 2d sup D2y +√2sup Dy <1,
0
W | | ¶ W | |
then by integrating along a path in W , we have sup Df =sup Dy <1 initially. Hence, by
W 0 ¶ W
| | | |
theabovearguments, wehave Df <1 forany0 t T ¥ .
t
| | ≤ ≤ ≤
Step 3. By Theorem 2.1 (3) and (4), we know that the norms of the second fundamental form
and all of its derivativesare bounded, and the flow (2.1) exists for all time. Hence, the solutionto
thespacelikeMCF (2.3)existssmoothlyin[0,¥ ).
Step 4. By [6, Corollary 6.1], we know that there exists a time sequence t ¥ such that
n
→
sup H 0whent ¥ . Sincewealsohaveagradientbound(seeTheorem2.1(3)),wecan
S 1k tnk→ n→
extract a subsequence t such that the graph M converges to a Lipschitz graph with sup H =0
i ti k k
and Df <1,which impliesthatthelimitsubmanifoldis aminimalspacelikesubmanifold.
| |
Step5. Interiorregularityofthelimitfollowsfrom[13,Theorem4.1],sincethesingularvalues
l of Df satisfy l l 1 1/h 2 almost everywhere for any i= j. Boundary regularity follows
i | i j|≤ − 0 6
from [5,Theorem 2.3]. Ourproofisfinished. (cid:3)
Remark 3.1. Clearly, our condition (1.1) is better than that in [13, Theorem 1.1] provided h =
0
max coshq >√2.
M
0
Acknowledgments
Theauthorwas supportedbythestarting-upresearch fund(Grant No. HIT(WH)201320)supplied
by Harbin Institute of Technology (Weihai), the project (Grant No. HIT.NSRIF.2015101) sup-
ported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology,
and the NSF of China (Grant No. 11401131). The author would like to thank the anonymous
referee for his or her careful reading and valuable comments such that the article appears as its
present version.
J.Mao 9
References
[1] A. L. Albujer and J. L. Alias, Calabi-Berstein results for maximal surfaces in Lorentzian
productspaces, J.Geom. Phys. 59(2009)620–631.
[2] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order
elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985)
261–301.
[3] E. De Giorgi, Sulla differenziabilita` e l’analiticita` delle estremali degli integrali multipli re-
golari,Mem. Accad. Sci.Torino.Cl. Sci. Fis.Mat. Nat.(3) 3(1957)25–43.
[4] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher
dimensions,J. ReineAngew.Math.229(1968)170–187.
[5] H. B. Lawson Jr. and R. Osserman, Non-existence, non-uniqueness and irregularity of solu-
tionsto theminimalsurfacesystem,ActaMath.139 (1977)1–17.
[6] G. Li and I. Salavessa, Mean curvature flow of spacelike graphs, Math. Z. 269 (2011) 697–
719.
[7] G.LiandI. Salavessa,GraphicBernsteinresultsincurvedpesudo-Riemannianmanifolds,J.
Geom.Phys. 59(2009)1306–1313.
[8] G.M.Lieberman,Secondorderparabolicdifferentialequations.WorldScientific,RiverEdge,
N.J.,1996.
[9] J. Moser, A new proof of De Giorgis theorem concerning the regularity problem for elliptic
differentialequations,Comm. PureAppl.Math.13 (1960)457–468.
[10] M. H. Protter and H. F. Weinberger, Maximumprinciplesin differential equations, Prentice-
Hall,EnglewoodCliffs,N.J.,1967.
[11] T. Rado´, On the problem of Plateau. Subharmonic functions, Reprint, Springer, New York–
Heidelberg,1971.
[12] N. Smale, Geometric P.D.E.s with isolated singularities, J. Reine Angew. Math. 440 (1993)
1–41.
[13] M.-T. Wang, The Dirichlet problem for the minimal surface system in arbitrary dimensions
and codimensions,Comm. PureAppl.Math.57 (2004)267–281.
[14] M.-T. Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential
Geom.57 (2001)301–338.
[15] M.-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbi-
trary codimension,Invent. Math.148(2002)525–543.