A Liouville type theorem for special Lagrangian Equations with constraints
aa r X i v : . [ m a t h . A P ] J a n A LIOUVILLE TYPE THEOREM FOR SPECIALLAGRANGIAN EQUATIONS WITH CONSTRAINTS
MICAH WARREN AND YU YUAN
Abstract.
We derive a Liouville type result for special Lagrangianequations with certain “convexity” and restricted linear growth assump-tions on the solutions. Introduction
In this note, we show the following
Theorem 1.1.
Let u be a smooth solution to the special Lagrangian equation (1.1) n X i =1 arctan λ i = c on R n , where λ i s are the eigenvalues of the Hessian D u ( x ) . Suppose that (1.2) 3 + (1 − ε ) λ i ( x ) + 2 λ i ( x ) λ j ( x ) ≥ for all i, j, x and any small fixed ε > and the gradient ∇ u ( x ) satisfies (1.3) |∇ u ( x ) | ≤ δ ( n ) | x | for large | x | and any fixed δ ( n ) < / √ n − . Then u must be a quadraticpolynomial. The special Lagrangian equation (1.1) arises in the calibrated geometry[HL]. A Lagrangian graph M = ( x, ∇ u ( x )) ⊂ C n = R n × R n is called special when the calibrating n -formΩ c = Re( e −√− c dz ∧ dz ∧ ... ∧ dz 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 )) ⊂ C n is a (volume minimizing) minimal surface in R n × R n [HL, Theorem 2.3,Proposition 2.17].By Fu’s classification result [F], any global solution to (1.1) on R is ei-ther quadratic or harmonic; a harmonic function with any linear growthcondition on the gradient is certainly quadratic; see also [Y3] for a unique-ness result for the global solutions to (1.1) with | c | > ( n − π . In the case n = 3 , other Liouville-Bernstein type results hold true for (1.1) under the Date : November 21, 2018.Y.Y. is partially supported by an NSF grant. following conditions respectively: λ i ≥ − K [Y2]; λ i λ j ≥ − K [Y4]; or c = π and the solution is strictly convex with quadratic growth [BCGJ]. Whileboundedness of the Hessian alone is sufficient in dimension three, certainboundedness and convexity are both needed for Liouville-Bernstein type re-sults to be valid for (1.1) in the general dimension ( n ≥ c = kπ and the solution is convex with lineargrowth [B]; with the almost convex assumption λ i ≥ − ε ( n ) [Y2]; with thesemi-convex assumption λ i ≥ − √ + γ everywhere, or with the (“equiva-lent”) assumption | λ i | ≤ √ − γ ′ everywhere [Y4]; or with the assumption λ i λ j ≥ − − ε ( n ) [Y4]. (It is straightforward that any convex solution witha bounded Hessian to (1.1) is a quadratic polynomial, by the well-known C α 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 argumentof [HJW].) A Liouville-Bernstein type result with the assumption | λ i | ≤ K and λ i λ j ≥ const > − was stated in [TW].The more general “convexity” condition (1.2) does not alone lead to anyHessian bound for the solutions to (1.1), but does guarantees that the volumeelement 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 Hessianestimate in terms of the gradient, for solutions to the special Lagrangianequation (1.1). The unpleasant technical assumption δ ( n ) < / √ n − whole process (for n ≥ . (Unlike [JX], we could not generalize the iteration argument in [HJW] toget a Liouville type result for now the larger image set (1.2) of the corre-sponding harmonic Gauss map to the Lagrangian Grassmanian.) The simpleconstraints | λ i | ≤ K like | λ i | ≤ | λ i | ≤ √ − γ are easily shown to beavailable in the blow-down process. An extra effort is needed to justify thatthe nonlinear constraints (1.2) or others like λ i λ j ≥ const are preserved un-der the C ,α convergence of the scaling process u k ( x ) = u (cid:0) k x (cid:1) /k . Takingadvantage of the single elliptic equation (1.1), we apply the W ,δ estimatesfor solutions in terms of the supreme norm of the solution to extract a W ,δ sub-convergent sequence, as in [Y1]. Then we extract another subsequencewith the Hessians converging almost everywhere. This justifies that the con-straints (1.2) are preserved in the above blow-down process. Another routeof the justification is through Allard’s regularity result (cf. [S, Section 36]). LIOUVILLE TYPE THEOREM 3
Actually, Theorem 1.1 holds true for n = 3 without any growth conditionlike (1.3) . The condition (1.2) implies λ i λ j ≥ − K, so as in [Y4] we can finda bound on the Hessian (possibly for a new potential), and then draw theconclusion. Note that the boundedness on the Hessian alone for n = 3 isenough for one to run the blow-down process to obtain a Liouville type result;see [F-C, Theorem 5.4] of Fischer-Colbrie. In general dimension n ≥ , wederive yet another Liouville-Bernstein type result for the solutions to (1.1)with the bounded Hessian satisfying weaker constraints (3.1); see Theorem3.1 in the appendix. One consequence of Theorem 3.1 coupled with the DeGiorgi-Allard ε -regularity theory is an improvement of the above mentionedLiouville-Bernstein type result in [Y4], namely, any global solution to (1.1)with λ i ≥ − √ − ε ( n ) everywhere or | λ i | ≤ √ ε ′ ( n ) everywhere is aquadratic polynomial (for n ≥ − Notation. ∂ i = ∂∂x i , ∂ ij = ∂ ∂x i ∂x j , u i = ∂ i u, u ji = ∂ ij u, etc.2. Proof Of Theorem 1.1
Taking the gradient of both sides of the special Lagrangian equation (1.1),we have(2.1) n X i,j g ij ∂ ij ( x, ∇ u ( x )) = 0 , where (cid:0) g ij (cid:1) is the inverse of the induced metric g = ( g ij ) = I + D uD u onthe surface ( x, ∇ u ( x )) ⊂ R n × R n . Simple geometric manipulation of (2.1)yields the usual form of the minimal surface equation △ g ( x, ∇ u ( x )) = 0 , where the Laplace-Beltrami operator of the metric g is given by △ g = 1 √ det g n X i,j ∂ i (cid:16)p det gg ij ∂ j (cid:17) . MICAH WARREN AND YU YUAN
Because we are using harmonic coordinates △ g x = 0 , we see that △ g alsoequals the linearized operator of the special Lagrangian equation (1.1) at u, △ g = n X i,j g ij ∂ ij . The gradient and inner product with respect to the metric g are ∇ g v = n X k =1 g k v k , · · · , n X k =1 g nk v k ! h∇ g v, ∇ g w i g = n X i,j =1 g ij v i w j , in particular |∇ g v | g = h∇ g v, ∇ g v i g . We begin by demonstrating a Jacobi inequality for the volume element V = p det g = n Y i =1 (1 + λ i ) . Lemma 2.1.
Suppose that u is a smooth solution to (1.1) satisfying (1.2).Then △ g ln V ≥ εn |∇ g ln V | g or equivalently (2.2) △ g V εn ≥ |∇ g V εn | g V εn . Proof.
By differentiating the minimal surface equation (2.1) again and per-forming some long and tedious computation, one gets the standard formulafor △ g ln V ; see for example [Y2, Lemma 2.1]. (The general formula for min-imal submanifolds of any dimension or codimension originates in Simons [Ss,p. 90].) At any fixed point, we assume that D u is diagonalized, then △ g ln V = n X i,j,k =1 (1 + λ i λ j ) h ijk , where h ijk = p g ii p g jj p g kk u ijk . Gathering all terms containing h ijj = h jij = h jji for a fixed i , we have(1 + λ i ) h iii + X j = i (1 + λ j ) h jji + X j = i (1 + λ i λ j ) h ijj + X j = i (1 + λ j λ i ) h jij = (1 + λ i ) h iii + X j = i (3 + λ j + 2 λ i λ j ) h jji . LIOUVILLE TYPE THEOREM 5
Thus △ g ln V = n X i =1 (1 + λ i ) h iii + X j = i (3 + λ j + 2 λ i λ j ) h jji (2.3) + 2 X i Combining (1.2) with (2.6) and (2.7) we have △ g ln V − εn |∇ g ln V | ≥ n X i =1 ( (cid:2) − ε ) λ i (cid:3) h iii + X j = i ( (cid:2) − ε ) λ j + 2 λ i λ j (cid:3) h jji ≥ . (2.8)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-fying condition (1.2) and |∇ u | ≤ δ < √ n − . Then | D u (0) | ≤ C ( n, δ, ε ) . Proof. Set v = u + α |∇ u (0) | h∇ u (0) , x i or u + αx if ∇ u (0) = 0 , where α = (cid:16) √ n − − δ (cid:17) / . Now v satisfies in B the following D v = D u, |∇ v (0) | ≥ α, and |∇ v | ≤ α + δ < √ n − . Set b = V εn , and consider the function w = ηb = h |∇ v | − ( α + δ ) | x | i + b ≥ . A positive maximum for w will be attained at a point p on the interior, since w (0) > w ( x ) vanishes on the boundary ∂B . At this point p, ∇ g ( ηb ) = 0 or ∇ g η = − ηb ∇ g b, ≥ △ g ( ηb ) = η △ g b + 2 h∇ g η, ∇ g b i g + b △ g η = η △ g b − |∇ g b | g b ! + b △ g η ≥ b △ g η, LIOUVILLE TYPE THEOREM 7 by the inequality (2.2) in Lemma 2.1. This last inequality implies a boundon | D v ( p ) | as the following. We have0 ≥ △ g η = △ g h |∇ v | − ( α + δ ) | x | i = n X i,j =1 g ij " n X k =1 ( v ki v kj + v k ∂ ij v k ) − ( α + δ ) ∂ ij | x | = 2 n X i =1 λ i − ( α + δ ) λ i ≥ " λ − ( α + δ ) λ − ( n − 1) ( α + δ ) , using the minimal surface equation (2.1) and assuming | λ | ≥ | λ i | for all i. It follows that 1 + λ ( p ) ≤ α + δ ) − ( n − 1) ( α + δ ) . We get α b (0) ≤ |∇ v (0) | b (0) ≤ η ( p ) b ( p ) ≤ ( α + δ ) " α + δ ) − ( n − 1) ( α + δ ) ε , then(2.9) 1 + λ i (0) ≤ (cid:18) δα (cid:19) nε " α + δ ) − ( n − 1) ( α + δ ) n . Therefore, we conclude the estimate | D u (0) | ≤ C ( n, δ, ε ) in Lemma 2.2. (cid:3) Lemma 2.3. Let u ∈ C ∞ ( R n \ { } ) be a solution to the special Lagrangianequation (1.1) and homogeneous of order ; that is, u ( x ) = | x | u ( x/ | x | ) . Suppose that the eigenvalues λ i of the Hessian D u ( x ) satisfy (1.2). Then u must be quadratic.Proof. Lemma 2.3 follows from Proposition 3.1; nonetheless we give a directproof in the following. Considering (1.2), (2.4), and (2.5), we observe thatthe coefficients of h ijk in (2.3) are strictly positive. Accordingly,(2.10) △ g ln V ≥ c ( λ ) n X i,j,k =1 h ijk with c ( λ ) > . Since u is homogeneous of order 2 , the homogeneous order 0 functionln V attains its maximum along a ray. We infer from the strong maximumprinciple that ln V ≡ const. It follows from (2.10) that D u ≡ 0. Therefore, u must be quadratic, 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 ofTheorem 1.1, as in [Y2]. MICAH WARREN AND YU YUAN Step 1. From the assumption that |∇ u ( x ) | ≤ δ | x | for large x, we have onthe ball B R ( p ) with any fixed p ∈ R n |∇ u ( x ) | ≤ δ ( | p | + R ) = (cid:18) δ + δ | p | R (cid:19) R. A rescaled version of Lemma 2.2 with R going to ∞ then leads to a Hessianbound, | D u ( p ) | ≤ C ( n, δ, ε ) , K, which must hold at each point p ∈ R n . Step 2. Repeating verbatim the argument in [Y1, p.263–264], we showthat we can find a tangent cone of the special Lagrangian graph ( x, ∇ u ( x ))at ∞ whose potential function is C , , homogenous order 2 , and still satisfiesthe “convexity” condition (1.2).Without loss of generality, we assume u (0) = 0 , ▽ u (0) = 0 . We “blowdown” u at ∞ . Set u k ( x ) = u ( kx ) k , k = 1 , , , · · · . We see that k u k k C , ( B R ) ≤ C ( K, R ) , so there exists a subsequence, still denoted by { u k } and a function u R ∈ C , ( B R ) such that u k → u R in C ,α ( B R ) as k → ∞ , and (cid:12)(cid:12) D u R (cid:12)(cid:12) ≤ K. By the fact that the family of viscosity solution is closed under C uniformlimit, we know that u R is also a viscosity solution of F (cid:0) D u (cid:1) = n X i =1 arctan λ i = c on B R . Applying the W ,δ estimate (cf. [CC] Proposition 7.4) to the difference u k − u R , we have (cid:13)(cid:13) D u k − D u R (cid:13)(cid:13) L δ ( B R/ ) ≤ C ( K, R ) k u k − u R k L ∞ ( B R ) → k → ∞ . Note that (cid:12)(cid:12) D u k (cid:12)(cid:12) , (cid:12)(cid:12) D u R (cid:12)(cid:12) ≤ K, so also (cid:13)(cid:13) D u k − D u R (cid:13)(cid:13) L n ( B R/ ) → k → ∞ . By a standard fact from real analysis, there exists another subsequence and C , function on B R , still denoted by { u k } and u R/ such that D u k → D u R/ almost everywhere as k → ∞ . So D u R still satisfies (1.2) almosteverywhere on B R/ . The diagonalizing process yields yet another subsequence, again denotedby { u k } and v ∈ C , ( R n ) such that u k → v in W ,nloc ( R n ) as k → ∞ ; v isa viscosity solution of (1.1) on R n ; (cid:12)(cid:12) D v (cid:12)(cid:12) ≤ K ; and D v still satisfies (1.2)almost everywhere on R n . The surfaces ( x, ▽ u k ( x )) are minimal in R n × R n and their potentials u k converge to v in W ,nloc ( R n ) , so by the monotonicity formula (cf. [S, p.84,Theorem 19.3] ), we conclude that M v = ( x, ▽ v ( x )) is a cone. LIOUVILLE TYPE THEOREM 9 Step 3. We claim that M v is smooth away from the vertex. Suppose M v is singular at P away from the vertex. We blow up M v at P to get atangent cone, which is a lower dimensional special Lagrangian cone crossinga line; repeat the procedure if the resulting cone is still singular away fromthe vertex. Finally we get a special Lagrangian cone which is smooth awayfrom the vertex, and the bounded eigenvalues of the Hessian of the potentialfunction satisfies (1.2), by a similar W ,δ argument as in Step 2. By Lemma2.3, the cone is flat. This is a contradiction to Allard’s regularity result (cf.[S, Theorem 24.2]).Applying Lemma 2.3 to M v , we see that M v is flat.Step 4. Now with the flatness of M v , a final application of the monotonic-ity formula yields that the original gradient graph ( x, ∇ u ( x )) is also a plane(cf. [Y2, p.123]). Therefore, u is a quadratic polynomial. (cid:3) Appendix We include here a uniqueness result for global solutions to the special La-grangian equation (1.1) with bounded Hessian satisfying certain “convexity”constraints (3.1). The constraints are only needed for n ≥ . Theorem 3.1. Let u be a smooth solution to the special Lagrangian equation(1.1). Suppose that the eigenvalues λ i of the Hessian D u ( x ) are bounded | λ i ( x ) | ≤ K and satisfy (3.1) 3 + λ i ( x ) + 2 λ i ( x ) λ j ( x ) ≥ 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 2.3replaced by the following proposition. (cid:3) Proposition 3.1. Let u ∈ C ∞ ( R n \ { } ) be a solution to the special La-grangian equation (1.1) and homogeneous of order , that is u ( x ) = | x | u ( x/ | x | ) . Suppose that the eigenvalues λ i of the Hessian D u ( x ) satisfy (3.1) for all i, j, and x = 0 . Then u must be quadratic.Proof. By (3.1), we certainly have (2.8) with ε = 0 in Lemma 2.1, that is △ g ln V ≥ n X i =1 (cid:0) λ i (cid:1) h iii + X j = i (cid:0) λ j + 2 λ i λ j (cid:1) h jji = n X i =1 (cid:0) λ i (cid:1) u iii + X j = i (cid:16) λ j + 2 λ i λ j (cid:17)(cid:16) λ j (cid:17) (cid:0) λ i (cid:1) u jji ≥ . (3.2)Since u is homogeneous of order 2, the Hessian D u ( x ) is homogeneousof order 0, hence ln V must attain its maximum along a ray. The strongmaximum principle yields that ln V is constant, so in fact(3.3) 0 = △ g ln V. We claim now that(3.4) △ u = const on R n \ { } . At any point p compute the derivative(3.5) ∂ i ( △ u ) = X j u jji for all i. Still assuming that D u is diagonalized at p , an inspection of (3.2),together with (3.3) shows that for all j with u jji = 0,(3.6) 3 + λ j + 2 λ i λ j = 0 . From 3 + λ i + 2 λ i λ j ≥ , we see that λ i ≥ λ j . Solving (3.6) for λ j we get λ j = − λ i − q λ i − , if λ i < λ j = − λ i + q λ i − , if λ i > p then reads0 = △ g u i p = X j 11 + λ j u jji = 11 + (cid:16) − λ i ± q λ i − (cid:17) X j u jji . Hence ∂ i ( △ u ) = 0 and △ u is constant.Differentiating (3.4), we see that each u ij satisfies △ u ij = 0 . 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