Lagrangian Mean Curvature flow for entire Lipschitz graphs II
aa r X i v : . [ m a t h . DG ] M a y LAGRANGIAN MEAN CURVATURE FLOW FOR ENTIRELIPSCHITZ GRAPHS II
ALBERT CHAU, JINGYI CHEN, AND YU YUAN
Abstract.
We prove longtime existence and estimates for solutions to a fully non-linear Lagrangian parabolic equation with locally C , initial data u satisfyingeither (1) − (1 + η ) I n ≤ D u ≤ (1 + η ) I n for some positive dimensional constant η , (2) u is weakly convex everywhere or (3) u satisfies a large supercritical La-grangian phase condition. introduction When a family of smooth entire Lagrangian graphs in C n evolve by the meancurvature flow their potentials u : R n × [0 , T ) → R will evolve, up to a time dependentconstant, by the following fully nonlinear parabolic equation:(1) ∂u∂t = n X i =1 arctan λ i u ( x,
0) = u ( x )where λ i ’s are the eigenvalues of D u . Conversely, if u ( x, t ) solves (1), then thegraphs ( x, Du ( x, t )) in R n will evolve by the mean curvature flow up to tangentialdiffeomorphism. The main result of the paper is the following. Theorem 1.1.
There exists a small positive dimensional constant η = η ( n ) such thatif u : R n → R is a C , function satisfying (2) − (1 + η ) I n ≤ D u ≤ (1 + η ) I n then (1) has a unique longtime smooth solution u ( x, t ) for all t > with initialcondition u such that the following estimates hold: (i) −√ I n ≤ D u ≤ √ I n for all t > . (ii) sup x ∈ R n | D l u ( x, t ) | ≤ C l /t l − for all l ≥ , t > and some C l depending onlyon l . (iii) Du ( x, t ) is uniformly H¨older continuous in time at t = 0 with H¨older exponent / . Date : June 17, 2018.2000 Mathematics Subject Classification. Primary 53C44, 53A10.The first two authors are partially supported by NSERC grants, and the third author is partiallysupported by an NSF grant.
In [1] Theorem 1.1 was proved for η any negative constant in which case it wasshown that (2) is preserved for all t >
0. In particular, a priori estimates wereestablished for any solution to (1) with D u so bounded. The estimates combinedmaximum principle arguments for tensors and a Bernstein theorem for entire specialLagrangians [11] via a blow up argument. The estimates depended on the negativityof η and could not be applied to the more general case of Theorem 1.1 even for η = 0.We overcome this through recent estimates in [13] for solutions to (1) satisfying certainHessian conditions (cf. Theorem 2.1 which is Theorem 1.1 in [13]). A particular caseof Theorem 1.1 (similarly for Theorems 1.2 and 1.3) is where Du : R n → R n is alift of a map f : T n → T n and T n is the standard n -dimensional flat torus. In this“periodic case”, our estimates together with the results in [7] imply that the graphs( x, Du ( x, t )) immediately become smooth after initial time and converge smoothlyto a flat plane in R n (cf. [1, 7, 8, 9]). In the hypersurface case, the global andlocal behavior of mean curvature flow of Lipschitz continuous initial graphs has beenstudied in [4, 5].After a coordinate rotation described in § − I n < D u
Let u : R n → R be a locally C , weakly convex function. Then (1) has a unique longtime smooth and weakly convex solution u ( x, t ) with initial condition u such that (i) either D u ( x, t ) > for all x and t > or there exists coordinates x , ..., x n on R n in which u ( x, t ) = w ( x k , ..., x n , t ) on R n × [0 , ∞ ) where k > and w isconvex with respect to x k , ..., x n for all t > , (ii) sup x ∈ R n |∇ lt A ( x, t ) | ≤ C l /t l +1 for all l ≥ , t > and some constant C l depending only on l where ∇ lt A ( x, t ) is the l th covariant derivative of thesecond fundamental form of the embedding F t : R n → R n given by x → ( x, Du ( x, t )) , (iii) the Euclidean distance from each point of F t ( R n ) to F ( R n ) in R n is H¨oldercontinuous in time at t = 0 with H¨older exponent / . We also prove the following
Theorem 1.3.
Let u : R n → R be a locally C , function satisfying (3) n X i =1 arctan λ i ≥ ( n − π . Then (1) has a unique longtime smooth solution u ( x, t ) with initial condition u suchthat (3) is satisfied with either strict inequality for all t > or equality for all t ≥ AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 3 in which case u must be quadratic. Moreover, u ( x, t ) also satisfies (ii) and (iii) inTheorem 1.2.Remark . Note that if u satisfies (3) then u must be convex.As discussed above, after a coordinate rotation we may assume D u in Theorem 1.2satisfies the strict inequality − I n ≤ D u < I n in which case Theorem 1.1 immediatelyprovides a longtime solution u ( x, t ) to (1). In order for this to correspond to thedesired longtime solution in the original coordinates we must first show − I n ≤ D u λ i ( D u ( x, t )) = 1for some ( x, t ) which would correspond to a non-graphical (vertical) Lagrangian inthe original coordinates. The second main difficulty comes from showing that either − I n < D u for all t >
0, or the solution splits off a quadratic term as in Lemma 4.2and this will give (i) in Theorem 1.2 after rotating back to the original coordinates.As for Theorem 1.3, by Remark 1.1, if u satisfies (3) then it is automaticallyconvex hence Theorem 1.2 guarantees a longtime convex solution u ( x, t ) to (1). Thedifficulty in showing (3) is preserved for all t > u is only C , with possibly unboundedHessian. Performing a similar but small σ coordinate rotation, we can assume that − K ( σ ) I n < D u < /K ( σ ) I n , for some constant K ( σ ) which approaches zero as σ →
0, and satisfies(4) n X i =1 arctan λ i ≥ ( n − π − nσ . We then observe that the set of positive semi-definite real n × n matrices satisfy-ing (4) is a convex set S , and we approximate u by convolution with the standardheat kernels, which has the effect of averaging elements in S , thus producing smoothapproximations with bounded derivatives (of order 2 and higher) and Hessians be-longing to S . We perform a further π/ n X i =1 arctan λ i ≥ ( n − π − n π − nσ . By Theorem 1.1 we then apply a maximum principle argument to show (5) is preservedstarting from each approximate initial data.The outline of the rest of the paper is as follows. In § § §
4. 2. preliminaries
In this section we establish some preliminary results.
ALBERT CHAU, JINGYI CHEN, AND YU YUAN
Proposition 2.1 ( [1]; Proposition 5.1 ) . Suppose u : R n → R is a smooth functionsuch that sup | D l u | < ∞ for each l ≥ . Then (1) has a smooth solution u ( x, t ) on R n × [0 , T ) for some T > such that sup x ∈ R n | D l u ( x, t ) | < ∞ for every l ≥ and t ∈ [0 , T ) .Remark . In Proposition 5.1 in [1] it was shown that the non-parametric meancurvature flow equation(6) ∂f a ∂t = n X i,j =1 g ij ( f )( f a ) ij f ( x,
0) = Du ( x )where g ij ( f ) is the matrix inverse of g ij ( f ) := δ ij + P na =1 f ai f aj , has a short timesolution f ( x, t ) provided u satisfies the conditions in Proposition 2.1. As explainedin [1] (see Lemma 5.2), this in fact provides a short time solution u ( x, t ) to (1) as inProposition 2.1 such that f ( x, t ) = Du ( x, t ) and the proof of Proposition 5.1 in [1]can also be adapted directly to (1) to establish Proposition 2.1. For convenience ofthe reader and completeness, we provide the details of this argument below. Proof.
Let C k + α,k/ α/ denote the standard parabolic H¨older spaces on R n × [0 , B = (cid:8) v ∈ C α, α | v ( x,
0) = 0 (cid:9) and define a map F : B → C α, α by F ( v ) = ∂v∂t − Θ( v )where Θ( v ) := P ni =1 arctan λ i ( D ( u + v )). Then the differential DF v at any v ∈ B is given by DF v ( φ ) = ∂φ∂t − n X i,j =1 g ij ( u + v ) φ ij where g ij ( u + v ) is the matrix inverse of I n + [ D ( u + v )] . Claim 1: DF v is a bijection from T v B onto T F ( v ) C α, α . This follows from the general theory of linear parabolic equations on R n × [0 , f , ..., f on R n recursively by f := Θ( u ) f := g ij ( u ) ∂ ij f . (7)Then we see that sup R n | D l f i | < ∞ for every i and l , and if we let w = F ( v ) where v = tf + t / f , then a straightforward computation gives(8) ∂ lt F ( v )( x,
0) = 0
AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 5 for l ≤ R n × [0 , | D lx D mt w | < ∞ for every l, m ≥
0. In particular w ∈ C α, α . By the inverse function theorem thereexists ǫ > || w − w || α, α < ǫ implies F ( v ) = w for some v ∈ B .For any 0 < τ <
1, define w τ by(10) w τ ( x, t ) = (cid:26) , t ≤ τw ( x, t − τ ) , τ < t < . Claim 2: || w τ − w || α, α < ǫ for sufficiently small τ > w τ ∈ C α, α and k w τ k α, α is bounded uniformly andindependently of τ . From this and the fact that w τ − w converges uniformly to 0 in C as τ →
0, it is not hard to show the claim follows.Hence by the inverse function theorem we have F ( v ) = w τ for some 0 < τ < v ∈ C α, α/ . In particular u + v solves (1) on R n × [0 , τ ]. Now the higherregularity of u can be shown as follows. For any x ∈ R n , consider the function˜ u ( x, t ) := u ( x + x , t ) − u ( x , − Du ( x , · x. Then ˜ u ( x, t ) ∈ B and still solves (1) on R n × [0 , τ ]. Now we can write (1) as ∂ ˜ u∂t = n X i =1 arctan λ i ( D ˜ u )= Z ∂∂s n X i =1 arctan λ i ( D ( s ˜ u )) ! ds = (cid:18)Z g ij ( s ˜ u ) ds (cid:19) ∂ ij ˜ u. (11)Notice that D ˜ u (0 ,
0) = ˜ u (0 ,
0) = 0 and that D ˜ u ( x, t ) = D u ( x + x , t ) is uni-formly bounded on R n × [0 , τ ]. Now if we let B (1) be the unit ball in R n it fol-lows from (1) that ˜ u ( x, t ) and thus D ˜ u ( x, t ) is uniformly bounded on B (1) × [0 , τ ],giving ˜ u ( x, t ) ∈ C α, + α ( B (1) × [0 , τ ]). In particular, by freezing the symbol a ij := R g ij ( s ˜ u ) ds in (11), we can view (11) as a linear parabolic equation for ˜ u with coefficients uniformly bounded in C α, + α ( B (1) × [0 , τ ]). Now applying the localparabolic Schauder estimates (Theorem 8.12.1, [6]) and a standard bootstrapping ar-gument to (11) we may then bound the C l + α norm of ˜ u ( x, t ) on B (1) by a constantdepending only on t and l .Now the fact that v is smooth with bounded derivatives as in the theorem followsby repeating the above argument for any x ∈ R n . (cid:3) Lemma 2.1 ([1]; Lemma 5.1) . Let u : R n → R be a C , function satisfying − C I n ≤ D u ≤ C I n for some constant C > . Then there exists a sequence of smoothfunctions u k : R n → R such that (i) u k → u in C α ( B R (0)) for any R and < α < , ALBERT CHAU, JINGYI CHEN, AND YU YUAN (ii) − C I n ≤ D u k ≤ C I n for every k , (iii) sup x ∈ R n | D l u k | < ∞ for every l ≥ and k .Proof. Let(12) u k ( x ) = Z R n u ( y ) K (cid:18) x, y, k (cid:19) dy where K ( x, y, t ) is the standard heat kernel on R n × (0 , ∞ ). Conditions (i) andsmoothness of u k are easily verified. By assumption, D y u ( y ) is a well defined anduniformly bounded function almost everywhere on R n and we may write D lx u k ( x ) = Z R n D y u ( y ) D l − x K (cid:18) x, y, k (cid:19) dy for every l ≥ (cid:3) Theorem 2.1 ([13]; Theorem 1.1 ) . Let u ( x, t ) be a smooth solution to (1) in Q ⊂ R n × ( −∞ , . When n ≥ we also assume that at least one of the following conditionsholds in Q (i) n X i =1 arctan λ i ≥ ( n − π , (ii) 3 + λ i + 2 λ i λ j ≥ for all ≤ i, j ≤ n .Then we have (13) [ u t ] , ; Q / + [ D u ] , ; Q / ≤ C ( k D u k L ∞ ( Q ) ) . Here Q r ( x, t ) = B r ( x ) × [ t − r , t ] ⊂ R n × ( −∞ , Q r := Q r (0 , Lemma 2.2.
Suppose u : R n → R is a C , function satisfying (14) − I n ≤ D u ≤ I n and that u ( x, t ) ∈ C ∞ ( R n × (0 , T )) T C ( R n × [0 , T )) is a solution to (1) and satisfies u ( x,
0) = u . Then (14) is preserved for all t .Proof. We begin by establishing the following special case
Claim: If u ( x, t ) is a smooth solution of (1) on R n × [0 , T ) satisfying(i) sup R n | D l u ( x, t ) | < ∞ for every t ∈ [0 , T ) and l ≥ , (ii) u ( x,
0) satisfies ( − δ ) I n ≤ D u ( x, ≤ (1 − δ ) I n for some δ > , then u ( x, t ) satisfies ( − δ ) I n ≤ D u ( x, t ) ≤ (1 − δ ) I n for each t ∈ (0 , T ).This was established in Lemma 4.1 in [1] and we provide a different proof of thishere. We begin by describing a change of coordinate which we will use at variousplaces throughout the paper. Let z j = x j + √− y j and w j = r j + √− s j ( j = 1 , ..., n )be two holomorphic coordinates on C n related by(15) z j = e √− σ w j AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 7 for some constant σ . Then as described in [12] (see p.1356), if L = { ( x, u ( x )) | x ∈ R n } in C n is represented as L = { ( r, v ( r )) | r ∈ R n } in the coordinates w j , then v satisfies(16) arctan λ i ( D v ) = arctan λ i ( D u ) − σ. Now by (ii) in the claim, as described in [11] we may choose σ = − π/ L and the new potential function will satisfy(17) δ − δ I n ≤ D v ≤ − δδ I n . The claim will be established once we show (17) is preserved for any δ >
0. Differen-tiating (1) twice with respect to any coordinate direction x k yields(18) n X i,j =1 g ij ∂ ij v kk − ∂ t v kk = n X l,m =1 g ll g mm ( λ l + λ m ) v lmk ≥ v denote partial differentiation. Now fix any vector V ∈ R n and any point ( r , t ) note that V T D v ( r , t ) V = v V V ( r, t ) where v V V ( r, t ) is just thesecond derivative of v ( r , t ) in the direction V . It follows from (18) that the function f ( r, t ) = V T (cid:18) D v ( r, t ) − − δδ I (cid:19) V satisfies n X i,j =1 g ij ∂ ij − ∂ t ! f ( r, t ) ≥ r, t ) in R n × [0 , T ). Now note that by our assumption on the derivatives of u we have that g ij ( r, t ) is uniformly equivalent to the Euclidean metric on R n uniformlyfor t ∈ [0 , T ] with T < T , while g ij ( r, t ) and f ( r, t ) are also continuous on R n × [0 , T )the maximum principle (Theorem 9, p.43, [3]) then implies f ( r, t ) ≤ t . Wecan similarly prove that f ( r, t ) ≥ t . This establishes the claim.Now let u and u ( x, t ) be as in the lemma, and let u k be a sequence as in Lemma2.1. Fix some sequence δ k → v k = (1 − δ k ) u k . Thenby Proposition 2.1 there exists a positive sequence T k such that for each k thereis a smooth solution v k ( x, t ) of (1) on R n × [0 , T k ) with initial condition v k andsup x ∈ R n | D l v k ( x, t ) | < ∞ for every l ≥ t ∈ [0 , T k ). For each k , assume that T k is the maximal time on which the solution v k exists. By the above claim we also have( − δ k ) I n ≤ D v k ( x, t ) ≤ (1 − δ k ) I n for each t ∈ [0 , T k ) and the main theorem in[1] then implies sup x ∈ R n | D l v k ( x, t ) | ≤ C l,k /t l − for all l ≥
3, and some constant C l,k depending only on l and δ k and it follows that T k = ∞ . In fact, the local estimates inTheorem 2.1 can be used to remove the dependence on δ k in these bounds. Indeed,fix some k , T ∈ (0 , ∞ ) and x ′ ∈ R n and let(19) w k ( y, s ) = 1 T (cid:16) v k ( y √ T + x ′ , sT + T ) − v k ( x ′ , T ) − Dv k ( x ′ , T ) · y (cid:17) . Then we have w k (0 ,
0) = Dw k (0 ,
0) = 0, and w k ( y, s ) solves (1) on R n × [ − ,
0] andsatisfies − (1 − δ k ) I n ≤ D w k ≤ (1 − δ k ) I n for all ( y, s ) ∈ R n × [ − , ALBERT CHAU, JINGYI CHEN, AND YU YUAN
Theorem 2.1 then gives(20) sup ( x,t ) ∈ B √ T/ ( x ′ ) × [(3 T/ ,T ] (cid:12)(cid:12) D v k ( x, t ) (cid:12)(cid:12) = sup ( y,s ) ∈ Q / T (cid:12)(cid:12) D w k ( y, s ) (cid:12)(cid:12) ≤ CT where B √ T / ( x ′ ) is the ball of radius √ T / x ′ ∈ R n and C is some constantindependent of k . Noting that x ′ ∈ R n and T ∈ (0 , ∞ ) were arbitrary we obtain(21) sup x ∈ R n (cid:12)(cid:12) D v k ( x, t ) (cid:12)(cid:12) ≤ Ct for all t ∈ (0 , ∞ ) and it follows from a scaling argument, described in the proof ofLemma 5.2 in [1], that for every t ∈ (0 , ∞ ) and l ≥ x ∈ R n (cid:12)(cid:12) D l v k ( x, t ) (cid:12)(cid:12) ≤ C l t l − for some constant C l depending only on l .From (22) we conclude that the v k ( x, t )’s have a subsequence converging to a func-tion v ( x, t ) on R n × [0 , ∞ ) where the convergence is smooth on compact subsets of R n × (0 , ∞ ). In particular, by construction we have that v ( x, t ) is smooth and solves(1) on R n × (0 , ∞ ), satisfies (14) for every t ∈ [0 , ∞ ) and v ( x,
0) = u ( x ). Moreover,by (1) we have | ∂ t v ( x, t ) | ≤ nπ for all ( x, t ) ∈ R n × [0 , ∞ ) from which we concludethat v ∈ C ( R n × [0 , T )).It now follows by the uniqueness result in [2] that u ( x, t ) = v ( x, t ) for all t ∈ [0 , T ),and thus u ( x, t ) also satisfies (14) for every t ∈ [0 , T ). This completes the proof ofthe lemma. (cid:3) We now apply the above results to prove the following proposition.
Proposition 2.2.
There exists a dimensional constant η = η ( n ) > such that forevery T > the following holds: if u ( x, t ) is a smooth solution to (1) on R n × [0 , T ) such that − (1 + η ) I ≤ D u ≤ (1 + η ) I at t = 0 and sup x ∈ R n | D l u | < ∞ for each t ∈ [0 , T ) and l ≥ , then u ( x, t ) satisfies −√ I n ≤ D u ≤ √ I n for all t ∈ [0 , T ) .Proof. Suppose otherwise. Then there exists a sequence η k → u k ( x, t ) each solving (1) on R n × [0 , T k ) where T k >
0, and each satisfying(a) − (1 + η k ) I n ≤ D u k ≤ (1 + η k ) I n at t = 0,(b) sup x ∈ R n (cid:12)(cid:12) D l u k (cid:12)(cid:12) < ∞ for each t ∈ [0 , T k ) and l ≥ | λ i ( D u k ( x k , t k )) | > √ x k , t k ) ∈ R n × [0 , T k ) and some i .Then by (a) and (b) it is not hard to show that there exists a sequence R k with R k ∈ (0 , T k ) satisfying(A) −√ I n ≤ D u k ≤ √ I n for all t ∈ [0 , R k ),(B) | λ i ( D u k ( x k , t k )) | = p / x k , t k ) ∈ R n × [0 , R k ) and some i .Now consider the sequence v k ( x, t ) := 1 t k (cid:0) u k ( xt k + x k , t k t + t k ) − u k ( x k , t k ) − Du k ( x k , t k ) · x (cid:1) AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 9 each solving (1) on R n × [ − ,
0] and each satisfying(i) − (1 + η k ) I n ≤ D v k ≤ (1 + η k ) I n at t = 0 and −√ I n ≤ D v k ≤ √ I n ∀ t > (cid:12)(cid:12) λ i ( D v k )(0 , (cid:12)(cid:12) = p / i ,(iii) v k (0 ,
0) = Dv k (0 ,
0) = 0.Then as assumption (ii) in Theorem 2.1 is satisfied, we may apply the estimatesthere as in the proof of Lemma 2.2 to show that the v k ( x, t ) ′ s have a subsequenceconverging to a function v ( x, t ) ∈ C ∞ ( R n × ( − , T C ( R n × [ − , x ∈ R n | D v ( x, t ) | bounded independent of t ∈ [ − / , v ( x, t ) solves (1) and satisfies − I n ≤ D v ( x, − ≤ I n in the L ∞ senseand | λ i ( D v (0 , | = p / i . Together, these facts contradict Lemma2.2. (cid:3) Remark . Noting that (13) in Theorem 2.1 holds in general when n ≤
3, we observethat when n ≤ √ C >
0. 3.
Proof of Theorem 1.1
Proof of Theorem 1.1.
Let u be as in Theorem 1.1 where η > u k be a sequence of approximations as in Lemma 2.1. By Proposition 2.1 wehave smooth short time solutions u k ( x, t ) to (1) with initial condition u k ( x, t ) = u k ( x ).Moreover, by Proposition 2.2 we have −√ I n ≤ D u ≤ √ I n for all ( x, t ). We willlet R n × [0 , T k ) be the maximal space time domain on which u k ( x, t ) is defined.Then by a rescaling argument and applying Theorem 2.1 as in the the proof ofLemma 2.2, we can show that for each k , T k = ∞ and u k ( x, t ) satisfies the estimatesin (22) for all l ≥ t >
0. In particular, we argue as in the last two paragraphs ofthe proof of Lemma 2.2 that some subsequence of the u k ( x, t )’s converge to a function u ( x, t ) solving (1) on R n × [0 , ∞ ) satisfying (i) and (ii) in the conclusions of Theorem1.1.We now show that Du ( x, t ) satisfies conclusion (iii) in Theorem 1.1. By differen-tiating (1) once in space and using (i) and the estimates in (ii) for l = 3 we mayestimate as follows for any x ∈ R n and t > t ′ > | Du ( x, t ) − Du ( x, t ′ ) | ( t − t ′ ) / ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z tt ′ ∂ s Du ( x, s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ( t − t ′ ) / ≤ C Z tt ′ (cid:12)(cid:12) D u ( x, s ) (cid:12)(cid:12) ds ( t − t ′ ) / ≤ C Z tt ′ s − / ds ( t − t ′ ) / ≤ C (23) for some constant C independent of x , t and t ′ . The uniqueness of u ( x, t ) follows fromthe uniqueness result in [2]. (cid:3) Proofs of Theorem 1.2 and Theorem 1.3
We begin by establishing the following lemmas
Lemma 4.1.
Let v ( r, t ) be a solution to (1) as in Theorem 1.1 and assume − I n ≤ D v ( r, t ) ≤ I n for all r and t . Then if λ ( r ′ , t ′ ) = 1 at some point where t ′ > , then λ ( r, t ) = 1 for all ( r, t ) ∈ R n × [0 , t ′ ) . Similarly if λ ( r ′ , t ′ ) = − at some point where t ′ > , then λ ( r, t ) = − for all ( r, t ) ∈ R n × [0 , t ′ ) .Proof. In [14], the authors consider a solution v to the elliptic equation correspondingto (1):(24) n X i =1 arctan λ i = C where C is some constant. By twice differentiating (24) and the characteristic equa-tion det( D v + λ i I n ) = 0 they obtained a formula for P na,b =1 g ab ∂ ab ln p λ i atany point where λ i is a non-repeated eigenvalue for D v ([14]; Lemma 2.1). By es-sentially the same calculations we may differentiate the parabolic equation (1) andthe characteristic equation det( D v + λ i I n ) = 0 twice in space, and also differenti-ate the characteristic equation once in time, to obtain the exact same formula for( P na,b =1 g ab ∂ ab − ∂ t ) ln p λ i at any point where λ i is a non-repeated eigenvalue for D v . Namely, if λ i is a non-repeated eigenvalue of D v at a point ( r , t ) then thefollowing holds at ( r , t ) (after making a linear change of coordinates on R n so that D v ( r , t ) is diagonal): n X a,b =1 g ab ∂ ab − ∂ t ! ln q λ i = (cid:0) λ i (cid:1) h iii + X α = i λ i λ i − λ α (1 + λ i λ α ) h ααi + X α = i (cid:20) λ i + 2 λ i λ i − λ α (1 + λ i λ α ) (cid:21) h iiα + X α<βα,β = i λ i (cid:18) λ i λ α λ i − λ α + 1 + λ i λ β λ i − λ β (cid:19) h αβi (25)where h αβγ ( r, t ) is the second fundamental form of the embedding F ( r, t ) = ( r, Dv ( r, t ))of R n to R n . Claim: If λ ( r ′ , t ′ ) = 1 at some point where t ′ >
0, then λ ( r, t ) = 1 for all( r, t ) ∈ R n × (0 , t ′ ].We will always assume that 1 ≥ λ ≥ λ ≥ · · · ≥ λ n ≥ − AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 11 multiplicity k and consider the function f = P ki =1 ln p λ i . Then f is a smoothfunction in a space-time neighborhood U × ( t ′ − ǫ, t ′ + ǫ ) of ( r ′ , t ′ ) (see [10]) andattains a maximum value in U × ( t ′ − ǫ, t ′ + ǫ ) at ( r ′ , t ′ ). Now we want to computethe evolution of f in U × ( t ′ − ǫ, t ′ + ǫ ). We illustrate how to do this first at somepoint where λ , ..., λ k are all distinct. In this case we may apply (25) separately toeach term in f , and after some computation we obtain n X a,b =1 g ab ∂ ab − ∂ t ! k X i =1 ln q λ i = X γ ≤ k (cid:0) λ γ (cid:1) h γγγ + I + II ≥ I + II (26)where I = X α<γ ≤ k (cid:0) λ α + 2 λ α λ γ (cid:1) h ααγ + X α ≤ k<γ λ α − λ γ + λ α ( λ α + λ γ ) λ α − λ γ h ααγ + X α<γ ≤ k (cid:0) λ γ + 2 λ γ λ α (cid:1) h γγα + X α ≤ k<γ λ α (1 + λ α λ γ ) λ α − λ γ h γγα II = 2 X α<β<γ ≤ k (3 + λ α λ β + λ β λ γ + λ γ λ α ) h αβγ + 2 X α<β ≤ k<γ (cid:20) λ α λ β + λ β (cid:18) λ β λ γ λ β − λ γ (cid:19) + λ α (cid:18) λ α λ γ λ α − λ γ (cid:19)(cid:21) h αβγ + 2 X α ≤ k<β<γ λ α (cid:18) λ α λ β λ α − λ β + 1 + λ α λ γ λ α − λ γ (cid:19) h αβγ (27)where I corresponds to summing the second and third term on the right hand sideof (25) for i = 1 , ..., k and II corresponds to summing the fourth term on the righthand side of (25) for i = 1 , ..., k . Our derivation above only applies at a point where λ , ..., λ k are all distinct, and thus cannot be used directly to calculate the evolution of f = P ki =1 ln p λ i at ( r ′ , t ′ ). We now remove this assumption on the distinctnessof eigenvalues by the approximation argument below.Consider the function v m ( r, t ) := v ( r, t ) − m k X j =1 jr j . Then for sufficiently large m , in some space-time neighborhood of ( r ′ , t ′ ) which westill denote as U × ( t ′ − ǫ, t ′ + ǫ ) the eigenvalues λ i,m of D v m will be between − k largest eigenvalues will be non-repeated. Thus the function ln q λ i,m is smooth in U × ( t ′ − ǫ, t ′ + ǫ ) for each i . On the other hand, by (1) and the definition v m we have(28) ∂v m ∂t = n X i =1 arctan λ i,m + w m where w m = n X i =1 arctan λ i ( v ) − n X i =1 arctan λ i ( v m ) . Note that w m approaches zero smoothly and uniformly on compact subsets of U × ( t ′ − ǫ, t ′ + ǫ ) as m → ∞ . By (28), the above referenced derivation of (25) and by(27) we have n X a,b =1 g abm ∂ ab − ∂ t ! k X i =1 ln q λ i,m = X γ ≤ k (cid:0) λ γ,m (cid:1) h γγγ + I m + II m + k X i =1 λ i,m λ i,m ( w m ) ii ≥ k X i =1 λ i,m λ i,m ( w m ) ii (29)in U × ( t ′ − ǫ, t ′ + ǫ ) where I m is obtained by replacing λ α and λ β in I by λ α,m and λ β,m respectively, and II m is obtained similarly. We have also used the fact that I m , II m is nonnegative. Letting m → ∞ , we conclude that ( P na,b =1 g ab ∂ ab − ∂ t ) f ≥ U × ( t ′ − ǫ, t ′ + ǫ ) and thus f = k ln √ U × ( t ′ − ǫ, t ′ ] by the strong maximumprinciple (Theorem 1, p.34, [3]).Now for any ( r ′′ , t ′′ ) ∈ R n × [0 , t ′ ] let γ ( s ) : [0 ,
1] be a line segment in space-timesuch that γ (0) = ( r ′ , t ′ ) and γ (1) = ( r ′′ , t ′′ ). Let A be the set of ¯ s ∈ [0 ,
1] for which λ ( γ ( s )) = 1 for all s ∈ [0 , ¯ s ]. Then the above argument shows that A is in fact openand non-empty. Moreover, A is clearly closed by continuity and we then concludethat A = [0 ,
1] and in particular, λ ( r ′′ , t ′′ ) = 1. This established the claim and thusthe first statement in the conclusion of the lemma.By considering the solution − v ( r, t ) to (1), we likewise conclude the second state-ment in the concslusion of the lemma is true. (cid:3) Lemma 4.2.
Let v ( r, t ) be a solution to (1) as in Theorem 1.1 and assume that − I n < D v ( r, t ) ≤ I n for all r ∈ R n and t ∈ [0 , ∞ ) . Then either D v ( r, t ) < I n forall r and t > or there exist coordinates r , ..., r n on R n in which we have v ( r, t ) = r + · · · + r k + w ( r k +1 , ..., r n , t ) on R n × [0 , ∞ ) where − I n < D w ( r, t ) < I n for all r , t > and k > .Proof. We begin by establishing the following claims.
AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 13
Claim 1: If v ( r ′ , t ′ ) = 1 at some point ( r ′ , t ′ ) with t ′ >
0, then v = 1 on R n × [0 , t ′ ].By a rotation of coordinates on R n , we may assume that D v ( r ′ , t ′ ) is diagonal.Since D v > − I n there exists some space time neighborhood U × ( t ′ − ǫ, t ′ + ǫ ) of ( r ′ , t ′ )in which − (1 − δ ) I n ≤ D v ( r, t ) ≤ I n for some ǫ, δ >
0. By (16) it follows that for somechoice of σ ∈ (0 , π/ C n (from w j to z j ) using (15) sothat the local family of Lagrangian graphs L = { ( r, Dv ( r, t )) | ( r, t ) ∈ U × ( t ′ − ǫ, t ′ + ǫ ) } is represented in the new coordinates as L = { ( x, Du ( x, t )) | ( x, t ) ∈ U × ( t ′ − ǫ, t ′ + ǫ ) } for some space time neighborhood U × ( t ′ − ǫ, t ′ + ǫ ) in which 0 ≤ D u ( x, t ) ≤ M I n with u ( x ′ , t ′ ) = M at some interior point ( x ′ , t ′ ) with respect to coordinates x , ..., x n given by (15). It follows from (18) and the strong maximum principle (Theorem 1,p.34, [3]) that u = M in U × [ t ′ − ǫ, t ′ ] and thus v = 1 in U × [ t ′ − ǫ, t ′ ].Now for any ( r ′′ , t ′′ ) ∈ R n × [0 , t ′ ] and let γ ( s ), s ∈ [0 , γ (0) = ( r ′ , t ′ ) and γ (1) = ( r ′′ , t ′′ ). Let A be the set of ¯ s ∈ [0 , v ( γ ( s )) = 1 for all s ∈ [0 , ¯ s ]. Then the above argument shows that A isin fact open and non-empty. Moreover, A is clearly closed by continuity and we thenconclude that A = [0 ,
1] and in particular, v ( r ′′ , t ′′ ) = 1. This established the claim. Claim 2:
In Claim 1, we in fact have v ( r, t ) = r + w ( r , ..., r n , t ) on R n × [0 , ∞ ).Integrating v twice with respect to r gives v ( r, t ) = r r w ( r , ..., r n , t ) + w ( r , ..., r n , t )on R n × [0 , t ′ ] for some functions w and w . It follows that w must in fact be linearwith respect to x , ..., x n as otherwise D v would be unbounded on R n × [0 , t ′ ) thuscontradicting our assumption on that − I n < D v ( r, t ) ≤ I n for all r and t . Ourassumption that D v ( r ′ , t ′ ) is diagonal then implies that w must in fact be constantin space. Finally, as the right hand side of (1) is uniformly bounded in absolute valuefrom which we further conclude that is in fact constant in time as well and thus aftera possible translation of the coordinate r we have v ( r, t ) = r w ( r , ..., r n , t )on R n × [0 , t ′ ] for some function w . Now observe that up to the addition of a timedependent constant, w ( r , ..., r n , t ) solves (1) on R n − × [0 , t ′ ] and by Theorem 1.1this extends to a smooth longtime solution which we still denote as w ( r , ..., r n , t ). Inparticular r + w ( r , ..., r n , t ) is also a longtime solution to (1) and it follows from theuniqueness result in [2] that the above representation of v ( r, t ) holds on R n × [0 , ∞ ).The lemma follows by iterating the arguments above starting with the function w in Claim 2. (cid:3) Proof of Theorem 1.2.
Now let u be a C , locally weakly convex function as inTheorem 1.2. Using σ = π/ C n and noting (16)(see also [11]), we represent the Lagrangian graph L = { ( x, u ( x )) | x ∈ R n } in thecoordinates z j as L = { ( r, v ( r )) | r ∈ R n } in the coordinates w j where v satisfies − I n ≤ D v < I n . Let v ( r, t ) be the long time solution to (1) with initial condition v given by Theorem 1.1. Then from Lemma 2.2 and Lemma 4.1 we have − I n ≤ D v ( r, t ) < I n for all r , t ≥
0. Moreover, applying Lemma 4.2 to − v ( r, t ) we furtherconclude that either − I n < D v ( r, t ) < I n r , t > v ( r, t ) = − r · · · − r k w ( r k +1 , ..., r n , t )on R n × [0 , T ) where k > − I n < D w ( r, t ) < I n for all r ∈ R n , t ≥
0. Let L t = { ( r, v ( r, t )) | ( r, t ) ∈ R n × [0 , ∞ ) } be the corresponding family of Lagrangiangraphs in C n . Then by (16), L t will correspond to a family of Lagrangian graphs { ( x, Du ( x, t )) | ( x, t ) ∈ R n × [0 , ∞ ) } such that u ( x, t ) is a longtime solution to (1)satisfying (i) in Theorem 1.2. Now note that as v ( x, t ) satisfies (ii) and (iii) in Theorem1.1 it also satisfies (ii) and (iii) in Theorem 1.2. It follows that u ( x, t ) must thenalso satisfy (ii) and (iii) in Theorem 1.2. The uniqueness of u ( x, t ) follows from theuniqueness result in [2]. (cid:3) Proof of Theorem 1.3.
Let u be a locally C , function satisfying (3). Then u isautomatically convex and by Theorem 1.2 there exists a longtime convex solution u ( x, t ) to (1) with initial condition u . In particular, note that u ( x, t ) satisfies (ii)and (iii) in Theorem 1.2. It will be convenient here to define the operatorΘ( A ) := n X i =1 arctan λ i ( A )on symmetric real n × n matrices A where the λ i ’s are the eigenvalues of A . A directcomputation shows that as u ( x, t ) solves (1), Θ( D u ( x, t )) evolves according to(31) ∂ t Θ = n X i,j =1 g ij ∂ ij Θ . We would like to use (31) and the maximum principle (Theorem 1, p.34, [3]) to con-clude that (3) is thus preserved for all t >
0. One difficulty here is that Θ( D u ( x, t ))is not necessarily continuous at t = 0. Another difficulty is that D u ( x, t ) is notneccesarily bounded above, and thus the symbol g ij is not necessarily bounded below(by a positive constant) on R n for t >
0. To overcome this we will need to transformand approximate our solution u ( x, t ) through the following sequence of steps. Step 1 (small rotation):
We begin using (15), with σ = σ ∈ (0 , π/
2) to bechosen in a moment, to change coordinates on C n and represent the Lagrangiangraphs L t = { ( x, u ( x, t )) | x ∈ R n } in the coordinates z j as L t = { ( r, v ( r, t )) | r ∈ R n } in the coordinates w j for some family v ( r, t ) with ( r, t ) ∈ R n × [0 , ∞ ). By (16) wehave(32) Θ( D v ( r, ≥ ( n − π − nσ and by (16) and the convexity of u ( x, t ) we have(33) − K ( σ ) ≤ D v ( r, t ) ≤ /K ( σ )for all r, t where K ( σ ) → σ → AGRANGIAN MEAN CURVATURE FLOW FOR ENTIRE LIPSCHITZ GRAPHS II 15
Step 2 (approximation):
Let v k be the sequence of approximations of v = v ( r, k we have − K ( σ ) ≤ D v k ≤ /K ( σ ) by(33). Moreover, sup r ∈ R n | D l v k ( r ) | < ∞ for all l ≥
3. Now we show that(34) Θ( D v k ) ≥ ( n − π − nσ is satisfied for all k .Fix r ∈ R n and k . By (12) we have D v k ( r ) = Z R n D v ( y ) K (cid:18) r, y, k (cid:19) dy. Approximating by the Riemann sums, we can find a double sequence { p ij } ⊂ R n anda sequence { j i } ⊂ Z + for which(35) D v k ( r ) = lim i →∞ j i X j =1 D v ( p ij ) K (cid:18) r, p ij , k (cid:19) i n . On the other hand, Z R n K (cid:18) r, y, k (cid:19) dy = 1and we may then further assume B i := j i X j =1 K (cid:18) r, p ij , k (cid:19) i n → i → ∞ . By (35) we then have(36) D v k ( r ) = lim i →∞ j i X j =1 D v ( p ij ) A ij where A ij = K (cid:0) r, p ij , k (cid:1) / ( i n B i ) and in particular P j i j =1 A ij = 1 while A ij ≥ i, j . Now since ( n − π − nσ > ( n − π by our choice of σ , the results in [12]assert that the set of symmetric n × n matrices A for which Θ ≥ ( n − π − nσ is aconvex set S in the space of real n × n symmetric matrices. This, (36) and the factthat D v ( p ij ) ∈ S for all i, j imply D v k ( r ) ∈ S . Thus (34) holds for each k . Step 3 ( π/ rotation): Now we use (15) as in Step 1, but with σ = π/
4, toobtain from v ( r, t ) and the v k ( r )’s a corresponding family w ( p, t ) and sequence w k ( p ).In particular, w ( p, t ) is a longtime solution to (1) and the w k ’s will satisfy (2) inTheorem 1.1, provided σ > σ has been made. They will also satisfy(37) Θ( D w k ( p )) ≥ ( n − π − nσ − n π k , Theorem 1.1 gives a longtime solution w k ( p, t ) to (1) withinitial condition w k satisfying sup r ∈ R n | D l w k ( p, t ) | < ∞ for all l ≥ t ≥
0. It follows from (37), (31), (16) and the weak maximum principle (Theorem 9, p.43, [3])that(38) Θ( D w k ( p, t )) ≥ ( n − π − nσ − n π p, t ). Now using Theorem 1.1 and arguing as in the beginning of the proof ofTheorem 1.1, we see that some subsequence of the w k ( p, t )’s converge smoothly anduniformly on compact subsets of R n × (0 , ∞ ) to a smooth limit solution to (1) on R n × (0 , ∞ ). By the uniqueness result in [2] and the definition of w k , we see this limitsolution is in fact the solution w ( p, t ). In particular, w ( p, t ) must satisfy (38) for all( p, t ).Rotating back to the original coordinates, we conclude from the last statementabove that u ( x, t ) must satisfy (3) for all t ≥
0. Thus either (3) holds with strictinequality for all t > x ′ , t ′ ) ∈ R n × (0 , ∞ ) at which equalityholds in (3) in which case (31) and the strong maximum principle (Theorem 1, p.34,[3]) give(39) Θ( D u ( x, t )) = ( n − π R n × (0 , t ′ ]. In this case, integrating (1) in t and noting the continuity of u ( x, t ) in t (for all t ≥
0) we obtain u ( x, t ) = u ( x, t ′ ) + ( n − π t − t ′ )for all t ∈ [0 , t ′ ], and thus for all t ∈ [0 , ∞ ) by the uniqueness result in [2]. Inparticular, D u ( x, t ) satisfies (39) for all t ≥
0. On the other hand, u ( x, t ′ ) is smoothin x and it follows that u ( x ) = u ( x,
0) is a smooth convex solution to the specialLagrangian equation Θ( D u ( x )) = ( n − π on R n and is thus quadratic by theBernstein theorem in [11]. This concludes the proof of Theorem 1.3. (cid:3) References
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Department of Mathematics, University of British Columbia, Vancouver, B.C.,V6T 1Z2, CanadaDepartment of Mathematics, University of Washington, Seattle, WA 98195, U.S.A.
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