Complete Translating Solitons to the Mean Curvature Flow in R3 with Nonnegative Mean Curvature

COMPLETE TRANSLATING SOLITONS TO THE MEAN CURVATURE FLOW IN R3 WITH NONNEGATIVE MEAN CURVATURE

JOEL SPRUCK AND LING XIAO

ABSTRACT. We prove that any complete immersed two-sided mean convex translating soliton Σ ⊂ R3 for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in R3 is the axisymmetric “bowl soliton”. We also show that if the mean curvature of Σ tends to zero at infinity, then Σ can be represented as an entire graph and so is the “bowl soliton”. Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.

1. INTRODUCTION

A complete immersed hypersurface f :Σn → Rn+1 with trivial normal bundle (two- sided for short) is called a translating soliton for the mean curvature flow, with respect to a unit direction en+1, if its mean curvature is given by H =< N, en+1 > where N is a global unit normal field for Σ. Then F (x, t) := f(x) + ten+1 satisfies

Σ ⊥ ⊥ ∆ F = HN =< N, en+1 > N = (en+1) = Ft , thus justifying the terminology. Translating solitons form a special class of eternal solutions for the mean curvature ﬂow that besides having their own intrinsic interest, are models of slow singularity formation. Therefore there has been a great deal of effort in trying to classify them in the case H > 0. In this paper, we shall always assume our translating solitons are mean convex which by abuse of language we take to mean H > 0. The abundance of glueing constructions for translating solitons with high genus and H changing sign (see [19],[20],[21], [9], [7], [23]) suggests a general classiﬁcation is unlikely. π For n = 1 the unique solution is the grim reaper curve Γ: x2 = log sec x1, |x1| < 2 , while for n ≥ 2 we have the one parameter family of convex grim cylinders (see Lemma 5.1 for the n = 2 case) n n x1 X X π (1.1) x = λ2 log sec + α x , α2 = λ2 − 1, |x | < λ, λ ≥ 1 , n+1 λ k k k 1 2 k=2 k=2

Research of the first author is partially supported in part by the NSF. 1 2 JOEL SPRUCK AND LING XIAO which can be obtained from the standard grim cylinders Γ×Rn−1 by a rotation and scaling. The family of grim graphical solitons in R3: x √ (1.2) uλ(x , x ) = λ2 log sec 1 ± Lx ,L = λ2 − 1, λ ≥ 1 1 2 λ 2 λ π defined over the strip S := {(x1, x2): |x1| < R := λ 2 } will play a central role in our classification of mean convex graphical translating solitons in R3. m+1 Similarly if xm+1 = v(x1, . . . , xm) is a graphical translating soliton in R , then for 0 x = (x1, . . . , xn) = (x , xm+1, . . . , xn), n n x0 X X (1.3) vλ(x) := λ2v( ) + α x , α2 = λ2 − 1, λ ≥ 1 , λ k k k k=m+1 k=m+1 is a graphical translating soliton in Rn+1 = Rm × Rn−m × R. There is as well a unique (up to horizontal translation) axisymmetric solution called the “bowl soliton” [1], [4] which 1 2 has the asymptotic expansion as an entire graph u(x) = 2(n−1 )|x| − log |x| + O(1). White [30] tentatively conjectured that all convex translating solutions have the form

j n−j {(x, y, z) ∈ R × R × R : z = f(|x|)} for j ≥ 2; for j = 1, f defines the grim reaper curves so is defined on an interval. White remarks that even if this conjecture is false, it may be true for blow up limits of mean convex mean curvature flows. In [28] Wang proved that in dimension n = 2, any entire convex graphical translating soliton must be rotationally symmetric, and hence the bowl soliton. He also showed there exist entire locally strictly convex graphical translating solitons for dimensions n > 2 that are not rotationally symmetric (thus disproving one conjecture of White [30]) as well as complete locally strictly convex graphical translating solitons defined over strip regions in Rn. Wang also conjectured that for n = 2, any entire graphical translating soliton must be locally strictly convex and a similar statement in dimension n > 2 under the additional assumption that H tends to zero at infinity. More recently, Haslhofer [12] proved the uniqueness of the bowl soliton in arbitrary dimension under the assumption that the translating soliton Σ is α-noncollapsed and uniformly 2-convex. The α-noncollapsed condition means that for each P ∈ Σ, there are ± α + − closed balls B disjoint from Σ−P of radius at least H(P ) with B ∩B = {P }. It figures prominently in the regularity theory for mean convex mean curvature flow [29], [30], [15],

[25]. The 2-convex condition (automatic if n = 2) means that if κn ≤ κn−1 ≤ . . . κ1 are the ordered principal curvatures of Σ, then κn + κn−1 ≥ βH for some uniform β > 0. The α-noncollapsed condition is a deep and powerful property of weak solutions of the mean COMPLETE TRANSLATING SOLUTIONS 3 convex mean curvature ﬂow [30],[13] which implies that any complete (α-noncollapsed) mean convex translating soliton Σ is convex with uniformly bounded second fundamental form. The main result of this paper is a proof of a more general form of the n = 2 conjecture of Wang.

Theorem 1.1. Let Σ ⊂ R3 be a complete immersed two-sided translating soliton for the mean curvature ﬂow with nonnegative mean curvature. Then Σ is convex. By Sacksteder’s theorem [23], the condition H > 0 and Corollary 2.1 of [28], we may conclude

Corollary 1.2. Σ is the boundary of a convex region in R3 whose projection on the plane spanned by e1, e2 is (after rotation of coordinates) either a strip region {(x1, x2): |x1| < 2 R} or R . Moreover Σ is the graph of a function u(x1, x2) which satisﬁes the equation ! ∇u 1 (1.4) div = , p1 + |∇u|2 p1 + |∇u|2 or in nondivergence form, 2 2 2 2 (1.5) (1 + ux2 )ux1x1 − 2ux1 ux2 ux1x2 + (1 + ux1 )ux2x2 = 1 + ux1 + ux2 . Combining Theorem 1.1 with Theorem 1.1 in [28] we have

Corollary 1.3. Any entire solution in R2 to the equation ! ∇u 1 div = p1 + |∇u|2 p1 + |∇u|2 must be rotationally symmetric in an appropriate coordinate system and hence is the bowl soliton.

A necessary and sufﬁcient condition for translating solitons to be graphical over R2 and thus the bowl soliton is given in the next theorem.

Theorem 1.4. Let Σ ⊂ R3 be a complete immersed two-sided translating soliton for the mean curvature flow with nonnegative mean curvature and suppose that H(P ) → 0 as P ∈ Σ tends to infinity. Then Σ is after translation the axisymmetric bowl soliton. The existence of the grim family uλ (see 1.2 ), the convexity Theorem 1.1 and a global curvature bound (see Theorem 2.8 in section 2) are the key tools that allows us to classify locally strictly convex graphical translating solitons defined over strips. 4 JOEL SPRUCK AND LING XIAO

Theorem 1.5. Let Σ = graph(u) be a complete locally strictly convex translating soliton λ π deﬁned over a strip region S := {(x1, x2): |x1| < R := λ 2 }. Then (after possibly relabeling the e2 direction) i. For λ ≤ 1 there is no locally strictly convex solution in Sλ. √ 2 ii. limx2→+∞ ux2 (x1, x2) = L := λ − 1, λ > 1. iii. limx2→−∞ ux2 (x1, x2) = −L. iv. λ lim (u(x1, x2 + A) − u(0,A)) = u (x1, x2) A→±∞ (1.6) x1 lim ux (x1,A) = λ tan . A→±∞ 1 λ

The above limits are uniform in |x1| ≤ R − ε for any ε > 0. λ v. For P = (x1, x2, u(x1, x2)) ∈ Σ, (x1, x2) ∈ S ,

(1.7) H(P ) ≤ R − |x1|,H(P ) ≥ θ(ε) if |x1| ≤ R − ε .

vi. u(x1, x2) = u(−x1, x2) and ux1 (x1, x2) > 0 for x1 > 0. vii. For any λ > 1 there exists a complete locally strictly convex translating soliton Σ = λ graph(u) deﬁned over S . Moreover, u(x1, x2) = u(−x1, x2), u(x1, x2) = u(x1, −x2).

Remark 1.6. At the moment it is not clear if the solution obtained in Theorem 1.5 part vii. is unique.

The organization of the paper is as follows. In section 2 we show that any immersed two-sided mean convex translating soliton is stable. This allows us to use the method of Choi-Schoen [3] as modiﬁed by Colding-Minicozzi [5], [6] to prove a global curvature bound (Theorem 2.8). This is needed for the proof of Theorem 1.1 in section 3, which is based on a delicate maximum principle argument. In section 4 we give the proof of Theorem 1.4. This result also allows us to start the proof of Theorem 1.5 which is long and detailed. The proof of parts i.-vi. is contained in section 5 and the proof of existence part vii. by a direct construction is contained in section 6.

2. STABILITY AND CURVATURE ESTIMATES. We will need the the following well-known identities that hold on any translating soliton in Rn+1 (see for example [18]). COMPLETE TRANSLATING SOLUTIONS 5

Lemma 2.1. Let Σ be a two-sided immersed translating soliton in Rn+1 with second fundamental form A. Let A = (hij) be the second fundamental form of Σ, u = xn+1 Σ and f Σ ∆ := ∆ + < ∇, en+1 > be the drift Laplacian on Σ. Then i. |∇u|2 = 1 − H2, ∆Σu = H2 ii. ∆f A + |A|2A = 0, iii. ∆f H + H|A|2 = 0, iv. ∆f (|A|)2 − 2|∇A|2 + 2|A|4 = 0.

n+1 It is well known that a translating soliton Σ with respect to the direction en+1 in R is a critical point of the weighted area functional Z A˜(Σ) = exn+1 dv Σ and is in fact minimal with repect to the weighted Euclidean metric exn+1 δ on Rn+1. The second variation of A˜ with respect to a compactly supported normal variation ηN is easily computed to be Z Z A˜00(0) = (|∇η|2 − |A|2η2)exn+1 dv = −ηLη exn+1 dv , Σ Σ where Lη = e−xn+1 divΣ(exn+1 ∇η) + |A|2η is the associated stability operator.

Proposition 2.2. i. Let Σ ⊂ Rn+1 be a complete immersed two-sided translating soliton with respect to en+1 with H ≥ 0. Then Σ is strictly stable, that is, λ1(−L)(D) > 0 on any compact subdomain D ⊂ Σ. ii. The following are equivalent: a. There exists a positive solution v of Lv = 0 for every bounded D. b. λ1(−L)(D) ≥ 0 for every bounded D. c. λ1(−L)(D) > 0 for every bounded D .

Proof. i. We may assume that H > 0. Then w = log H satisﬁes ∆w+ < ∇w, en+1 > +|A|2 = −|∇w|2. Hence Z Z (η2|A|2 − |∇η|2)exn+1 dv = − (η2|∇w|2 − 2η < ∇η, ∇w > +|∇η|2)exn+1 dv Σ Σ Z = − |η∇w − ∇η|2exn+1 dv < 0. Σ ii. The proof is a straightforward modiﬁcation of that of Fischer-Colbrie and Schoen [8] and will not be given. 6 JOEL SPRUCK AND LING XIAO

We will need the following corollary for the case at hand n = 2.

Corollary 2.3. Let Bρ(P ) be an intrinsic ball in Σ with ρ < 2π. Then Bρ(P ) is disjoint from the conjugate locus of P and Z Z (2.1) f 2|A|2dv ≤ e2ρ |∇f|2 dv . Σ Σ

1 for f ∈ H0 (Bρ(P )).

H2 1 Proof. Since K ≤ 4 ≤ 4 , the ﬁrst statement follows from standard comparison geometry. 2 2 Also |∇x3| = 1 − H ≤ 1, so |x3(Q) − x3(P )| ≤ ρ for any Q ∈ Bρ(P ). Hence −ρ x x ρ x e e 3 (P ) ≤ e 3 (Q) ≤ e e 3 (P ) and the result follows from Proposition 2.2.

We now follow the Colding-Minicozzi method [5, 6] with appropriate modiﬁcation, to prove intrinsic area bounds and then curvature bounds. For two dimensional graphs, such curvature estimates follow immediately from the work of Leon Simon [27], see also [24].

Proposition 2.4. Let Σ ⊂ R3 be a two-sided immersed translating soliton with H ≥ 0 and let Bρ(P ) be a topological disk in Σ. Then Bρ(P ) is disjoint from the cut locus of P for e2ρ < 2 and A(B (P )) i. ρ ≤ 2π , ρ2 (2.2) Z 1 1 ii. |A|2 dv ≤ 2π{(log )−2 + 2(log )−1} for µ ∈ (0, 1). µ µ Bµ2ρ(P )

Proof. We first prove Bρ(P ) is disjoint from the cut locus C(P ) of P, that is the injectivity radius of P satisfies r0 := injp(Σ) > ρ. Suppose for contradiction that Q ∈ ∂Br0 (P ) is a cut locus of P and r0 ≤ ρ . We know by Corollary 1.2 that Q is not in the conjugate locus of P so by Klingenberg’s lemma (see for example Lemma 5.7.12 of [22]) there are two mininimizing geodesics from P to Q which fit together smoothly at Q with a possible corner at P and bounding a domain D ⊂ Br0 (P ). By Gauss-Bonnet, 1 Z Z A(D) ≥ K dv = 2π − κg dσ ≥ π , 4 D ∂D hence A(B (P )) > A(D) ≥ 4π. On the other hand by (2.2) with ρ = r (proved below), r0 √ 0 2 1 A(Br0 (P )) < 2πr0. Therefore r0 > 2, contradicitng r0 ≤ ρ < 2 log 2. COMPLETE TRANSLATING SOLUTIONS 7

We now prove the stated inequalities. Let l(s) be the length of ∂Bs(P ) and K(s) = R K dv. By Gauss-Bonnet, Bs(P ) Z 0 (2.3) l (s) = κg dσ = 2π − K(s) . ∂Bs(P )

For r = d(P, x), f(x) = η(r), η, −η0 ≥ 0, η(ρ) = 0, we use the stability inequality (2.1) and write

|A|2 = H2 − 2K ≥ −2K, to obtain (for ρ ≤ r0) Z Z (2.4) −2 Kf 2 ≤ 2 |∇f|2 Bρ(P ) Bρ(P )

By the coarea formula,

Z Z ρ Z Z ρ Kf 2 = η2(s) Kdσds = η2(s)K0(s)ds Bρ(P ) 0 ∂Bs(P ) 0 (2.5) Z Z ρ Z Z ρ |∇f|2 = |∇f|2dσds = (η0)2(s)l(s)ds Bρ(P ) 0 ∂Bs(P ) 0

r For η(r) = 1 − ρ , using (2.3)-(2.5) this gives

Z ρ 4 0 s A(Bρ(P )) (2.6) − (2π − l (s))(1 − )ds ≤ 2 2 ρ 0 ρ ρ By integration by parts,

Z ρ s A(B (P )) (2.7) (2π − l0(s))(1 − )ds = πρ − ρ 0 ρ ρ Finally combining (2.6), (2.7) and simplifying gives inequality i. in (2.2) For part ii. we use the logarithmic cutoff

 1 if s ≤ µ2ρ   log s (2.8) η(s) = ρ − 1 if µ2ρ < s < µρ log µ   0 if s > µρ 8 JOEL SPRUCK AND LING XIAO in (2.1). Then Z Z µρ 2 2 l(s) |A| ≤ 2 2 ds (log µ) 2 s Bµ2ρ(P ) µ ρ µρ 2 A(B (P )) Z µρ A(B (P )) (2.9) ≤ s + 2 s ds 2 2 3 (log µ) s µ2ρ s µ2ρ 1 2 ≤ 4π{ 2 + 1 } (log µ) log µ by (2.2) part i. For later use we will need two well known lemmas; the ﬁrst is an extrinsic mean value inequality (for a proof see [6] p. 26-27) and the second one says that curvature bounds implies graphical with intrinsic and extrinsic balls related (see [6] Lemma 2.4).

3 Lemma 2.5. Let Σ ⊂ R be an embedded surface with x0 ∈ Σ,Bs(x0)∩∂Σ = ∅. Suppose the mean curvature of Σ satisﬁes |H| ≤ C and f ≥ 0 is a smooth function on Σ satisfying ∆Σf ≥ −λs−2f. Then ( λ +Cs) e 4 Z f(x0) ≤ 2 fdv . πs Bs(x0)∩Σ

Proposition 2.7. (Choi-Schoen type curvature bound) Let Σ ⊂ R3 be a two-sided immersed translating soliton and let B (P ) be disjoint from the cut locus of P. Then there √ ρ ε R 2 exists ε, τ < < ρ such that if for all r0 ≤ τ, there holds |A| ≤ ε, then for all 2π Br0 (P ) 2 −2 0 < σ ≤ r0, y ∈ Br0−σ(P ) we have |A| (y) ≤ σ .

2 2 Proof. Deﬁne F (x) = (r0 −r(x)) |A| (x) on Br0 (P ) and suppose F assumes its maximum 2 2 2 2 at x0. Note that if F (x0) ≤ 1, then σ |A| (y) ≤ (r0 − r(y)) |A| (y) ≤ F (x0) ≤ 1 and we 2 2 are done. If not, deﬁne σ by 4σ |A| (x0) = 1. Then by the triangle inequality on Bσ(x0), 1 r − r(x) ≤ 0 ≤ 2 , 2 r0 − r(x0) COMPLETE TRANSLATING SOLUTIONS 9

2 2 −2 which implies supBσ(x0) |A| ≤ 4|A| (x0) = σ . On Σ we have the Simons’ type equation 2 2 4 2 1 2 2 1 L(|A| ) − 2|∇A| + 2|A| = 0 which implies ∆(|A| + 2 ) ≥ −2|A| (|A| + 2 ). Hence 2 1 −2 σ for f(x) := |A| + 2 , ∆f ≥ −2σ f on Bσ(x0). Using Lemmas 2.6 2.5 with s = 4 , λ = 1 6 ,C = 1 and Proposition 2.4 part i. (a = 0), we ﬁnd

1 1 2 1 16 ( 1 + σ ) 2 (2.10) ( + ) = |A| (x ) + ≤ e 24 4 {ε + 2πr } . 4σ2 2 0 2 πσ2 0 Multiplying (2.10) by σ2 we ﬁnd 2 1 1 σ 16 ( 1 + σ ) 2 32e (2.11) < + ≤ e 24 4 {ε + 2πr } < ε 4 4 2 π 0 π π which is impossible for ε ≤ 128e . We are now in a position to prove the curvature estimate we will need in the next section.

Theorem 2.8. Let Σ ⊂ R3 be a complete immersed two-sided translating soliton with 2 respect to e3 with H ≥ 0. Then there is a universal constant C such that |A| (P ) ≤ C for P ∈ Σ.

Proof. For any P ∈ Σ, we ﬁx ρ > 0 such that eρ < 2 as in Proposition 2.4 so that

Bρ(P ) is disjoint from the cut locus of P. We may assume Bρ(P ) is a topological disk since by Proposition 2.2, the universal cover of Bρ(P ) endowed with pull-back metric is also a stable translating soliton with nonnegative mean curvature. Thus using Proposition 2.4 part − 6π 2 ii. with µ = e ε , the conditions of Proposition 2.7 are satisﬁed for τ = µ ρ. We can 2 −2 choose σ = τ and obtain |A| (P ) ≤ τ .

3. PROOFOF THEOREM 1.1. We restate for the readers convenience our main result.

Theorem 3.1. Let Σ ⊂ R3 be a complete immersed two-sided translating soliton for the mean curvature ﬂow with nonnegative mean curvature. Then Σ is convex.

Proof. Without loss of generality, we assume Σ satisﬁes

H = hν, e3i > 0. Let " #1/2 x + x x − x 2 (3.1) f(x , x ) = 1 2 + 1 2 , 1 2 2 2 10 JOEL SPRUCK AND LING XIAO

( 2 r4e−1/r if r < 0 (3.2) φ(r) = 0 if r ≥ 0, and 2 X zi (3.3) g(z) = f(z) φ . f(z) i=1

It’s easy to see that g is smooth when z1 6= z2. Now denote

G(A) = g(κ(A)) and F (A) = f(κ(A)), where A is a 2 × 2 symmetric matrix and κ(A) are the eigenvalues of A. Now let A be the second fundamental form of Σ. Then G/F ≤ 1 and therefore G/F achieves its maximum either at an interior point or “at inﬁnity”. By a straight forward calculation, we have

G ∇F G G ∆Σ + 2 , ∇ + ∇ , e F F F F 3 (3.4) Gijh |A|2 GF ijh |A|2 Gpq,rs GF pq,rs = − ij + ij + − h h . F F 2 F F 2 pqk rsk

We next compute the last term in (3.4).

2 1 pq,rs pq g − g 2 (3.5) G hpqkhrsk = g hppkhqqk + 2 h12k, κ2 − κ1 where 2 X zi zi zi gpq = f pq φ − φ˙ f f f (3.6) i=1 2 1 X zi zi zi + φ¨ δp − f p δq − f q . f f i f i f i=1 It follows that FGpq,rs − GF pq,rs g2 − g1 f 2 − f 1 (3.7) = (fgpq − gf pq) + 2 f − g κ2 − κ1 κ2 − κ1 = I + II. COMPLETE TRANSLATING SOLUTIONS 11

We proceed to calculate the terms I and II.

" 2 # 2 X zi X zi zi zi I = f pq g − z φ˙ + φ¨ δp − f p δq − f q − gf pq i f f i f i f (3.8) i=1 i=1 2 2 X zi X zi zi zi = −f pq z φ˙ + φ¨ δp − f p δq − f q , i f f i f i f i=1 i=1

2 z2 X zi zi zi g2 − g1 = φ˙ + f 2 φ − φ˙ f f f f (3.9) i=1 2 z1 X zi zi zi − φ˙ − f 1 φ − φ f f f f i=1 and κ − κ 2 1 II = f(g2 − g1) − g(f 2 − f 1) 2 " 2 # z2 z1 X zi = f φ˙ − φ˙ + f 2 g − z φ˙ f f i f i=1 (3.10) " 2 # X zi − f 1 g − z φ˙ − g f 2 − f 1 i f i=1 2 z2 z1 X zi = f φ˙ − φ˙ + z φ˙ (f 1 − f 2). f f i f i=1

Assume κ1 > 0 > κ2; then (3.11) G ∇F G G ∆Σ + 2 , ∇ + ∇ , e F F F F 3 Gpq,rs GF pq,rs 1 = ( − )h h = (FGpq,rs − GF pq,rs)h h F F 2 pqk rsk F 2 pqk rsk 1 ¨ κ2 p κ2 p q κ2 q 2 ˙ κ2 κ1 + κ2 2 = 2 φ δ2 − f δ2 − f hppkhqqk + 2 φ h12k ≥ 0. κ1 κ1 κ1 κ1 κ1 κ1 κ2 − κ1 pq ˙ κ1 2 Here we used f = 0, φ f = 0, and f = 0 when κ1 > 0 > κ2. It follows that if G/F achieves its maximum at an interior point, then by the strong maximum principle we have 12 JOEL SPRUCK AND LING XIAO

G |A|2 F ≡ constant. If κ2/κ1 6= 0, then ϕ := log H2 is constant and so ∇|A|2 ∇H 0 = ∇ϕ = − 2 |A|2 H Σ 0 = ∆ ϕ + h∇ϕ, e3i (3.12) |∇A|2 |∇|A|2|2 |∇H|2 = {2 − + 2 } |A|2 |A|4 H2 2 = {|∇A|2 − |∇|A||2} |A|2 2 2 Hence |∇A| = |∇|A|| and so h12,k = 0, k = 1, 2. By the Codazzi equations, h11,2 = h22,1 = 0. Since h22 = r0h11, we deduce ∇A = 0. Thus, M is a complete surface with constant mean curvature. Since M satisfies H = hν, e3i, we conclude that M is a plane, a G contradiction. Therefore in this case we must have F ≡ 0, and thus κ2 ≥ 0. If G/F achieves its maximum at infinity, by Theorem 2.8 we may apply the Omori-Yau maximum principle and conclude that there exists a sequence Pn tending to infinity with rn := κ2/κ1(Pn) → r0, where −1 ≤ r0 < 0. Moreover we have at Pn: (3.13) 2 1 ¨ rnh11k h22k 2 ˙ 1 + rn 2 ∇F G ≥ φ(rn) − + 2 φ(rn) h12k − 2 , ∇ n κ1 κ1 κ1 rn − 1 F F 2 ¨ rnh11k h22k 2 ˙ 1 + rn 2 ˙ h11k h22k h11k = φ(rn) − + 2 φ(rn) h12k − 2φ(rn) , − rn κ1 κ1 κ1 rn − 1 h11 h11 h11 and

h22k h11k (3.14) Cn,k := − rn → 0 as n → ∞. h11 h11 Note that ∇ H −κ hτ , e i C˜ := k = k k 3 n,k κ κ (3.15) 1 1 h22k h11k h11k h11k = − rn + (1 + rn) = Cn,k + (1 + rn) h11 h11 h11 h11 From (3.14) we have,

h22k h11k (3.16) = −|rn| + Cn,k, k = 1, 2 κ1 κ1 and 2 h11kh22k h11k h11k (3.17) 2 = −|rn| + Cn,k . κ1 κ1 κ1 COMPLETE TRANSLATING SOLUTIONS 13

h11k Claim: (1 + rn) → 0 k = 1, 2. h11 If not, we can choose a subsequence, still denoted by {Pn}, so that for n ≥ N

h112 h221 (3.18) (1 + rn) (Pn) ≥ ε0, (1 + rn) (Pn) ≥ ε0. κ1 κ1 Then from (3.13) we have ˙ 2 2 1 ¨X 2 ˙ X h11k 2φ h112 h221 ≥ φ Cn,k − 2φ Cn,k − (1 + rn) 2 + 2 n κ1 1 + |rn| κ1 κ1 ¨X 2 ˙ 1 h221 Cn,1 = φ Cn,k − 2φ Cn,1 − + |rn| κ1 |rn| 2 2 h112 1 + rn h112 h221 +Cn,2 + 2 + 2 κ1 1 + |rn| κ1 κ1 C2 ¨X 2 ˙ n,1 = φ Cn,k − 2φ |rn| (3.19) 2 2 ˙ 1 + rn h112 h112 1 + rn h221 Cn,1 h221 − 2φ 2 + Cn,2 + 2 − 1 + |rn| κ1 κ1 1 + |rn| κ1 |rn| κ1 ˙ 2φ h112 h112 ≥ − (1 + rn) − Cn,2(1 + |rn|) 1 + |rn| κ1 κ1 h221 h221 Cn,1 + (1 + rn) − (1 + |rn|) κ1 κ1 |rn| ˙ φ ε0 h112 h221 ≥ −2 + , 1 + |rn| 2 κ1 κ1 which leads to a contradiction for n ≥ N by (3.18). Thus the claim is proven and so ˜ Cn,k → 0. Therefore ν(Pn) converges to e3 and so H(Pn) → 1.

Now let Σn = Σ − Pn be the surface obtained from Σ by translation of Pn to the origin.

Since Σ has bounded principle curvatures, so do the Σn. Choosing a subsequence which we still denote by Σn, the Σn converge smoothly to Σ∞. Thus Σ∞ is a translating soliton which satisﬁes H = hν, e3i , and H(0) = 1. Moreover, we have κ κ inf 2 = 2 (0). x∈Σ∞ κ1 κ1

As before we conclude that G/F = constant, and Σ∞ has constant mean curvature one, which is impossible. Therefore Σ is convex. 14 JOEL SPRUCK AND LING XIAO

4. PROOFOF THEOREM 1.4. In this section we give the proof of

Theorem 4.1. Let Σ ⊂ R3 be a complete immersed two-sided translating soliton for the mean curvature ﬂow with positive mean curvature and suppose that H(P ) → 0 as P ∈ Σ tends to inﬁnity. Then Σ is after translation the axisymmetric bowl soliton.

Proof. Suppose for contradiction that Σ = graph(u) projects onto the strip |x1| < R. A Consider for any A, the convex curve γ (x1) = (x1, A, u(x1,A): −R ≤ x1 ≤ R).

Since u(x1,A) → +∞ as x1 → ±R, there is a smallest value x1 = x1(A) so that ux1 (x1(A),A) = 0. Since Σ is not a grim cylinder we may assume by a suitable change of coordinates that ux2 (x1(0), 0) ≥ δ > 0. We normalize u(x1(0), 0) = 0 ; then by convexity of u,

u(x1(A),A) ≥ u(x1(0), 0) + Aux2 (x1(0), 0) ≥ Aδ, (4.1) 0 = u(x1(0), 0) ≥ u(x1(A),A) − Aux2 (x1(A),A) ≥ A(δ − ux2 (x1(A),A)), which implies ux2 (x1(A),A) ≥ δ. By the assumption that H(P ) → 0 as P ∈ Σ tends to inﬁnity, we have that |∇u(x1(A),A)| = |ux2 (x1(A),A)| = ux2 (x1(A),A) → ∞ as A → ∞. Therefore

(4.2) u(x1, x2) ≥ u(x1(A),A) + (x2 − A)ux2 (x1(A),A) ≥ Aδ + (x2 − A)ux2 (x1(A),A), B and so choosing x2 = B,A = 2 ,

u(x1,B) δ 1 B B (4.3) lim ≥ + lim ux (x1( ), ) = +∞ . B→∞ B 2 2 B→∞ 2 2 2

Hence u grows superlinearly as x2 → ∞.

We now compare Σ with a tilted cylinder of radius R. Consider the parametrized family of graphs √ q t 2 2 2 x3 = v (x1, x2) := − 1 + t R − x1 + t(x2 − A), |x1| ≤ R, x2 ≥ A 1 of constant mean curvature H = R with respect to upward normal direction. Since t t v (x1, x2) ≤ t(x2 − A), for any choice of t ≥ 0, u > v for x2 sufﬁciently large by (4.3). Also, u > vt for |x | = R and for x = A. For t ≤ δ, u > vt in x ≥ A, |x | ≤ R 1 2 √ 2 1 t 2 by (4.2). Note that limt→∞ v (x1, 3A) ≥ limt→∞(2tA − 1 + t R) = +∞ for A > R. We can therefore increase t until there is a ﬁrst contact of Σ = graph(u) and graph(vt), which must occur over an interior point of the half-strip {(x1, x2), |x1| < R, x2 > A}. COMPLETE TRANSLATING SOLUTIONS 15

1 This gives a P := (x1, x2, u(x1, x2)) ∈ Σ, x2 > A with H(P ) ≥ R . Since A is arbitrary we have a contradiction.

5. ASYMPTOTIC BEHAVIOR OF COMPLETE LOCALLY STRICTLY CONVEX TRANSLATING SOLITONS

In this section we study the asymptotic behavior of complete locally strictly convex translating solitons.

Lemma 5.1. Let Σ = graph(u) be a complete mean convex translating soliton in R3 and suppose that the smallest principle curvature κ2 vanishes at a point of Σ. Then κ2 ≡ 0 everywhere and after translation, Σ is grim cylinder of the form Σ = graph(uλ) deﬁned in a strip {(x1, x2): |x1| < R} where x √ π uλ(x , x ) := λ2 log sec ( 1 ) ± λ2 − 1 x ,R = λ, λ ≥ 1 . 1 2 λ 2 2 In particular if Σ contains a line, then Σ is a grim cylinder of the above form.

Proof. If we choose an orthonormal frame τ1, τ2 so that κ2(P ) = h22(P ), then κ2 ≡ 0 < κ1 on Σ by Lemma 2.1 part ii. and the maximum principle. Thus the Gauss curvature KΣ ≡ 0 and so Σ has a representation (see for example [11]) Σ = graph(z) where z(x1, x2) =

η(x1) + αx2 for a constant α and a scalar function η deﬁned in a simply connected region containing the projection of P on the x1, x2 plane. Therefore (5.1) (1 + α2)η00 = 1 + α2 + η02

−2 2 2 Set η˜(x1) = λ η(λx1), where λ = 1 + α . Then (5.2) η˜00 = 1 +η ˜02 and so (up to translation of coordinates and an additive constant)

(5.3) η˜(x1) = log sec x1 , which proves the lemma. We have the following Harnack inequality for the mean curvature H on Σ (compare Hamilton [10], Corollary 1.2).

Lemma 5.2. For any two points P1,P2 ∈ Σ,

Σ −d (P1,P2) (5.4) H(P2) ≥ e H(P1) . 16 JOEL SPRUCK AND LING XIAO

Proof. Since H =< N, e3 >, ∇kH = −κk < τk, e3 > so that |∇H|2 ≤ |A|2 = H2 − 2K ≤ H2 .

Therefore, |∇ log H| ≤ 1 and (5.4) follows.

Lemma 5.3. Let Σ = graph(u) be a complete mean convex translating soliton in R3 deﬁned in the strip {(x1, x2): |x1| < R}. Then

H(x1, x2) := H((x1, x2, u(x1, x2)) ≤ R − |x1| .

P 2 uij 2 2 1 2 2 2 Proof. We use that W 6 ≤ |A| ≤ H = W 2 where W = 1 + |∇u| . Then |DW | ≤ W 1 or |DH| = |D( W )| ≤ 1. By the convexity of u and the fact that u(x1, x2) → ∞ as R − |x1| → 0, x2 ﬁxed, we see that H(x1, x2) → 0 as R − |x1| → 0, x2 ﬁxed. Hence H(x1, x2) ≤ min (R − x1, x1 + R) = R − |x1|.

Lemma 5.4. Let Σ = graph(u) be a complete mean convex translating soliton in R3 deﬁned over R2. Then H(P ) → 0 for P ∈ Σ tending to inﬁnity.

Proof. We slightly modify the argument of Haslhofer [12]. Fix P0 ∈ Σ and suppose there is a sequence Pn ∈ Σ tending to inﬁnity with lim infn→∞ H(Pn) > 0. Passing to a subse- Pn−P0 quence, we may assume converges to a unit direction ω. Let Σn = Σ − Pn be the |Pn−P0| surface obtained from Σ by translation of Pn to the origin. Since Σ has bounded principle curvatures, so do the Σn. Choosing a subsequence which we still denote by Σn, the Σn converge smoothly to Σ∞, a convex translating soliton. Moreover, Σ∞ is an entire graph Pn−P0 since the Σn are all entire graphs. Since the region K above Σ is convex and → ω, |Pn−P0| the limit Σ∞ contains a line. Therefore by Lemma 5.1, Σ∞ is a grim cylinder and thus is a graph over a strip, a contradiction. Lemma 5.5. Let Σ = graph(u) be a complete locally strictly convex translating soliton λ π deﬁned over a strip region S := {(x1, x2): |x1| < R := λ 2 }. Then there exist sequences n n n n n n n n n n Pn = (x1 , x2 , u(x1 , x2 )), P n = (x1 , x2 , u(x1 , x2 )) ∈ Σ with x2 → ∞, x2 → −∞ and H(Pn),H(P n) ≥ θ > 0.

n n n n Proof. By Theorem 1.4 there is a sequence of points Pn = (x1 , x2 , u(x1 , x2 )) ∈ Σ n where (after relabeling axes if necessary) lim infn→∞ H(Pn) ≥ θ > 0 and x2 → +∞. n Note that by Lemma 5.3, |x1 | ≤ R − θ. We claim there is also a sequence P n = n n n n n (x1 , x2 , u(x1 , x2 )) ∈ Σ with x2 → −∞ and H(P n) ≥ θ > 0. If not, then for x2 → −∞,H(x1, x2) := H(x1, x2, u(x1, x2)) → 0. For any B ≤ 0 let x1(B) be the smallest COMPLETE TRANSLATING SOLUTIONS 17

value so that ux1 (x1(B),B) = 0. Since Σ is not a grim cylinder, we may assume by a translation of coordinates that ux2 (x1(0), 0) = δ > 0. We normalize u(x1(0), 0) = 0; then

u(x1(B),B) ≥ u(x1(0), 0) + Bux2 (x1(0), 0) = Bδ , and

0 = u(x1(0), 0) ≥ u(x1(B),B) − Bux2 (x1(B),B) ≥ B(δ − ux2 (x1(B),B)) .

Hence ux2 (x1(B),B)) ≤ δ. Since H(x1, x2) → 0 as x2 → −∞, we conclude

(5.5) ux2 (x1(B),B)) → −∞ as B → −∞. Therefore

(5.6) u(x1, x2) ≥ u(x1(B),B)+(x2 −B)ux2 (x1(B),B) ≥ Bδ +(x2 −B)ux2 (x1(B),B) Λ Now choose x2 = Λ < 0 and B = 2 . Then δΛ Λ Λ Λ u(x , Λ) ≥ + u (x ( ), ) , 1 2 2 x2 1 2 2 hence

u(x1, Λ) δ 1 Λ Λ (5.7) lim ≤ + ux (x1( ), ) → −∞ Λ→−∞ Λ 2 2 2 2 2 by (5.5). Thus u(x , Λ) → ∞ superlinearly as Λ → −∞. We now compare Σ with a tilted 1 √ t 2p 2 2 cylinder of radius R. Consider x3 = v (x1, x2) := − 1 + t R − x1 + t(x2 − B) in the B t half-strip S := {(x1, x2): |x1| ≤ R, x2 ≤ B < 0}. Since v ≤ t(x2 − B), for any choice t t of t ≤ 0, u > v for x2 sufficiently small (i.e. x2 large negative) by (5.7). Also u > v for t B |x1| = R and x2 = B < 0, as soon as u(x1,B) > 0. For t ≥ −δ, u > v in S and since √ t 2 lim v (x1, 3B) ≥ lim (2Bt − 1 + t R) → ∞ , t→−∞ t→−∞ we can decrease t until there is a first contact point P ∈ SB of graph(vt) and Σ. At 1 P ∈ Σ,H(P ) ≥ R , a contradiction. We now analyze the asymptotic behavior of complete locally strictly convex translating solitons Σ = graph(u) in R3 that are defined over the strip region Sλ. Theorem 5.6. Let Σ = graph(u) be a complete locally strictly convex translating soliton λ π defined over a strip region S := {(x1, x2): |x1| < R := λ 2 , λ ≥ 1}. Then (after possibly relabeling the e2 direction) i. For λ ≤ 1 there is no locally strictly convex solution in Sλ. 18 JOEL SPRUCK AND LING XIAO √ 2 ii. limx2→+∞ ux2 (x1, x2) = L := λ − 1, λ > 1. iii. limx2→−∞ ux2 (x1, x2) = −L. iv. λ lim (u(x1, x2 + A) − u(0,A)) = u (x1, x2) A→±∞ (5.8) x1 lim ux (x1,A) = λ tan . A→±∞ 1 λ ε λ The above limits hold uniformly on S = {(x1, x2) ∈ S : |x1| ≤ R − ε}. ε v. For P = (x1, x2, u(x1, x2)) ∈ Σ, (x1, x2) ∈ S , (5.9) H(P ) ≥ θ(ε) .

Proof. We will prove parts i,ii; the argument for iii. is the same by Lemma 5.5. By

Lemma 5.5 there is a sequence of points Pn ∈ Σ where (after relabeling axes if neces- n n sary) lim infn→∞ H(Pn) ≥ θ > 0 and x2 → +∞. Note that by Lemma 5.3, |x1 | ≤ R − θ. Arguing as in lemma 5.4, we find as above Σn converge smoothly to Σ∞, a convex trans- λ ∞ lating soliton defined over the strip S − (x1 , 0) passing through the origin. By Lemma n n n n 5.1, λ > 1 and u(y1 + x1 , y2 + x2 ) − u(x1 , x2 ) converges smoothly to y + x∞ √ x∞ λ2 log sec ( 1 1 ) + λ2 − 1 y − λ2 log sec 1 λ 2 λ √ as n → ∞. In particular u (y + xn, y + xn) converges smoothly to λ2 − 1 as n → ∞. x2 1 1 2 2 √ ∞ 2 Replacing y1 by x1 − x1 we find lim supx2→∞ ux2 (x1, x2) ≥ λ − 1. On the other hand, n ux2 (x1, x2) ≤ ux2 (x1, x2 + x2 ) for n large so parts i,ii iii. are proven.

We will prove iv. for x2 → +∞ as the argument in the other case is identical. Note that by part ii, for x2 = A, x1 = x1(A) (recall ux1 (x1(A),A) = 0), H((x1(A), A, u(x1(A),A)) ≥ √ 1 . Hence we may choose xn → ∞ arbitrary and xn = x (xn). Arguing as in part ii., 1+L2 2 1 1 2 we have ∞ n n n n λ ∞ 2 x1 i. lim (u(y1 + x1 , y2 + x2 ) − u(x1 , x2 )) = u (y1 + x1 , y2) − λ log sec , n→∞ λ (5.10) ∞ n n (y1 + x1 ) ii. lim ux1 (y1 + x1 , y2 + x2 ) = λ tan . n→∞ λ ∞ ∞ in |y1 + x1 | < R. Setting y1 = y2 = 0 in (5.10) ii., we conclude x1 = 0. We claim that n n λ i. lim (u(x1, x2 + x2 ) − u(0, x2 )) = u (x1, x2) , n→∞ (5.11) n x1 ii. lim ux1 (x1, x2 + x2 ) = λ tan n→∞ λ COMPLETE TRANSLATING SOLUTIONS 19 in S. Since |D(1/W )| ≤ 1,

n n n n |1/W (x1 + x1 , x2 + x2 ) − 1/W (x1, x2 + x2 )| ≤ |x1 | , and so by (5.10) ii. (since limx2→∞ ux2 (x1, x2) = L), n x1 (5.12) lim W (x1, x2 + x2 ) = λ sec . n→∞ λ P n 2 n Using |uij(x1, x2 )| ≤ W (x1, x2 ) (as in Lemma 5.3), we have by (5.12) n n n n n n |u1(x1 + x1 , x2 + x2 ) − u1(x1, x2 + x2 )| ≤ |x1 ||u11(x1 + O(|x1 |), x2 + x2 )| (5.13) n 2 n n ≤ |x1 |W (x1 + O(x1 ), x2 ) → 0 as n → ∞. Therefore we obtain

n x1 lim ux1 (x1, x2 + x2 ) → λ tan , n→∞ λ proving (5.11) ii. Now since

n n n n n n |u(x1 + x1 , x2 + x2 ) − u(x1, x2 + x2 )| ≤ |x1 ||ux1 (x1 + O(|x1 |), x2 + x2 )| → 0 , and n n n n n n |u(x1 , x2 ) − u(0, x2 )| ≤ |x1 ||ux1 (O(|x1 |), x2 )| → 0 by (5.11) ii , n (5.11) i. follows from (5.10) i. Since the choice of x2 is arbitrary, part iv. follows. To prove part v. we will use Lemma 5.2 with 1 P2 = (x1, x2, u(x1, x2)),P1 = (x1(x2), x2, u(x1(x2), x2)),H(P1) ≥ √ . 1 + L2

Normalize u by |∇u(x1(0), 0)| = 0. We observe that Z R−ε q Σ 2 d (P2,P1) ≤ L(x2) := 1 + ux1 (t, x2) dt −R+ε and by (5.8), L(x2) ≤ M(ε) for |x2| ≥ A(ε) sufﬁciently large. Since u is a convex solution to the translating soliton equation, 2 2 2 2 (1 + ux2 )ux1x1 + (1 + ux1 )ux2x2 = 2ux1 ux2 ux1x2 + (1 + ux1 + ux2 ) √ ≤ 2|u ||u | u u + (1 + u2 + u2 ) (5.14) x1 x2 x1x1 x2x2 x1 x2 u2 u2 ≤ (1 + u2 )u + x1 x2 u + (1 + u2 + u2 ) . x1 x2x2 2 x1x1 x1 x2 (1 + ux1 ) This implies 2 2 2 ux1x1 ≤ 1 + ux1 and by symmetry ux2x2 ≤ 1 + ux2 ≤ 1 + L (5.15) √ q √ 2 2 |ux1x2 | ≤ ux1x1 ux2x2 ≤ 1 + L 1 + ux1 . 20 JOEL SPRUCK AND LING XIAO √ 0 2 Therefore using (5.15), |L (x2)| ≤ 1 + L L(x2) and so √ √ 2 2 Σ 1+L (A(ε)−|x2|) 1+L A(ε) (5.16) d (P2,P1) ≤ L(x2) ≤ e M(ε) ≤ e M(ε) =: M(ε)

−M(ε) for |x | ≤ A(ε). Therefore H(P ) ≥ θ(ε) := √e , completing the proof. 2 1 1+L2 An immediate application of Theorem 5.6 is the following symmetry result.

Theorem 5.7. Let Σ = graph(u) be a complete locally strictly convex translating soliton λ deﬁned over a strip region S . Then u(x1, x2) = u(−x1, x2) and ux1 (x1, x2) > 0 for x1 > 0.

Proof. We employ the method of moving planes; see for example [2]. Set St = {(x1, x2) ∈ λ S : t < x1 < R} for t ∈ (0,R). We want to show that the function

t (5.17) v (x1, x2) := u(x1, x2) − u(2t − x1, x2) > 0 in St for all t ∈ (0,R) .

Once (5.17) is proven, the conclusion of Theorem 5.7 follows easily. Indeed letting t → 0 in (5.17) implies by continuity

(5.18) u(x1, x2) ≥ u(−x1, x2) for 0 ≤ x1 ≤ R.

Since we may replace x1 by −x1 in (5.18), we have equality and thus we have symmetry in x1. From (5.17) we may also conclude that ux1 ≥ 0 for 0 < x1 < R. Since ux1 satisﬁes an elliptic equation without zeroth order term, we have from the maximum principle either ux1 > 0 or ux1 ≡ 0. Since the latter possibility is impossible, Theorem 5.7 follows. t t Set u = u(2t − x1, x2). Since u and u both satisfy the same elliptic equation (1.4) in St, the difference vt = u − ut satisﬁes a linear elliptic equation

2 2 X ij t X k t (5.19) A vij + B vk = 0 in St , i,j=1 i=1

t t t and so v cannot have an interior minimum in St. Also v = 0 for x1 = t, limx1→R v = +∞, and by Theorem 5.6 part iv.,

x1 t 2 sec λ (5.20) lim v = λ log { (2t−x ) } ≥ 0. x2→±∞ 1 sec λ

Therefore vt ≥ 0 and so by the maximum principle (5.17) holds completing the proof. COMPLETE TRANSLATING SOLUTIONS 21

6. EXISTENCE OF A COMPLETE LOCALLY STRICTLY CONVEX TRANSLATING SOLITON Σ = GRAPH(w) INTHESTRIP Sλ

In this section we prove existence of a complete locally strictly convex translating solitons in a strip and show there are no other locally strictly convex solutions.

Theorem 6.1. There exists a doubly symmetric complete translating soliton Σ = graph(w) in Sλ, λ > 1.

Before we start our construction, we recall a well known compactness theorem (see for example Theorem 2.6 of [17]) for the mean curvature ﬂow.

n Proposition 6.2. (compactness of mean curvature flow) Let {(Σi ,Xi(t)), −1 < t < 1} be n+1 a sequence of mean curvature flow properly immersed in Bρ(0) ⊂ R . Suppose that sup |A|(x, t) ≤ Λ, ∀t ∈ (−1, 1) Σi,t∩Bρ(0) ∞ for some Λ > 0. Then a subsequence of {Σi,t ∩ Bρ(0), −1 < t < 1} converges in the C topology to a smooth mean curvature flow {Σ∞,t, −1 < t < 1} in Bρ(0).

Consider the family of rectangles π R = {x ∈ 2 : |x | < R, |x | < A, R = λ , λ > 1,A ≥ 1}. A R 1 2 2 and let uA be the solution of the Dirichlet problem  X uiuj  δ − u = 1 in R ,  ij 1 + |Du|2 ij A (6.1) i,j   u = 0 on ∂RA.

The existence of a unique smooth solution is RA is standard (we can round the corners to form a smooth convex doubly symmetric domain but this is not essential). Moreover by the method of moving planes, uA is symmetric in both x1, x2 with its negative minimum at (0,0). Let MA = − inf uA; then by the maximum principle MA is positive and strictly increasing.

Lemma 6.3. MA → ∞ as A → ∞.

Proof. The proof is by contradiction. Assume MA → C as A → ∞. Then we have uA → u (since the family uA + MA is monotone increasing with all derivatives uniformly bounded) and u is a solution of (6.1) in Sλ. Moreover, we have inf u = −C and u = 0 π on ∂S. Let v(x) = log sec x1 be the graph of the (standard) grim cylinder in |x1| < 2 . 22 JOEL SPRUCK AND LING XIAO

n Then v(x) > u(x, y) and C1 = inf(v(x) − u(x, y)) is bounded. Now let u˜ (x, y) = u(x, y + yn) + C1 where (v(xn) − u(xn, yn)) → C1. By the maximum principle we must π have |(xn, yn)| → ∞. Assume xn → x0 and note |x0| < 2 . Since u is bounded and u = constant on the boundary of the strip, all derivative of u˜ are uniformly bounded. Hence a subsequence of u˜n converges smoothly to a solution u∞(x, y) ≤ v(x) with equality at (x0, 0). This contradicts the maximum principle. A direct consequence of Lemma 6.3 is the following

Lemma 6.4. For any k > 1, there exist A = A(k) such that MA(k) = k.

Now let {uAk } be a sequence of solution of equation (6.1), such that inf uAk = −k.

Denote wk = uAk + k; then wk ≥ wk(0) = 0. We claim that a subsequence of the wk converges to a solution Σ = graph(w) deﬁned on the strip region Sλ. This follows from Propositions 2.4, 2.7, and the Compactness Theorem, Proposition 6.2. It remains show that Σ = graph(w) is complete.

λ Lemma 6.5. For any x0 ∈ ∂S we have lim w(x) = ∞. x→x0 λ λ Proof. We argue by contradiction. Fix x0 ∈ ∂S and suppose there is a sequence xn ∈ S such that xn → x0 and w(xn) ≤ C. Then for ﬁxed k > C such that x0 is in the domain of wk, wk(xn) ≤ w(xn) ≤ C. But wk is continuous at x0 so k = limn wk(xn) ≤ C, a contradiction. By Theorem 1.1, Σ = graph(w) is convex and since w ≥ 0, w 6= uλ so w is locally strictly convex, completing the proof of Theorem 5 part vii.

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DEPARTMENT OF MATHEMATICS,JOHNS HOPKINS UNIVERSITY,BALTIMORE, MD 21218 E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS,RUTGERS UNIVERSITY,PISCATAWAY, NJ 08854 E-mail address: [email protected]