Convergence of Gauss curvature flows to translating solitons
aa r X i v : . [ m a t h . DG ] O c t CONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS
BEOMJUN CHOI, KYEONGSU CHOI, AND PANAGIOTA DASKALOPOULOS
Abstract.
We address the asymptotic behavior of the α -Gauss curvature flow, for α > /
2, with initialdata a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section.We show that the flow converges, as t → + ∞ , locally smoothly to a translating soliton which is uniquelydetermined by the asymptotic cylinder of the initial hypersurface. Contents
1. Introduction 12. Preliminaries 33. Local speed estimate 64. Local convexity estimate 105. Convergence to translating soliton 14Appendix A. Monotonicity formula 17References 201.
Introduction
Given α >
0, the α -Gauss curvature flow ( α -GCF in abbreviation) is a one-parameter family of embeddings F : M n × [0 , T ) → R n +1 such that for each t ∈ [0 , T ), F ( M n , t ) = Σ t is a complete convex hypersurface in R n +1 , and F ( · , t ) satisfies(1.1) ∂∂t F ( p, t ) = − K α ( p, t ) ν ( p, t ) , where K ( p, t ) is the Gauss curvature of Σ t at F ( p, t ), and ν ( p, t ) is the unit normal vector of Σ t at F ( p, t )pointing outward of the convex hull of Σ t .The classical Gauss curvature flow (GCF), the α = 1 case, was first introduced by W. Firey [21] todescribe the shape of worn stones and the asymptotic behavior when it disappears. In [21], W. Firey provedthat if a closed strictly convex solution to the GCF in R has the central symmetry, then it converges toa round sphere after rescaling. Later, B. Andrews [3] removed the central symmetry condition. In higherdimensions n ≥
3, P. Guan and L. Ni [22] obtained the convergence to a self-shrinking soliton after rescaling,and K. Choi and P. Daskalopoulos [13] showed the uniqueness of self-shrinking soliton. Namely, a closedstrictly convex solution to the GCF converges to a round sphere after rescaling in R n +1 .In addition to the standard case α = 1, the asymptotic behavior of the α -GCF has been widely studied.In particular, in the α = n +2 case, an affine transform of a solution remains as a solution, and thus wecall the n +2 -GCF as the affine normal flow. E. Calabi [10] showed that a self-shrinking soliton to the affinenormal flow is an ellipsoid. (See also [8] for an alternative proof.) B. Andrews [2] obtained the convergenceof the closed affine normal flow to an ellipsoid after rescaling. In the range of α > n +2 , the convergence of the closed α -GCF to a round sphere after rescaling has beenshown by B. Chow [16] for α = n , and by B. Andrews and X. Chen [6] for ≤ α ≤ n = 2.Later, for the all α > n +2 B. Andrews, P. Guan and L. Ni [7] showed the convergence to a self-similarsoliton after rescaling. Moreover S. Brendle, K. Choi, and P. Daskalopoulos [8] proved the uniqueness ofself-shrinking solitons. Namely, a closed strictly convex solution to the α -GCF with α > n +2 converges to around sphere after rescaling in R n +1 .Regarding small powers α ∈ (0 , n +2 ), the asymptotic behavior remains as an open problem. B. Andrewsclassified closed self-shriking solitons in the curve case n = 1 [5], and showed the existence of non-trivialclosed self-shrinking solitons in higher dimensions [4].Regarding the non-compact case, the translating solitons to the α -GCF have been classified for α = n +2 and α > . In the affine normal case α = n +2 , the translating solitons are paraboloids. The n = 2 caseshowed first by K. J¨orgens [26], and later by J.C.C. Nitsche [27] with another proof by using the complexanalysis. E. Calabi [9] extended the result for n ≤
5, and A.V. Pogorelov [28] proved for all dimensions. S.Y.Cheng and S.T. Yau [11] provided an alternative proof by using the affine geometry.In [31] and [32], J. Urbas showed that every translating soliton for α > is contained in a boundedcylinder Ω × R , namely Ω ⊂ R n is bounded. Moreover, if α > then given a convex bounded set Ω ⊂ R n there exists a translating soliton asymptotic to the cylinder Ω × R . Furthermore, for each convex boundedΩ, the translating soliton is unique up to translations. One the other hand, for α ∈ (0 , ] H. Jian and X.J.Wang [25] showed the existence of infinitely many entire translating solitons.Recently, the authors [12] showed the convergence to a translating soliton for n = 1 and α > . In thispaper, we establish its higher dimensional result for n ≥ Theorem 1.1.
For α ≥ , let Σ = ∂ ˆΣ ⊂ R n +1 be a non-compact convex hypersurface contained ina cylinder of bounded cross-section. Then, the weak α -GCF flow solution as defined in Definition 2.6,converges, as t → + ∞ , locally smoothly to a translating soliton. Moreover, the limiting soliton is uniquelydetermined by the cylinder asymptotic to Σ . For small α ∈ (1 / ,
1) we add the technical assumptions that M can be approximated by compacthypersurfaces with uniform R K α dg and ( α − R P K α dg bounds, where P is defined at (2.1). Notice that R K α dg and ( α − R P K α dg denote the total speed and total acceleration, respectively. See Lemma A.1.Our result for α ∈ (1 / ,
1) states as follows:
Theorem 1.2.
Let Σ = ∂ ˆΣ ⊂ R n +1 be a non-compact hypersurface contained in a cylinder of boundedcross-section. For given fixed α ∈ (1 / , , if there is a sequence of compact smooth strictly convex hyper-surfaces Σ i = ∂ ˆΣ i which increases to Σ (i.e. ˆΣ k ⊂ ˆΣ k +1 and ∪ i ˆΣ i = ˆΣ ) with uniform upper bounds on R Σ i K α dg and ( α − R Σ i P K α dg , then the weak α -GCF converges, as time t → + ∞ , locally smoothly to atranslating soliton. Moreover, the limiting soliton is uniquely determined by the cylinder asymptotic to Σ . The weak flow Σ t is asymptotic to the initial asymptotic cylinder, say Ω × R , for all time by Theorem 2.7and thus Σ t can be written as convex graphs on a fixed domainΣ t = ∂ { x n +1 > u ( x, t ) : x ∈ int(Ω) } . Smooth convergence in the statements of the theorems above means the C ∞ loc (Ω) convergence of the functions u ( · , t ) − inf x ∈ int(Ω) u ( x, t ) to u Ω ( · ), which represents the soliton asymptotic to Ω × R . If the boundary ofdomain Ω is weakly convex and not strictly convex, then the corresponding translating soliton may touchthe boundary of the cylinder and have flat sides (c.f in the work by K. Choi, P. Daskalopoulos, and K.A.Lee in [14]). Therefore, a smooth convergence up to boundary is not expected. ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 3
We introduce the notion of weak α -GCF solution to state and prove the convergence of flows with generalweakly convex initial hypersurfaces. The existence and uniqueness of the weak flow is shown in Theorem2.7. Note that if Σ is weakly convex and has flat sides, the solution Σ t preserves the flat sides for a certainamount of time by the result of R. Hamilton [23]. See also the optimal regularity of an evolving flat sidefor short time [18] and for long time [19]. Even in this case, our theorem shows that the solution restrictedin K × R , for each K ⊂ Int(Ω), becomes smooth and strictly convex at large times and converges to atranslating soliton in K × R .Let us remark that in order to converge to a translating soliton, the initial hypersurface Σ must becontained in a bounded cylinder. Jointly with L. Kim and K.A. Lee, the second and third authors in [15]showed by a barrier argument that if Σ is a graph over a (possibly non-compact) domain Ω ⊂ R n , thenany solution Σ t running from Σ must remain as a graph over the same domain Ω . On the other hand,every translating soliton to the α -GCF with α > / is contained in a bounded cylinder.The following monotonicity formula will be used to identify the limit as a soliton. The technical assump-tions in Theorem 1.2 were made so that this inequality can be applied. Theorem 1.3.
Given α ≥ n − n , compact strictly convex smooth solution Σ t to the α -GCF satisfies ddt Z Σ t P K α dg ≥ (cid:0) n − + 2 α − (cid:1) Z Σ t P K α dg ≥ . We notice that B. Chow [17] obtained the above monotonicity formula for the GCF ( α = 1); (see theproof of Lemma 4.3 in [17]). In the same paper, B. Chow also obtained a monotonicity formula (Lemma5.2 in [17]) for the rescaled GCF. In [1] B. Andrews generalized the monotonicity formula for the rescaled α -GCF. We prove Theorem 1.3 in the Appendix for the readers who are only interested in the formula.2. Preliminaries
Definition 2.1. (i) Σ ⊂ R n +1 is a convex hypersurface if it is a boundary of convex set with non-emptyinterior, say ˆΣ . Note convex hypersurface Σ = ∂ ˆΣ is complete and embedded.(ii) For C convex hypersurface Σ = ∂ ˆΣ, we say it is strictly convex at p ∈ Σ if the second fundamentalform with respect to the inner normal is positive definite.Throughout this paper, h ij denotes the second fundamental form. For a strictly convex solution, one mayconsider the inverse b ij of the second fundamental form h ij , which satisfies b ik h kj = δ ij . We also denoteby dg := √ det g dx the volume form induced from the ambient Euclidean metric. We let S := h F, ν i and S x := h F − x , ν i denote the support functions with respect to the origin and x ∈ R n +1 , respectively.Moreover, we recall the following tensor P ij and the quantity P defined by B. Chow in [17]:(2.1) P ij := ∇ ij K α − b mn ∇ m h ij ∇ n K α + K α h ki h kj and P := b ij P ij . Note that, for solutions to α -GCF, (2.15) implies(2.2) P = 1 αK α ( ∂ t K α − b ij ∇ i K α ∇ j K α ) . Let us recall the unique existence of translating solitons by J. Urbas and denote them as follows.
Definition 2.2 (Theorem of J.Urbas [31, 32]) . For α > / ⊂ R n ,we define by u Ω : Ω → R to be the graph function of the unique translating soliton which is asymptotic to ∂ Ω × R , it moves in the positive e n +1 direction, and satisfies inf u Ω ( · ) = 0. In other words, the hypersurfacegiven by ∂ { ( x, x n +1 ) ∈ R n +1 : x n +1 > u ( x ) } defines the translating soliton. BEOMJUN CHOI, KYEONGSU CHOI, AND PANAGIOTA DASKALOPOULOS
Remark . In the case where Ω is not a strictly convex domain, it is possiblethat lim sup x → x u Ω ( x ) < ∞ , for some x ∈ ∂ Ω, hence the hypersurface { x n +1 = u Ω ( x ) } may not bealways complete. This is the reason why we in the definition above we defined the translating soliton as ∂ { ( x, x n +1 ) ∈ R n +1 : x n +1 > u ( x ) } . Urbas [32] showed the existence of such solitons and their uniquenessamong solutions realized in certain generalized sense. To be more specific, Urbas [32] showed that if a convexfunction u ( x ) defined on Ω satisfies the translating soliton equation(2.3) det D u = β (1 + | Du | ) n +22 − α for some β > | R n − Du (Ω) | = 0, then u = u Ω + C , for some constant C . We will use this characterization of soliton in the proofs of Theorem 1.1.
Definition 2.4.
For α > / ⊂ R n , let us note the speed of associatedtranslating soliton by(2.4) λ := 1 | Ω | α "Z R n p | p | ) n +2 − α dp α . (The derivation of this formula follows from (5.7) and Du (Ω) = R n ). Moreover note, that when α = 1,(2.5) λ := 1 | Ω | "Z R n p | p | ) n +1 dp = ω n | Ω | where ω n = | S n | .We derive evolutions of basic geometric quantities under α -GCF. Proposition 2.5.
For a strictly convex hypersurface, we have ∇ m K = Kb ij ∇ m h ij (2.6) ∇ i ( b ij K ) = 0(2.7) ∇ l b ij = − b ip ∇ l h pq b qj . (2.8) For a smooth strictly convex solutions of α -GCF, we have ∂ t g ij = − K α h ij (2.9) ∂ t dg = − K α Hdg (2.10) ∂ t ν = ∇ K α = ∇ i K α ∇ i F (2.11) ∂ t h ij = ∇ ij K α − K α h ik h kj (2.12) = αK α b rs ∇ rs h ij + αK α ( αb kl b mn − b km b ln ) ∇ i h mn ∇ j h kl + αK α Hh ij − (1 + nα ) K α h ik h kj (2.13) ∂ t b pq = αK α b ij ∇ ij b pq − αK α b ip b jq ( αb kl b mn + b km b ln ) ∇ i h kl ∇ j h mn − αK α Hb pq + (1 + nα ) K α g pq (2.14) ∂ t K α = αK α b ij ∇ ij K α + αHK α (2.15) ∂ t | F | = αK α b ij ∇ ij | F | + 2( nα − K α S − K α b ij g ij (2.16) ∂ t S = αK α b ij ∇ ij S + αK α HS − (1 + nα ) K α (2.17) Proof. By K = (det g ij )(det h ij ) ∇ m K = K ∇ m log K = K ∇ m log(det h ij ) = Kb ij ∇ m h ij . Next, ∇ i ( b ij K ) = ( ∇ i b ij ) K + b ij ∇ i K = − b ik b jl ( ∇ i h kl ) K + b ij Kb kl ( ∇ i h kl ) = 0 . ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 5 (2.8) follows from taking a derivative on b ij h jk = δi k . The evolution equations (2.9) - (2.15) are shown inProposition 2.1 [15]. Note that( ∂ t − αK α b ij ∇ ij ) F = − K α ν − αK α b ij h ij ( − ν )= ( nα − K α ν. Thus, we have ( ∂ t − αK α b ij ) h F, F i = 2 h F, ( nα − ν i − αK α b ij h∇ i F, ∇ j F i = 2( nα − S − K α b ij g ij and, using ∇ ij ν = ∇ i ( h jk ∇ k F ) = − h jk h ki ν + ∇ k h ij ∇ k F , we obtain( ∂ t − αK α b ij ∇ ij ) h F, ν i = ( nα − K α + h F, ( ∂ t − αK α b ij ∇ ij ) ν i − αK α b ij h∇ i F, ∇ j ν i = ( nα − K α + h F, ∇ K α − αK α b ij ∇ k h ij ∇ k F + αK α Hν i − nαK α = − ( nα − K α + αK α HS. (cid:3)
Definition 2.6.
Suppose that ˆΣ t ⊂ R n +1 , t ∈ [0 , T ) is a one-parameter family of closed convex sets withpositive (or infinite) volume. The family of convex hypersurfaces Σ t = ∂ ˆΣ t ⊂ R n +1 , t ∈ [0 , T ) is called a weak sub-solution to the α -GCF if for any t ∈ [0 , T ) the following holds: if Σ ′ t = ∂ ˆΣ ′ t is any smooth strictlyconvex solution to the α -GCF with ˆΣ ′ t ⊂ ˆΣ t , then ˆΣ ′ t ⊂ ˆΣ t holds for all t ≥ t . Similarly, Σ t = ∂ ˆΣ t isa weak super-solution to the α -GCF if for any t ∈ [0 , T ) the following holds: if Σ ′ t = ∂ ˆΣ ′ t is any smoothstrictly convex solution to the α -GCF with ˆΣ t ⊂ ˆΣ ′ t , then ˆΣ t ⊂ ˆΣ ′ t for all t ≥ t . Σ t is a weak solution if itis both weak sub- and super-solution.The following result states the existence and uniqueness of a weak solution starting at any convex hy-persurface Σ = ∂ ˆΣ ⊂ R n +1 , compact or non-compact, and asymptotic to a cylinder. This result simplyfollows from known results on smooth solutions and an a standard approximation argument, but we includeits proof here for completeness. Theorem 2.7.
Let Σ = ∂ ˆΣ ⊂ R n +1 be a convex hypersurface. If Σ is compact, then there is a uniqueweak solution Σ t to the α -GCF running from Σ and defined over t ∈ [0 , T ) for some T < + ∞ . If Σ is noncompact and asymptotic to a cylinder Ω × R , then there is a unique weak solution Σ t to the α -GCF runningfrom Σ defined for all t ∈ [0 , + ∞ ) . Moreover, Σ t is non-compact and asymptotic to Ω × R for all t ∈ [0 , ∞ ) .Proof. Consider the first case that Σ is compact. Choose an increasing sequence convex bodies ˆΣ i, withsmooth strictly convex boundaries Σ i, (see Ch3.4 [29] for an approximation by smooth strictly convexhypersurfaces) and ∪ i ˆΣ i, = int( ˆΣ ). Let Σ i,t be the unique smooth solution to the α -GCF starting fromΣ i, (see in [16]). By the comparison principle the sequence Σ i,t is increasing in i , hence the limits ˆΣ t := ∪ i ˆΣ i,t and Σ t = ∂ ˆΣ t exist. We claim that Σ t is a weak solution with initial data Σ . By construction, Σ t is aweak super-solution. Let us next show that Σ t is a weak sub-solution as well. Assume without loss ofgenerality that 0 ∈ ˆΣ and, according to our definition above, suppose that Σ ′ t = ∂ ˆΣ t is a smooth strictlyconvex α -GCF flow with ˆΣ ′ ⊂ ˆΣ . Given small ǫ >
0, we consider the rescaled solution (1 − ǫ ) Σ ′ τ , with τ = (1 − ǫ ) − (1+ nα ) t starting at (1 − ǫ ) Σ ′ . For sufficiently large i , (1 − ǫ ) ˆΣ ′ ⊂ ˆΣ i, holds if i ≥ i . Thus,the comparison principle guarantees (1 − ǫ ) ˆΣ ′ τ ⊂ ˆΣ i,t . Taking the limit i → ∞ and then ǫ →
0, yields theinclusion ˆΣ ′ t ⊂ ˆΣ t proving that Σ t is a weak sub-solution. We then conclude that Σ t is a weak solution. Forthe uniqueness assertion, let us assume that we have another weak solution Σ ′′ t starting at Σ . Then, thesame argument as above, shows that each small ǫ >
0, there is i ≫ − ǫ ) ˆΣ i, (1 − ǫ ) − (1+ nα ) t ⊂ ˆΣ ′′ t ⊂ (1 + ǫ ) ˆΣ i, (1+ ǫ ) − (1+ nα ) t , for i ≥ i . BEOMJUN CHOI, KYEONGSU CHOI, AND PANAGIOTA DASKALOPOULOS
Taking the limits i → ∞ and ǫ →
0, we conclude that Σ t = Σ ′′ t . Finiteness of T follows by comparing thesolution with a huge spherical solution containing it.Consider the next case that Σ is non-compact and asymptotic to Ω × R . Choose a sequence of increasingcompact sets ˆΣ i, with smooth strictly convex boundaries Σ i, such that ∪ i ˆΣ i, = ˆΣ . Let Σ i,t , t ∈ [0 , T i ),be the unique smooth strictly convex solutions to the α -GCF and define ˆΣ t := ∪ i ˆΣ i,t and Σ i,t = ∂ ˆΣ i,t asbefore. Note Σ i,t exists for t ∈ [0 , T ), where T ≥ lim i →∞ T i = (0 , ∞ ]. By the construction, Σ t is already aweak super-solution. Let Σ ′ t = ∂ ˆΣ ′ t be a smooth strictly convex α -GCF with ˆΣ ′ ⊂ ˆΣ . When ˆΣ ′ is compact,one can use the same argument as before to show ˆΣ ′ t ⊂ ˆΣ t . Let us assume ˆΣ ′ be a non-compact. Then ˆΣ ′ has to be asymptotic to a cylinder Ω ′ × R with Ω ′ ⊂ Ω. By the same scaling and limiting argument, we mayassume Ω ′ ⊂⊂ Ω. For such a Σ ′ , it was also shown in [15] that the unique smooth solution (hence it is Σ ′ t )exists for all t ∈ [0 , ∞ ) and the solution is written on the fixed domain Ω ′ . Moreover, the construction in[15] shows ˆΣ ′ t can be approximated by an increasing sequences of compact smooth strictly convex α -GCFsΣ ′ i,t = ∂ ˆΣ ′ i,t . For each ˆΣ ′ j there is i j such that ˆΣ ′ j, ⊂ ˆΣ i j , . This implies ˆΣ ′ j,t ⊂ ˆΣ i j ,t ⊂ ˆΣ t . This provesˆΣ ′ t ⊂ ˆΣ t . i.e. ˆΣ t is a weak solution. This also proves T = ∞ since we may put a non-compact rotationallysymmetric strictly convex hypersurface which is asymptotic to a round cylinder in the inside of ˆΣ and applythe comparison. Finally, asymptotic cylinder of ˆΣ t can not shrink as we can insert such a barrier arbitrarilyclose to the boundary of Ω × R at initial time t = 0. (cid:3) Corollary 2.8. If ˆΣ i, is an increasing sequences of convex sets such that ∪ i, ˆΣ i, = ˆΣ and ˆΣ is compactor non-compact asymptotic to a cylinder, then ∪ i ˆΣ i,t = ˆΣ t . Here, ˆΣ i,t and ˆΣ t are weak solutions. In this paper, when Σ i,t = ∂ ˆΣ i,t is referred, it means approximating smooth compact strictly convexsolutions of Σ t from inside unless otherwise stated.3. Local speed estimate
We review the following Harnack estimate which was shown by B. Chow in [17].
Theorem 3.1 (B. Chow [17]) . Let Σ t be a smooth compact strictly convex solution of α - GCF with α > .Then, (3.1) 1 K α ( ∂ t K α − b ij ∇ i K α ∇ j K α ) ≥ − nα nα t . This has the following consequence:
Proposition 3.2.
Let x n +1 = u ( x ′ , t ) be a smooth strictly convex graphical solution to α - GCF , α > overa domain x ∈ U and assume it is part of a compact smooth solution or a smooth limit of such solutions.Then, (3.2) u tt ≥ − nα nα u t t and hence, for t ≥ t > , (3.3) u t ( · , t ) ≥ (cid:18) t t (cid:19) nα nα u t ( · , t ) . Proof.
For any 1-form V i , K α b ij ( V i + ∇ i log K α )( V j + ∇ j log K α ) ≥ ∂ t log K α + 2 K α b ij V i ∇ j log K α + K α b ij V i V j ≥ − nα nα t . ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 7
In other words, for any vector field U i = K α b ij V j , ∂ t K α + 2 U i ∇ i K α + h ij U i U j ≥ − nα nα K α t . For a graphical solution of α -GCF, x n +1 = u ( x ′ , t ), note that ∂ t u = K α h− ν,e n +1 i and ∂ tt u ( x ′ , t ) = ( ∂ t + W i ∇ i )( K α h− ν, e n +1 i ) with W = K α h− ν, e n +1 i e tan n +1 . Here e tan n +1 = e n +1 − h e n +1 , ν i ν . Using this and ∂ t ν = ∇ K α , we check( ∂ t + W i ∇ i )( K α h− ν, e n +1 i )= 1 h− ν, e n +1 i (cid:0) ∂ t K α + W i ∇ i K α (cid:1) + K α ( ∂ t + W i ∇ i ) 1 h− ν, e n +1 i = 1 h− ν, e n +1 i (cid:0) ∂ t K α + 2 W i ∇ i K α + h ij W i W j (cid:1) ≥ − nαn + α K α t h− ν, e n +1 i . This proves the first part and the rest follows by this differential inequality. (cid:3)
Suppose that Σ t is a non-compact weak α -GCF solution asymptotic to Ω × R and let { x n +1 = u i ( x ′ , t ) } bethe graph representation of the lower parts of the approximating compact smooth strictly convex solutionsΣ i,t as graphs. Let us denote by Ω i,t the spatial domain of u i ( · , t ), that is the projection of Σ i,t ⊂ R n +1 tothe hyperplane x n +1 = 0. Proposition 3.3.
For each Ω ′ ⊂⊂ Ω and t > there is L > so that the following holds: all T > t thereis i so that (3.4) ∂ t u i ( x ′ , t ) = K α h− ν, e n +1 i ≤ L for ( x ′ , t ) ∈ Ω ′ × [ t , T ] and i > i . Moreover, for each Ω ′ ⊂⊂ Ω there are positive constants t , δ , L so that the following holds: for all T > t there is i such that (3.5) 0 < δ ≤ ∂ t u i ( x ′ , t ) = K α h− ν, e n +1 i for ( x ′ , t ) ∈ Ω ′ × [ t , T ] and i > i . Proof.
Let us assume, without loss of generality, that Ω contains the origin and that the speed of thetranslating soliton defined on Ω, call it u Ω , is λ . Fix a small ǫ ∈ (0 , /
6) so that Ω ′ ⊂⊂ (1 + ǫ ) − nα Ω. SinceˆΣ t is asymptotic to Ω × R for all t ≥ ∪ i ˆΣ i,t = ˆΣ t , given T ′ there is i such that if i > i (3.6) (1 + ǫ ) − nα Ω ⊂⊂ Ω i,t for all t ∈ [0 , T ′ ] .T ′ is some number which will be chosen later.Using the scaling properties of our equation, if we defineˆ u ( x ′ ) := (1 + ǫ ) − nα u Ω ((1 + ǫ ) nα x ′ ) = u (1+ ǫ ) − nα Ω ( x ′ )then ˆ u is the translating soliton on (1+ ǫ ) − nα Ω which has speed (1+ ǫ ) λ . Similarly, we define the translatingsoliton ¯ u ( x ′ ) := (1 − ǫ ) − nα u Ω ((1 − ǫ ) nα x ′ ) = u (1 − ǫ ) − nα Ω ( x ′ )on (1 − ǫ ) − nα Ω which has speed (1 − ǫ ) λ . Depending on Σ , we may find a large L > u ( x ′ ) − L ≤ u ( x ′ ,
0) on Ω and u ( x ′ , ≤ ˆ u ( x ′ ) + L ǫ ) − nα Ω . BEOMJUN CHOI, KYEONGSU CHOI, AND PANAGIOTA DASKALOPOULOS
It follows that there is an i such that for i > i , then(3.7) ¯ u ( x ′ ) − L ≤ u i ( x ′ ,
0) on Ω i, and u i ( x ′ , ≤ ˆ u ( x ′ ) + L on (1 + ǫ ) − nα Ω . Furthermore, by (3.6) and (3.7), one can apply the comparison principle between x n +1 = u i ( x ′ , t ) and twobarriers and obtain, for i > i ,¯ u ( x ′ ) − L + (1 − ǫ ) λt < u i ( x ′ , t ) < ˆ u ( x ′ ) + L + (1 + ǫ ) λt on Ω ′ × [0 , T ′ ] . In particular, we have for t ∈ [0 , T ′ ] and i > i ≤ f ( x ′ , t ) := (ˆ u ( x ′ ) + L + (1 + ǫ ) λt ) − u i ( x ′ , t ) ≤ L + ǫ λt )and 0 ≤ g ( x ′ , t ) := u i ( x ′ , t ) − (¯ u ( x ′ ) − L + (1 − ǫ ) λt ) ≤ L + ǫ λt ) . Let us now prove the first upper bound (3.4). Choose T ′ in (3.6) by T ′ = 2 T . Suppose ∂ t u i ( x , t ) = C at some x ∈ Ω ′ and t ∈ [ t , T ]. Then by (3.3), ∂ t u i ( x , t ) ≥ C η for some η = η ( α, n ) ∈ (0 ,
1) and all t ∈ [ t , t ]. Then, we have0 ≤ f ( x , t ) = f ( x , t ) + Z t t ∂ t f ≤ L + ǫ λt ) + [(1 + ǫ ) λt − Cηt ]and hence ∂ t u i ( x , t ) = C ≤ (1 + 3 ǫ ) λη + 2 Lηt , proving that the bound from above in (3.4) holds for any t ∈ [ t , T ] and t fixed.We will next prove (3.5). To this end, suppose ∂ t u i ( x , t ) = c at some x ∈ Ω ′ and t >
0. Provided T ′ > t and i > i , (3.3) implies that ∂ t u i ( x , t ) ≤ γc for any t ∈ [ t / , t ] and some γ = γ ( α, n ) >
1. Then,0 ≤ g ( x , t ) = g ( x , t /
2) + Z t t / ∂ t g ≤ (cid:18) L + ǫ λ t (cid:19) + (cid:20) γc t − (1 − ǫ ) λ t (cid:21) implying that for any ǫ < / c ≥ − e γ λ − Lγt ≥ λ γ − Lγt . Hence, ∂ t u i ( x , t ) = c ≥ λ γ if t ≥ Lλ . Let us choose t := 16 L/λ >
0. Then for every T ≥ t , if we choose T ′ = T the previous yields the lower bound (3.5) for i > i . (cid:3) On a strictly convex smooth solution Σ t we may define the Gaussian curvature K as a function of thenormal vector ν at a point p , i.e. we define ¯ K ( ν, t ) := K ( p ( ν, t ) , t ) where p = p ( ν, t ) is chosen so that ν ( p ) = ν . Using the evolution of ν , one sees that ∂ t ¯ K α = ∂ t K α − b ij ∇ i K α ∇ j K α . Hence B. Chow’s Harnackinequality on K (see (3.1)) implies(3.8) ∂ t ¯ K α ≥ − nα nα ¯ K α t which after integrated in time t ∈ [ t , t ] gives(3.9) ¯ K α ( · , t ) ≥ (cid:18) t t (cid:19) nα nα ¯ K α ( · , t ) . An argument along the lines of Proposition 3.3 which combines a Harnack inequality in the form (3.9) andbarrier arguments applied on the support function S ( · , t ) instead of the height function u ( · , t ), was actuallyused by the authors in Section2 [12]. Following similar arguments as in Proposition 3.3 and [12], we obtainthe following: ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 9
Proposition 3.4.
Let Σ i,t be a sequence of compact smooth strictly convex solutions which approximatenon-compact weak solution Σ t asymptotic to a cylinder of bounded section Ω × R . For any small µ > , thereare positive constants t , δ depending on Σ ,and µ which make the following hold: for all T > there is i such that, for M i,t with i > i , δ ≤ K α ( p, t ) if t ≤ t ≤ T , and h− ν ( p, t ) , e n +1 i ≥ µ. For given t , there is M depending on Σ and t which make the following hold: for all T > there is i such that, for M i,t with i > i , K α ( p, t ) ≤ M if t ≤ t ≤ T , and h− ν ( p, t ) , e n +1 i ≥ . Proof.
Assume Ω ⊂ R n contains the origin. Define the support function ¯ S ( ν, t ) = sup x ∈ Σ t h x, ν i . Let ǫ > − ǫ ) λ where λ is speed of the translator u Ω on Ω. We can make that this translator contains our initial surface Σ (and hence all Σ i, ) by shifting the translator in − e n +1 direction. If ¯ S + ( ν, t ) is the support function of thistranslator, then Lemma 2.6 [12], the comparison between support functions, gives¯ S ( ν, t ) ≤ ¯ S + ( ν, t ) = C + (1 − ǫ ) λt h ν, e n +1 i on M i,t ∩ {h− ν, e n +1 i ≥ } by some C ( ǫ , Σ , α, n ) > . On the other hand, by inserting a translating soliton of speed (1 + ǫ ) λ inside, we know that the point( L + (1 + ǫ ) λt ) e n +1 (for some L >
0) is located inside of Σ t . Thus, h F − ( L + (1 + ǫ ) λt ) e n +1 , ν i ≥ T ′ > i with − C + (1 + ǫ ) λt h ν, e n +1 i ≤ ¯ S ( ν, t ) on M i,t if i > i by some C ( ǫ , Σ , α, n ) > . In particular, if i > i , we have0 ≤ f ( ν, t ) := ¯ S + ( ν, t ) − ¯ S ( ν, t ) ≤ C − ǫ λt h ν, e n +1 i ) for t ∈ [0 , T ′ ]and 0 ≤ g ( ν, t ) := ¯ S ( ν, t ) − C + (1 + ǫ ) λt h ν, e n +1 i ≤ C − ǫ λt h ν, e n +1 i ) for t ∈ [0 , T ′ ] . In the meantime, note that ∂ t ¯ S ( ν, t ) = ¯ K α ( ν, t ). In the estimates below, we assume i > i = i ( T ′ )Let us prove the upper bound. For given t > K α ( ν , t ) = a at some ν ∈ S n − := S n ∩ { x n +1 ≤ } and 0 < t < T ′ /
2. Then (3.9) impliesthat ( ¯ K α ) t ( ν , t ) ≥ η a for t ∈ [ t , t ] and some η = η ( α, n ) ∈ (0 , ≤ f ( ν , t ) = f ( ν , t ) + Z t t ∂ t f ≤ C + ǫ λt ) + [(1 + e ) λt − ηa t ]implies that the upper bound ¯ K α ( ν , t ) ≤ (cid:16) (1 + 3 ǫ ) λη + 2 Cηt (cid:17) =: M , where M depends on Σ and t . Thisproves the upper bound.Let us prove the lower bound. For given µ >
0, suppose ¯ K α ( ν , t ) = a at some ( ν , t ) with h− ν , e n +1 i ≥ µ > < t < T ′ . Then by (3.9), ¯ K α ( ν , t ) ≤ γ c for t ∈ [ t / , t ] and some γ = γ ( α, n ) >
1. Hence,for ǫ ∈ (0 ,
1) to be chosen later, there is C = C ( ǫ , Σ , α, n ) such that0 ≤ g ( ν , t ) = g ( ν , t /
2) + Z t t / ∂ t g ≤ (cid:16) C + ǫ λ t (cid:17) + (cid:16) γa t − (1 − e ) µλ t (cid:17) implying that a ≥ (1 − ǫ ) µ − ǫ γ λ − Cγt . Now by choosing ǫ := µ µ (hence (1 − ǫ ) µ = 3 ǫ ) we have a ≥ ǫ γ λ − Cγt = µ µ λγ − Cγt for some C = C ( µ, Σ , α, n ) . Therefore a = ¯ K α ( ν , t ) ≥ µ µ λ γ if t ≥ C µµλ . In summary, for given µ ∈ (0 , t = 8 C µµλ such that K α ( ν , t ) ≥ δ on M i,t with t > i if t ≤ t ≤ T ′ and h− ν, e n +1 i ≥ µ . δ > µ , Σ , α , and n . (cid:3) Local convexity estimate
In this section we will prove an estimate which gives a local bound from below on the minimum principalcurvature λ min of our solution Σ t in terms of upper and lower bounds of the speed K α . The estimate isimportant later in our proof of main theorem. We need some preliminary results and we begin with a simpleobservation on convex graphs. Lemma 4.1.
Let x n +1 = u ( x ′ ) be a C convex graph on {| x ′ | ≤ r } and assume there is δ > such that K h− ν, e n +1 i > δ , where ν = ( Du, − p | Du | denotes the normal vector to the graph. Then there is C = C ( δ r − n , n ) such that sup | x ′ |≤ r u − inf | x ′ |≤ r u ≤ C r.
Proof.
We may assume without loss of generality that r = 1 and thatinf | x ′ |≤ u = u ( x ′ ) = 0 and L := sup | x ′ |≤ u = sup | x ′ | =1 u = u ( x ′ ) > | x ′ | ≤ | x ′ | = 1. Since u is convex, the set A := { x ′ : u ( x ′ ) ≤ L with | x ′ | < } is convex, {| x ′ | ≤ } ⊂ A and x ′ ∈ ∂A . This implies that u ≥ L on B := { x ′ : h x ′ , x ′ i > | x ′ | < } . Also, theconvexity of u implies that for every for every x ′ ∈ B , | Du ( x ′ ) | ≥ u ( x ′ ) − u ( x ′ ) | x ′ − x ′ | ≥ L . It follows that the normal vectors ν = ( Du, − p | Du | , are contained in C := n v ∈ S n : 0 ≤ h v, − e n +1 i ≤ p L/ o . One can roughly bound | C | ≤ c n L − . On the other hand, note | B | = c ′ n > | ν [ B ] | = Z B K h− ν, e n +1 i dx ′ ≥ c ′ n δ. Since ν [ B ] ⊂ C , we conclude that c ′ n δ ≤ c n L − or L ≤ C n δ − . Recallingthat L := sup | x ′ |≤ u and that inf | x ′ |≤ u = 0, this finishes the proof of the lemma. (cid:3) Lemma 4.2.
Let
Σ = ∂ ˆΣ be a complete C convex hypersurface in R n +1 . Suppose ∈ Σ , h− ν (0) , e n +1 i > ,and that, around the origin, Σ can be represented as a convex graph over a disk D ρ := { x ′ ∈ R n : | x ′ | ≤ ρ } ,for some ρ > . i.e. there is a convex function u : D ρ → R such that { ( x ′ , u ( x ′ )) : x ′ ∈ D ρ } = { ( x ′ , x n +1 ) ∈ Σ : h− ν ( x ) , e n +1 i > and x ′ ∈ D ρ } =: Γ . ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 11
If we further assume that K h− ν, e n +1 i ≥ δ on Γ for some δ > , then there is C = C ( δ, ρ, n ) such that h− ν ( x ) , e n +1 i − ≤ C on { x ∈ Σ : h x, ν ( x ) i ≤ ρ and h− ν, e n +1 i ≥ } . Proof.
We have assumed that u (0) = 0. By Lemma 4.1, u ( x ′ ) = u ( x ′ ) − u (0) ≤ C ′ ρ on {| x ′ − | ≤ ρ } , forsome C ′ ( δ, ρ, n ). Therefore, the ball B ρ (( C ′ + 2) ρ e n +1 ) is located above to Σ ∩ {h− ν, e n +1 i ≥ } . Hencearound this center point x := ( C ′ + 2) ρe n +1 , we have h x − x , ν ( x ) i ≥ ρ , for all x ∈ Σ ∩ {h− ν, e n +1 i ≥ } .It follows that for all x ∈ Σ ∩ {h− ν, e n +1 i ≥ } satisfying h x, ν ( x ) i ≤ ρ , we have2 ρ ≤ h x − x , ν ( x ) i = h x, ν ( x ) i − h x , ν ( x ) i ≤ ρ − ρ ( C ′ + 2) h ν ( x ) , e n +1 i which implies the desired bound C ′ +2 ≤ h− ν ( x ) , e n +1 i . (cid:3) Combining the lemmas above, we have the following result which we need in order to apply our crucialestimate Theorem 4.4 and obtain Corollary 4.5.
Proposition 4.3.
Let
Σ = ∂ ˆΣ ⊂ R n +1 be a C convex hypersurface a part of which is a convex graph x n +1 = u ( x ′ ) on convex domain Ω ⊂ R n . For given x = ( x ′ , u ( x ′ )) ∈ Σ with x ′ ∈ Ω , suppose thatd ( x ′ , ∂ Ω) := 4 ǫ and K h− ν, e n +1 i ≥ δ > on { ( x ′ , u ( x ′ )) : | x ′ − x ′ | ≤ ǫ } . Then { x ∈ Σ : h x − x , ν ( x ) i ≤ ǫ, h− ν ( x ) , e n +1 i ≥ } is compact and, on this set, there is C = C ( δ, ǫ, n ) such that h− ν ( x ) , e n +1 i − ≤ C and | x − x | ≤ C diam(Ω) . Proof.
The first gradient bound follows directly from Lemma 4.1 and 4.2. The second is a consequence thegradient bound. (cid:3)
Next, we show our convexity estimate. The proof is independent of previous propositions, but they willcombined in Corollary 4.5 to give regularity estimates to the weak solutions asymptotic to a cylinder.
Theorem 4.4.
For a given α > , let M t = F ( · , t )( M n ) be a complete smooth strictly convex solution of α -GCF. For F := F ( p , t ) ∈ M t , suppose there exist constants ǫ , δ , L > such that δ ≤ K α ( p, t ) ≤ L and | F ( p, t ) − F | ≤ L on { ( p, t ) ∈ M n × [0 , t ] : h F ( p, t ) − F , ν ( p, t ) i ≤ ǫ } .Then there is C = C ( ǫ, δ, L, α, n ) so that λ − ( p , t ) ≤ C (cid:0) t − (cid:1) . Proof.
We may assume F = F ( p , t ) = 0. Let S := h F, ν i be the support function. Under the α -GCF, by(2.9) and (2.14) we have( ∂ t − Kb rs ∇ rs ) b = − αK α b i b j ( αb kl b mn + b km b ln ) ∇ i h kl ∇ j h mn − αK α Hb + (1 + nα ) K α − K α ≤ − αK α b i b j ( αb kl b mn + b km b ln ) ∇ i h kl ∇ j h mn Define the cut off function η := ( ǫ − S ) + and compute that ( ∂ t − Kb ij ∇ ij ) ln η = ( nα + 1) K α η − αK α HSη + αK α b ij ∇ i η ∇ j ηη . For some β > γ > w := η β b e γ | F | t and apply the maximum principle to bound the maximum of η β λ − e γ | F | t . Suppose that a positive max-imum of η β λ − e γ | F | t on M × [0 , t ] is obtained at ( p ′ , t ′ ). At this point, choose a local coordinate s.t. b ij = λ i δ ij , λ = λ min , and g ij = δ ij at ( p ′ , t ′ ). A direct calculation using (2.17) and (2.16) shows that atthe maximum point ( p ′ , t ′ ) we have0 ≤ ( ∂ t − αK α b ij ∇ ij ) ln w ≤ β (cid:20) ( nα + 1) K α η − αK α HSη + αK α b ij ∇ i η ∇ j ηη (cid:21) − b h αK α b i b j ( αb kl b mn + b km b ln ) ∇ i h kl ∇ j h mn i + αK α b ij ∇ i b ∇ j b ( b ) + 2( nα − γK α S − γαK α b ij g ij + 1 t ′ . Since 0 ∈ M t , note that S ≥ t ≤ t . Moreover, S ≤ ǫ on the support of η . Using these we have C ( L, n, α ) > ≤ ( ∂ t − αK α b ij ∇ ij ) ln w ≤ C ( βη + ǫγ ) − γαK α b ij g ij + 1 t ′ + β αK α b ij ∇ i η ∇ j ηη + 1 b [ − αK α b i b j ( αb kl b mn + b km b ln ) ∇ i h kl ∇ j h mn ] + αK α b ij ∇ i b ∇ j b ( b ) . On the other hand, at this maximum point we have ∇ ln w = β ∇ ηη + ∇ b b + γ ∇| F | = 0and therefore for fixed i (not summing over i )(4.1) β αK α b ii ∇ i η ∇ i ηη = 1 β αK α b ii (cid:18) ∇ i b b + γ ∇ i | x | (cid:19) (cid:18) ∇ i b b + γ ∇ i | x | (cid:19) ≤ β αK α b ii ∇ i b ∇ i b ( b ) + 2 γ β αK α b ii ∇ i | x | ∇ i | x | ≤ β αK α b ii ∇ i b ∇ i b ( b ) + 8(sup | x | ) γ β αK α b ii . We use the above for every i = 1 and plug them into (4.1). Then, there exists C = C ( L, δ, α, n ) such that0 ≤ ( ∂ t − αK α b ij ∇ ij ) ln w ≤ C ( βη + ǫγ ) − (2 γ − | F | ) γ β ) αK α b ii g ii + 1 t ′ + β αK α b ∇ η ∇ ηη + 1 b [ − αK α b b ( αb kl b mn + b km b ln ) ∇ h kl ∇ h mn ] + X i =1 (cid:18) β (cid:19) αK α b ii ∇ i b ∇ j b ( b ) + αK α b ∇ b ∇ b ( b ) . Choosing β = 2 and γ = 14(sup | F | ) , we obtain ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 13 ≤ ( ∂ t − αK α b ij ∇ ij ) ln w ≤ C ( 2 η + ǫγ ) − γαK α b ii g ii + 1 t ′ + 2 αK α b ∇ η ∇ ηη + 1 b [ − αK α b b ( αb kl b mn + b km b ln ) ∇ h kl ∇ h mn ] + X i =1 αK α b ii ( b ) ∇ i h ∇ i h + αK α ( b ) ∇ h ∇ h ≤ C ( 2 η + ǫγ ) − γαK α b ii g ii + 1 t ′ + 2 αK α h h F, ∇ F i η + 1 b [ − αK α b b ( b kk b ll ) ∇ h kl ∇ h kl ] + X i =1 αK α b ii ( b ) ∇ i h ∇ i h + αK α ( b ) ∇ h ∇ h ≤ C ( 2 η + ǫγ ) − γαK α b ii g ii + 1 t ′ + 2 αK α (sup | F | ) 1 η ( b η ) − . Combining the last inequality with the bound K α ≥ δ , we conclude that there is C = C ( ǫ, δ, L, sup | F | , α, n )such that b ≤ C (cid:0) t ′ + 1 η + ( b η ) − η (cid:1) . Note that 0 < t ′ ≤ t , η ≤ ǫ and 1 ≤ e γ | F | ≤ e / . Hence the last bound yields w ( p ′ , t ′ ) = η b e γ | x | t ′ ≤ C (cid:0) t + t w ( p ′ , t ′ ) (cid:1) from which we conclude the bound w ( p ′ , t ′ ) ≤ C t (cid:0) t (cid:1) . The result readily follows from w ( p , t ) := ǫ b ( p , t ) t ≤ w ( p ′ , t ′ ) . (cid:3) Corollary 4.5.
Let Σ t ⊂ R n +1 be a non-compact weak α -GCF asymptotic to Ω × R and x n +1 = u ( x ′ , t ) , bethe graphical representation of Σ t on Ω . Then, for any Ω ′ ⊂⊂ Ω there exists t > and L > such that u i → u in C ∞ (Ω ′ × [ t , ∞ )) , and h− ν, e n +1 i , λ − , λ max ≤ L on ( x ′ , t ) ∈ Ω ′ × [ t , ∞ ) . Proof.
Let us denote 4 ǫ := d (Ω ′ , ∂ Ω) >
0. We also fix an approximating sequence Σ i,t and denote the graphfunctions of Σ i,t ∩ {h− ν, e n +1 i ≥ } by u i ( x, t ).By Proposition 3.3, we can apply Proposition 4.3 and obtain T = T (Σ , Ω ′ , α, n ) with the following: forall T > T there is i so that for every ( x ′ , u i ( x ′ , t )) ∈ Σ i,t with i > i , x ′ ∈ Ω ′ and T ≤ t ≤ T ,(4.2) 1 h− ν ( x ) , e n +1 i ≤ C and | x − x | ≤ C diam (Ω) on { S ( x ′ ,u i ( x ′ ,t )) ( x ) ≤ ǫ } by some C = C (Σ , ǫ, α, n ). On the other hand, Proposition 3.4 gives upper and lower bounds on K α onthe region Σ i,t ∩ {h− ν, e n +1 i − ≤ C } for T ≤ t ≤ T . i.e. we have two bounds of K α on { S x ( x ) ≤ ǫ } for t ∈ [ T , T ] for large i > i . Now we may apply Theorem 4.4 and obtain the bound on λ − at ( x ′ , u i ( x ′ , t ) ∈ Σ i,t when x ′ ∈ Ω ′ and T + 1 / ≤ t ≤ T . The bound on λ max follows from the bounds on λ − and K α . In summary, if i > i , x n +1 = u i ( x ′ , t ) on ( x ′ , t ) ∈ Ω ′ × [ T + 1 / , T ] has uniform bounds on (1 + | Du i | ) / , λ ( x ′ ), and λ n ( x ′ ). One can use standard regularity estimates of uniformly parabolic equation to deducethat u i converges to u in C ∞ sense on the specified domain. (cid:3) Convergence to translating soliton
In this section we give the proof of our main convergence result Theorem 1.1. It will be based on thefollowing monotonicity formula which holds on compact solutions and is shown in Corollary A.4 in theAppendix. Recall the notation P := b ij P ij , where P ij is given by (2.1). Theorem 5.1.
Let Σ t be a smooth compact closed strictly convex solution of the α -GCF with α > . Then (5.1) ddt Z P K α dg = Z ( P ij P kl b ik b jl + (2 α − P ) K α dg ≥ (cid:0) n − + 2 α − (cid:1) ( R P K α dg ) R K α dg . In particular, when α = 1 the last term is n +1 nω n (cid:16)R Σ t P Kdg (cid:17) where ω n = | S n | = R Kdg .Proof.
Shown in Corollary A.4 in the Appendix. (cid:3)
Proposition 5.2.
For α ≥ , let Σ t = ∂ { x n +1 ≥ u ( x ′ , t ) } be a non-compact weak solution to α -GCF whichis asymptotic to Ω × R for some bounded convex domain Ω . Then for every τ > and U ⊂⊂ Ω , lim t →∞ Z t + τt Z { ( x ′ ,u ( x ′ ,s )) : x ′ ∈ U } P K α dg ds = lim t →∞ Z t + τt Z Σ s ∩{ x ′ ∈ U } P K α dg ds = 0 . Proof.
Let us consider an approximating sequence of smooth compact strictly convex solutions (from inside)Σ i,t with an additional assumption that Σ i. has reflection symmetry about { x n +1 = i } . By Corollary 4.5,Σ i,t converges locally smoothly to Σ t when their lower parts are viewed as graphs.The approximation of Σ t by Σ i,t shown above and the positivity of P K α , imply that it suffices to showthe following statement: for given τ > and ǫ > , there is t such that for each t ≥ t , we have (5.2) lim sup i →∞ Z t + τt Z Σ i,t P K α dg ds ≤ ǫ. Claim 5.1.
For any fixed finite time interval [1 , T ] , there is some large i such that K α − ≤ C < ∞ on Σ i,t for i ≥ i , t ∈ [1 , T ] . The constant C only depends on Σ .Proof of Claim. This is by Proposition 3.4 and the symmetry of Σ i,t with respect to { x n +1 = j } . (cid:3) By shifting t = 1 as the initial time we may assume the claim holds from time t = 0. We now continuewith the proof of the proposition, that is the proof of (5.2). The Harnack inequality (3.1) and the bound inthe previous claim yield that for any T >
0, we have(5.3) J ( i ) ( s ) := Z Σ i,s P K α dg ≥ − n nα s Z Σ i,s K α dg ≥ − nω n nα Ct for i ≥ i = i ( T ), and s ∈ [0 , T ].Let us choose t := nω n nα α − Cǫ . We have J ( i ) ( s ) ≥ − α − ǫ , for all t ≤ s ≤ T and i ≥ i = i ( T ). Themonotonicity formula (5.1) gives that ∂ t J ( i ′ ) ( t ) ≥ (2 α − Z Σ ( i ′ ) t P K α dg , for all t > ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 15
If there are t ≥ t and i ′ > i ( T ) ( T > t + τ will be determined later) such that R t + τt R Σ i ′ ,t P K α dg ds > ǫ ,then(5.4) J ( i ′ ) ( t + τ ) = J ( i ′ ) ( t ) + Z t + τt ∂ s J ( i ′ ) ( s ) ds ≥ − α − ǫ + (2 α − Z t + τt Z Σ i ′ ,s P K α dg ds ≥ α − ǫ. From (5.1), we have(5.5) ∂ s J ( i ′ ) ( s ) ≥ (2 α −
1) [ J ( i ′ ) ( s )] R Σ i ′ ,s K α dg ≥ α − ω n [ J ( i ′ ) ( s )] sup Σ i ′ ,s K α − . Under the assumption that K α − ≤ C and J ( i ′ ) ( t + τ ) ≥ α − ǫ , this ODE inequality blows up in finitetime T = T ( ǫ, α, C, n, t + τ ). If we choose this T and then the above argument shows there is no such i ′ > i = i ( T ). (cid:3) When α = 1, we don’t need Claim 5.1 and the previous proof shows the following slightly general version,which won’t be used in this present work. Proposition 5.3.
For any τ > and ǫ > , there is T ( τ, ǫ, n ) > such that the following holds: if x n +1 = u ( x, t ) on ( x, t ) ∈ ¯ U × [ − T, T ] for some bounded ¯ U ⊂ R n is a smooth graphical convex solution to theclassical GCF (with boundary) which is a smooth limit of (parts of ) smooth strictly convex closed solutions,then Z τ − τ Z { ( x,u ( x,s )) : x ∈ U } P K dgds ≤ ǫ. We will next establish the conclusion of Proposition 5.2 for α ∈ (1 / , . Proposition 5.4.
For α ∈ (1 / , , suppose that Σ satisfies the assumptions of Theorem 1.2. Then theconclusion of Proposition 5.2 holds.Proof. By the assumptions, we have approximating compact hypersurfaces Σ i, such that N ( i ) (0) ≤ C and J ( i ) (0) ≥ − C . Since (cid:0) N ( i ) ( t ) (cid:1) α − α is concave in time (by Corollary A.5) and ∂ t N ( i ) ( t ) = ( α − J ( i ) ( t ) (byLemma A.1), we conclude that (cid:0) N ( i ) ( t ) (cid:1) α − α ≤ M + M t for some M = M ( C, α ) >
0. Since 1 − αα <
1, it follows that N ( i ) ( t ) ≤ ( M + M t ) − αα , that is the function N ( i ) ( t ) has sublinear growth rate.By a similar argument to (5.3), J ( i ) ( t ) ≥ − n nα t Z Σ i,t K α dg ≥ − n nα ( M + M t ) − αα t . Hence there is t = t ( n, α, ǫ, M ) such that J ( i ) ( t ) ≥ − α − ǫ for all t ≥ t .If there exists t ≥ t and i for which(5.6) Z t + τt Z Σ i,t P K α dg ds > ǫ, then by the argument of (5.4) we have J ( i ) ( t + τ ) ≥ α − ǫ .From (5.5), we derive the following ODE inequality ∂ s J ( i ) ( s ) ≥ (2 α −
1) [ J ( i ) ( s )] ( M + M s ) − αα . By the sublinear growth of the denominator, it can be checked that the ODE blows up in finite time T = T ( ǫ, α, M, t + τ ) if J ( i ) ( t + τ ) ≥ α − ǫ > . Therefore we have the opposite inequality of (5.6) if i issufficiently large so that the maximum existence time T i of Σ i,t satisfies T i ≥ T . (cid:3) Next lemma shows that the α -GCF solution with P ≡ Lemma 5.5.
Let F : M n × [ − ǫ, ǫ ] → R n +1 be a strictly convex smooth immersion which satisfies P = 1 αK α ( ∂ t K α − b ij ∇ i K α ∇ j K α ) ≡ . Then F ( M n , has to be a translating soliton.Proof. We observe first that for such a solution the evolution of P in (A.7) implies that P ij ≡
0. Let usdefine T := b ij ∇ i K α ∇ j F + K α ν. Then ∇ m T = ∇ m b ij ∇ i K α ∇ j F + b ij ∇ im K α ∇ j F + b ij ∇ i K α ( − h jm ν ) + ∇ m K α ν + K α h mj ∇ j F. Using 0 = P im = ∇ im K α − b kl ∇ k h im ∇ l K α + K α h ik h km , we get ∇ m T = − b ik b jl ∇ m h kl ∇ i K α ∇ j F + b ij ∇ j F ( b kl ∇ k h im ∇ l K α − K α h ik h km ) + K α h mj ∇ j F = 0 . Namely, T is a constant vector. Note that h T, ν i = K α and this shows F ( M n ,
0) is a translating soliton witha velocity − T . (cid:3) Proof of Theorem 1.1.
In view of Corollary 4.5 and standard parabolic regularity theory, for any given τ i → ∞ , we may take a further subsequence (which we still denote by τ i ) so that u ( x ′ , t + τ i ) − inf Ω u ( x ′ , τ i ) → u ∞ ( x ′ , t ) in C ∞ loc (Ω × ( −∞ , ∞ )) . By Proposition 5.2 and Lemma 5.5, x n +1 = u ∞ ( x ′ , t ) on Ω × ( −∞ , ∞ ) has to be a (possibly incomplete)translating soliton. It suffices to show this is actually the unique translating soliton defined on Ω. i.e. u ∞ ( x ′ , ≡ u Ω ( x ′ ).Let us denote u ∞ , := u ∞ ( · , λ e n +1 .i.e. K α = λ h− ν, e n +1 i ⇐⇒ " det D u ∞ , (1 + | Du ∞ , | ) n +22 α = λ (1 + | Du ∞ , | ) − / on Ω . This implies(5.7) λ /α | Ω | = Z Ω det D u ∞ , (cid:16)p | Du ∞ , | (cid:17) n +2 − α = Z Du ∞ , (Ω) p | p | ) n +2 − α ≤ Z R n p | p | ) n +2 − α =: Λ( n, α ) < ∞ provided α > . ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 17
Note the equality holds if and only if | R n − Du ∞ , (Ω) | = 0. i.e. when u ∞ , = u Ω ; (see the characterizationof u Ω which is discussed after Definition 2.2).Assume wihtout loss of generality that Ω contains the origin. Since we can apply the previous argumentfor every subsequence of the sequence τ i , this implies(5.8) lim sup s →∞ u t (0 , s ) ≤ (cid:18) Λ( n, α ) | Ω | (cid:19) α =: λ Ω . In view of the argument in the first paragraph, we can always find a converging subsequence. Thus it sufficesto show lim inf s →∞ u t (0 , s ) ≥ λ Ω . On the contrary, suppose there is a sequence of time τ i → ∞ such that u t (0 , τ i ) ≤ λ Ω (1 − δ ) for some δ > c > u t (0 , s ) ≤ λ Ω (1 − δ ) on s ∈ [(1 − c ) τ i , τ i ]. By (5.8),for every fixed ǫ > u (0 , (1 − c ) τ i ) ≤ (1 + ǫ ) λ Ω (1 − c ) τ i + O (1) as i → ∞ . Thus u (0 , τ i ) ≤ u (0 , (1 − c ) τ i ) + λ Ω (1 − δ ) cτ i ≤ (1 + ( ǫ (1 − c ) − δc )) λ Ω τ i + O (1) . Choosing ǫ := cδ − c , we get u (0 , τ i ) ≤ (1 − δc ) τ i + O (1) as i → ∞ . On the other hand, by considering¯ u ( x ′ ) := (1 − δc ) − nα u Ω ((1 − δc ) nα x ′ ) = u (1 − δc ) − nα Ω ( x ′ )and use ¯ u ( · ) − L for large L as an initial barrier as we did in Proposition 3.3, we get u (0 , t ) ≥ (1 − δc ) t − O (1) as t → ∞ . This is a contradiction and finishes the proof. (cid:3)
Proof of Theorem 1.2.
In this case, we assume that N (0) := Z Σ K α dg ≤ C and also that J (0) := Z Σ t P ij b ij K α dg ≥ − C for some C < ∞ , with P ij := ∇ ij K α − b mn ∇ m h ij ∇ n K α + K α h ki h kj . Note that for compact solutions LemmaA.1 gives ∂ t N = ( α − J and hence this assumption corresponds to giving upper bounds on N and itsfirst time derivative at t = 0. Then the proof goes same as the proof of Theorem 1.1. Except that we useProposition 5.4 instead of Proposition 5.2. (cid:3) Appendix A. Monotonicity formula
Let F : M n × [0 , T ] → R n +1 be a parametrization of a smooth strictly convex closed solution Σ t of the α -GCF. We define the entropies(A.1) N ( t ) := Z Σ t K α dg and(A.2) J ( t ) := Z Σ t P ij b ij K α dg where(A.3) P ij := ∇ ij K α − b mn ∇ m h ij ∇ n K α + K α h ki h kj . Here, dg := √ det g dx is the intrinsic volume form inherited from the imbedding F . In this section we willsummarize and prove certain entropy identities and inequalities which are used in this work. Lemma A.1. (A.4) ddt N ( t ) = ( α − J ( t ) . Proof.
By equation (2.15) and (2.10),(A.5) ddt ( K α dg ) = ddt ( K α ) dg + K α ddt dg = (cid:0) αK α b ij ∇ ij K α + αHK α − HK α (cid:1) dg = (cid:0) α b ij ∇ ij K α + ( α − HK α (cid:1) K α dg. Hence ddt N = Z Σ t (cid:0) α b ij ∇ ij K α + ( α − HK α (cid:1) K α dg. Using the following integration by parts Z K α b ij ∇ ij K α dg = Z K α − Kb ij ∇ ij K α dg = − Z b ij K ∇ i K α − ∇ j K α dg (cid:0) by eq (2.7) (cid:1) = − Z α − α b ij ∇ i K α ∇ j K α dg we conclude the desired identity ddt N ( t ) = ( α − Z Σ t (cid:16) b ij ∇ ij K α − b ij αK α ∇ i K α ∇ j K α + HK α (cid:17) K α dg = ( α − J ( t ) . (cid:3) Theorem A.2.
We have ddt J ( t ) = Z Σ (cid:0) b ik b jl P ij P kl + (2 α − b ij P ij ) (cid:1) K α dg. Remark
A.3 . Note that(A.6) b ij P ij = b ij ∇ ij K α − b ij αK α ∇ i K α ∇ j K α + K α H which follows from (2.6). Proof of Theorem.
The evolution of b kl P kl , shown in Theorem 3.2 [17], is given by(A.7) ddt ( b kl P kl ) = αK α b ij ∇ ij (cid:0) b kl P kl ) + 2 αb ij ∇ i K α ∇ j ( b kl P kl ) + b ik b jl P ij P kl + α ( b kl P kl (cid:1) . By this and equation (A.5), we get(A.8) ddt J = Z ddt ( b kl P kl ) K α dg + Z (cid:0) b kl P kl (cid:1) (cid:0) αb ij ∇ ij K α + ( α − HK α (cid:1) K α dg = Z (cid:0) b ik b jl P ij P kl + α ( b kl P kl ) (cid:1) K α dg + I ONVERGENCE OF GAUSS CURVATURE FLOWS TO TRANSLATING SOLITONS 19 where(A.9) I := Z (cid:16) αK α b ij ∇ ij ( b kl P kl ) + 2 αb ij ∇ i K α ∇ j ( b kl P kl )+ α ( b ij ∇ ij K α )( b kl P kl ) + ( α − b kl P kl ) HK α (cid:17) K α dg. To finish the proof of the theorem it suffices to show that I = ( α − R ( b kl P kl ) K α dg . Note that for anytwo functions F and G we have the following integration by parts formula:(A.10) Z (cid:0) ∇ ij G (cid:1) (cid:0) b ij F K α (cid:1) dg = − Z ∇ j G ∇ i (cid:0) b ij F K α (cid:1) dg = − Z (cid:16) ∇ j G ∇ i (cid:0) ( b ij K ) F K α − (cid:1) dg = − Z b ij ∇ j G ∇ i F K α + F Kb ij ∇ j G ∇ i K α − dg (by ∇ i ( b ij K ) = 0) = − Z b ij ∇ j G ∇ i F K α dg − α − α Z F b ij ∇ j G ∇ i K α dg. Applying formula (A.10) with F := αK α and G := b kl P kl we obtain Z ∇ ij ( b kl P kl ) (cid:0) αK α b ij (cid:1) K α dg = ( − α + 1) Z b ij ∇ i K α ∇ j ( b kl P kl ) K α dg. Hence,
Z (cid:0) αK α b ij ∇ ij ( b kl P kl )+2 αb ij ∇ i K α ∇ j ( b kl P kl ) (cid:1) K α dg = Z b ij ∇ i K α ∇ j ( b kl P kl ) K α dg (by eq (A.10)) = Z (cid:16) − ( b kl P kl )( b ij ∇ ij K α ) K α − α − α ( b kl P kl ) b ij ∇ i K α ∇ j K α (cid:17) dg. Plugging this into (A.9), yields I = Z (cid:16) αK α b ij ∇ ij ( b kl P kl ) + 2 αb ij ∇ i K α ∇ j ( b kl P kl )+ α ( b ij ∇ ij K α )( b kl P kl ) + ( α − b kl P kl ) HK α (cid:17) K α dg = Z ( α − b ij ∇ ij K α − αK α b ij ∇ i K α ∇ j K α + HK α ]( b kl P kl ) K α dg (cid:0) by (A.6) (cid:1) = ( α − Z ( b kl P kl ) K α dg. This finishes the proof of the theorem. (cid:3)
Corollary A.4.
For α ≥ n − n , we have ddt Z ( b ij P ij ) K α dg ≥ (cid:0) n + 2 α − (cid:1) Z ( b ij P ij ) K α dg ≥ (cid:0) n + 2 α − (cid:1) (cid:0) R ( b ij P ij ) K α dg (cid:1) R K α dg ≥ . Proof.
The α = 1 case is proven in Lemma 4.3 [17]. In the more general case, the result readily follows bythe previous Theorem, the inequality b ik b jl P ij P kl ≥ n ( b ij P ij )
20 BEOMJUN CHOI, KYEONGSU CHOI, AND PANAGIOTA DASKALOPOULOS and H¨older’s inequality. (cid:3)
Corollary A.5.
For α > with α = 1 , we have d dt N α − α ≤ . Proof.
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Acknowledgements
K. Choi has been partially supported by NSF grant DMS-1811267.P. Daskalopoulos and B. Choi have been partially supported by NSF grant DMS-1600658.
Beomjun Choi:
Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON M5S2E4, Canada.
Email address : [email protected] Kyeongsu Choi:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave,Cambridge, MA 02139, USA. & Korea Institute for Advanced Study, 85 Hoegiro,Dongdaemun-gu, Seoul 02455,Republic of Korea.
Email address : [email protected] Panagiota Daskalopoulos:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY10027, USA.
Email address ::