CYLINDRICAL ESTIMATES FOR HYPERSURFACES MOVING BY CONVEX CURVATURE FUNCTIONS

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1 CYLINDRICAL ESTIMATES OR HYPERSURACES MOVING BY CONVEX CURVATURE UNCTIONS BEN ANDREWS AND MAT LANGORD Abstract. We prove a complete amily o cylindrical estimates or solutions o a class o ully non-linear curvature lows, generalising the cylindrical estimate o Huisken-Sinestrari [HS09, Section 5] or the mean curvature low. More precisely, we show that, or the class o lows considered, an m + 1-convex 0 m n 2 solution becomes either strictly m-convex, or its Weingarten map approaches that o a cylinder R m S n m at points where the curvature is becoming large. This result complements the convexity estimate proved in [ALM13] or the same class o lows. 1. Introduction Let M be a smooth, closed maniold o dimension n, and X 0 : M R n+1 a smooth hypersurace immersion. We are interested in smooth amilies X : M [0, T R n+1 o smooth immersions X, t solving the initial value problem { t Xx, t = Wx, tνx, t, C X, 0 = X 0. where ν is the outer normal ield o the evolving hypersurace X and W the corresponding Weingarten curvature. In order that the problem C be well posed, we require that W be given by a smooth, symmetric, degree one homogeneous unction : Γ R o the principal curvatures κ i which is monotone increasing in each argument. The symmetry o ensures that is a smooth, basis invariant unction o the components o the Weingarten map or an orthonormal rame invariant unction o the components o the second undamental orm [Gl63]. Monotonicity implies monotonicity with respect to the Weingarten curvature, which ensures that the low is weakly parabolic. This guarantees local existence o solutions o C, as long as the principal curvature n-tuple o the initial data lies in Γ [Ba, Main Theorem 5]. or technical reasons, we require the ollowing additional conditions Conditions. i that Γ is a convex cone 1, and is homogeneous o degree one; and ii that is convex Mathematics Subject Classiication. 53C44, 35K55, 58J35. This research was partially supported by Discovery grant DP o the Australian Research Council. The second author grateully acknowledges the support o an Australian Postgraduate Award during the completion o this work. 1 We remark that this condition can be slightly weakened. See [ALM13]. 1

2 2 BEN ANDREWS AND MAT LANGORD Then, since the normal points out o the region enclosed by the solution, we may assume that 1,..., 1 Γ, and we we lose no generality in assuming that is normalised such that 1,..., 1 = 1. The additional conditions i-ii have several consequences. Most importantly, they allow us to obtain a preserved cone o curvatures or the low [ALM13, Lemma 2.4]. This allows us to obtain uniorm estimates on any degree ero homogeneous unction o curvature along the low Lemma 2.2; in particular, we deduce uniorm parabolicity o the low Corollary 2.3. The convexity condition then allows us to apply the second derivative Hölder estimate o Evans [Ev82] and Krylov [Kr82] to deduce that the solution exists on maximal time interval [0, T, T <, such that max M {t} as t T, as in [ALM, Proposition 2.6]. This paper addresses the behaviour o solutions as. Let us recall the ollowing curvature estmate [ALM13] c. [HS99a, HS99b]: Theorem 1.1 Convexity Estimate. Let X : M [0, T R n+1 be a solution o C such that satisies Conditions i ii. Then or all ε > 0 there is a constant C ε < such that Gx, t ε x, t + C ε or all x, t M [0, T, where G is given by a smooth, non-negative, degree one homogeneous unction o the principal curvatures o the evolving hypersurace that vanishes at a point x, t i and only i W x,t 0. Theorem 1.1 implies that the ratio o the smallest principal curvature to the speed is almost positive wherever the curvature is large. Combining it with the dierential Harnack inequality o [An94b] and the strong maximum principle [Ha84] yields useul inormation about the geometry o solutions o C near singularities [ALM13] c. [HS99a, HS99b]: Corollary 1.2. Any blow-up limit o a solution o C is weakly convex. In particular, any type-ii blow-up limit o a solution o C about a type-ii singularity is a translation solution o C o the orm X : R k Γ n k R R n+1, k {0, 1,..., n 1}, such that X Γ n k is a strictly convex translation solution o C in R n k+1. Motivated by [HS09, Section 5], we apply Theorem 1.1 to obtain the ollowing amily o cylindrical estimates or solutions o C: Theorem 1.3 Cylindrical Estimate. Let X be a solution o C such that Conditions i ii hold. Suppose also that X is m + 1-convex or some m {0, 1,..., n 2}. That is, κ κ β or some β > 0. Then or all ε > 0 there is a constant C ε > 0 such that G m x, t ε x, t + C ε or all x, t M [0, T, where G m : M [0, T R is given by a smooth, non-negative, degree one homogeneous unction o the principal curvatures that vanishes at a point x, t i and only i κ 1 x, t + + κ x, t 1 c m κ 1 x, t,..., κ n x, t, where c m is the value takes on the unit radius cylinder, R m S n m. Theorem 1.3 implies that the ratio o the quantity K m := κ κ 1 c m to the speed is almost positive wherever the curvature is large. Observe that this quantity is non-negative on a weakly convex hypersurace Σ only i either Σ is strictly m convex,

3 CYLINDRICAL ESTIMATES 3 or Σ = R m S n m. In particular, we ind that whenever κ 1 x, t + + κ m x, t is small compared to the speed, the Weingarten curvature is close to that o a thin cylinder R m S n m. We obtain the ollowing reinement o Corollary 1.2: Corollary 1.4. Any blow-up limit o an m + 1-convex, 0 m n 2, solution o C is either strictly m-convex, or a shrinking cylinder R m S n m. In particular, i the blow-up is o type-ii, then this limit is a translation solution o C o the orm X : R k Γ n k R R n+1 or k {0, 1,..., m 1}, such that X Γ n k is a strictly convex translation solution o C in R n k+1. Huisken-Sinestrari obtained Theorem 1.3 or the mean curvature low in the case m = 1, making spectacular use o it through their surgery program [HS09], which yielded a classiication o 2-convex hypersuraces. Moreover, the m = 0 case produces an analogue o Huisken s curvature estimate or convex solutions o the mean curvature low [Hu84, Theorem 5.1]. This estimate implies that a convex solution o C becomes round at points o large curvature, which is crucial in proving that solutions contract to round points. This result was proved by dierent means or the class o lows considered here [An94a]. 2. Preliminaries We will ollow the notation used in [ALM13]. In particular, we recall that a smooth, symmetric unction g o the principal curvatures give rise to a smooth unction G o the components o the Weingarten map. Working in an orthonormal rame o eigenvectors o W, we may consider G to be a unction o the components h ij o the second undamental orm. To simpliy notation, we denote Gx, t G hx, t = gκx, t and use dots to denote derivatives o unctions o curvature as ollows: ġ k v k = d ds g + sv Ġ kl AB kl = d s=0 ds GA + sb s=0 g pq v p v q = d2 ds 2 g + sv Gpq,rs AB pq B rs = d2 s=0 ds 2 GA + sb. s=0 The derivatives o g and G are related in the ollowing way c. [Ge90, An94a, An07]: Lemma 2.1. Let g : Γ R be a smooth, symmetric unction. Deine the unction G : S Γ : R by GA := gλa, where λa denotes the eigenvalues o A up to order. Then or any diagonal A with eigenvalues in Γ we have Ġ kl A = ġ k λaδ kl, 2.1 and or any diagonal A with distinct eigenvalues lying in Γ, and any symmetric B GLn, we have G pq,rs AB pq B rs = g pq λab pp B qq + 2 ġ p λa ġ q λa 2 Bpq. 2.2 λ p>q p A λ q A In particular, in an orthonormal rame o eigenvectors o W we have Ġ kl =ġ k δ kl G pq,rs B pq B rs = g pq B pp B qq + 2 p>q ġ p ġ q κ p κ q Bpq 2,

4 4 BEN ANDREWS AND MAT LANGORD where we are denoting Ġ Ġ h, etc. We note that g 0 i and only i ġ p ġ q p q 0 [ALM13, Lemma 2.2], so Lemma 2.1 implies that G is convex i and only i g is convex. Lemma 2.2. Let X : M [0, T R n+1 be a solution o C such that satisies Conditions i ii. Let g : Γ R be a smooth, degree ero homogeneous symmetric unction. Then there exists c > 0 such that or all x, t M [0, T. I g > 0, then there exists c > 0 such that c g κ 1 x, t,..., κ n x, t c. 1 c g κ 1x, t,..., κ n x, t c. Proo. Let Γ 0 be a preserved cone or the solution X. Then K := Γ 0 S n is compact. Since g is continuous, the required bounds hold on K. But these extend to Γ 0 \ {0} by homogeneity. The claim ollows since κx, t Γ 0 \ {0} or all x, t M [0, T. By Condition i, the derivative o is homogeneous o degree ero. Since k > 0 or each k, we obtain uniorm parabolicity o the low: Corollary 2.3. There exists a constant c > 0 such that or any v T M it holds that 1 c v 2 ij v i v j c v 2, where is the time-dependent norm on M corresponding to the time-dependent metric induced by the low. We now recall the ollowing evolution equation see or example [AMZ13]: Lemma 2.4. Let X : M [0, T R n+1 be a solution o C such that satisies Conditions i ii. Let G : M [0, T R be given by a smooth, symmetric, degree one homogeneous unction g o the principal curvatures. Then G satisies the ollowing evolution equation: t LG = Ġkl pq,rs kl Gpq,rs h pq h rs + G W where L := kl k l is the linearisation o, and W 2 := kl h k r h rl. In particular, the speed unction satisies t L = W 2. As we shall see, in order to obtain Theorem 1.3, it is crucial to obtain a good lower bound on the term Q W, W := Ġkl pq,rs kl Gpq,rs k h pq l h rs or the pinching unctions G m which we construct in the ollowing section. The ollowing decomposition o Q is crucial in obtaining this bound.

5 CYLINDRICAL ESTIMATES 5 Lemma 2.5. or any totally symmetric T R n R n R n, we have Ġkl pq,rs kl Gpq,rs B T kpq T lrs = ġ k pq k g pq T kpp T kqq + 2 p ġ q ġ p q T pqq 2 + T qpp 2 p>q p q + 2 g kpq kpq kpq T kpq k>p>q at any diagonal matrix B with distinct eigenvalues i, where and are the three dimensional cross and dot product respectively, and we have deined the vectors kpq := k, p, q, g kpq := ġ k, ġ p, ġ q, and kpq := p q k p k q, k q k p p q, k p. p q k q Proo. Since B is diagonal, Lemma 2.1 yields supressing the dependence on B Ġkl pq,rs kl Gpq,rs T kpq T lrs = k,p,qġ k pq k g pq T kpp T kqq + 2 k ġ k p q p k ġp ġ q T kpq 2. q p q p>q

6 6 BEN ANDREWS AND MAT LANGORD We now decompose the second term into the terms satisying k = p, k = q, k > p, p > k > q, and q > k respectively: ġ k p q k p>q p k ġp ġ q T kpq 2 q p q = ġ p p q p>q p p ġp ġ q T ppq 2 + ġ q p q q p q p>q p q ġp ġ q T qpq 2 q p q ġ k p q p k ġp ġ q T kpq 2 q p q k>p>q p>k>q p>q>k = p ġ q ġ p q T pqq 2 + T qpp 2 + ġ k p p>q p q p k ġp ġ q q p q k>p>q + ġ p k k p ġk ġ q + ġ q k p q k q k q ġk ġ p T kpq 2 p k p = p ġ q ġ p q T pqq 2 + T qpp 2 + ġ p q p>q p 1 q ġ p 1 q k p k q k>p>q ġ k q 1 k ġ q ġ k p p q k 1 k ġ p 1 T kpq 2 p p q k q = p ġ q ġ p q T pqq 2 + T qpp 2 + g kpq p>q p kpq kpq T kpq 2. q k>p>q We complete this section by proving that m + 1-convexity is preserved by the low C, so that this assumption need only be made on initial data: Proposition 2.6. Let X be a solution o C such that Conditions i ii are satisied. Suppose that there is some m {1,..., n 1} and some β > 0 such that κ σ1 x, κ σm x, 0 β x, 0 or all x M and all permutations σ P n. Then this estimate persists at all later times. Proo. Denote by SM the unit tangent bundle over M [0, T and consider the unction Z deined on m SM by m Zx, t, ξ 1,... ξ m = hξ α, ξ α β x, t. Since we have α=1 in Zx, t, ξ 1,..., ξ m = κ σ1 x, t + + κ σm x, t β x, t ξ 1,...,ξ m S x,t M or some σ P n, it suices to show that Z remains non-negative. irst ix any t 1 [0, T and consider the unction Z ε x, t, ξ 1,... ξ m := Zx, t, ξ 1,... ξ m + εe 1+Ct, where C := sup M [0,t1 ] W 2. Note that C is inite since M is compact and is bounded. Observe

7 CYLINDRICAL ESTIMATES 7 that Z ε is positive when t = 0. We will show that Z ε remains positive on M [0, t 1 ] or all ε > 0. So suppose to the contrary that Z ε vanishes at some point x 0, t 0, ξ 0 1,... ξ0 m. We may assume that t 0 is the irst such time. Now extend the vector ξ 0 := ξ 0 1,... ξ0 m to a ield ξ := ξ 1,..., ξ n near x 0, t 0 parallel translation in space and solving ξ i α t = ξj αh j i. Then the resulting ields have unit length. Now recall see or example [ALM13] the ollowing evolution equation t h ij = Lh ij + pq,rs i h pq j h rs + W 2 h ij 2 h 2 ij, where L := kl k l and W 2 := kl h 2 kl. It ollows that m t L Z ε x, t, ξ = ε1 + Ce 1+Ct + pq,rs ξα h pq ξα h rs + Wx, t 2 Zx, t, ξ α=1 ε1 + Ce 1+Ct + Wx, t 2 Zx, t, ξ. Since the point x 0, t 0, ξ t=t0 is a minimum o Z ε, we obtain 0 t L x0,t 0 Z εx, t, ξ ε1 + Ce 1+Ct 0 Cεe 1+Ct 0 = εe 1+Ct 0 > 0. This is a contradiction, implying that Z ε cannot vanish at any time in the interval [0, t 1 ]. Since ε > 0 was arbitrary, we ind Z 0 at all times in the interval [0, t 1 ]. Since t 1 [0, T was arbitrary, we obtain Z Constructing the pinching unction. In this section we construct the pinching unctions G m satisying the conditions in Theorem 1.3. Let us irst introduce the pinching cones Γ m := { Γ : σ1 + + σ > c 1 m or all σ P n }, where P n is the group o permutations o the set {1,..., n}. Using the methods o [Hu84], and their adaptations to two-convex lows in [HS09] and ully non-linear lows in [ALM13], we will show that, in order to prove Theorem 1.3, it suices to construct a smooth unction g m : Γ R satisying the ollowing properties: Properties. i g m 0 or all Γ with equality i and only i Γ m Γ; ii g m is smooth an homogeneous o degree one; iii or every ε > 0 there exists c ε > 0 such that or all diagonal matrices B and totally symmetric 3-tensors T, it holds that m pq,rs kl Gpq,rs m T 2 B T kpq T lrs c ε Ġkl whenever G m B ε B, where G m is the matrix unction corresponding to g m as described in Section 2; and

8 8 BEN ANDREWS AND MAT LANGORD iv or every δ > 0, ε > 0, and C > 0 there exist γ ε > 0 and γ δ > 0 such that G m kl Ġkl m B B 2 kl γ ε 2 G m δ B + γ δ 2 B or all B {G m B ε B} { 1 δ B C i}, where 1 is the smallest eigenvalue o B. Our construction o the pinching unction g m will be independent o the choice o m. So let us ix m {0, 1,..., n 2} and assume that the low is m + 1-convex. We irst consider the preliminary unction g : Γ R deined by g := σi 1 c m, 3.1 ϕ where ϕ : R R is a smooth 2 unction which is strictly convex and positive, except on R + {0}, where it vanishes identically. Such a unction is readily constructed; or example, we could take ϕr = { r 4 e 1 r 2 i r < 0 0 i r 0. We note that such a unction necessarily satisies ϕr rϕ r 0 and ϕ r 0 with equality i and only i r 0. Now deine the scalar G : M [0, T R by Gx, t := gκ 1 x, t,..., κ n x, t. Then G is a smooth, degree one homogeneous unction o the components o the Weingarten map which is invariant under a change o basis. Moreover, G is non-negative and vanishes at, and only at, points or which the sum o the smallest m + 1-principal curvatures is not less than c 1 m. Thus Properties i and ii are satisied by g. We now show that property iii is satisied weakly by g: Lemma 3.1. Let G be the matrix unction corresponding to the unction g deined by 3.1. Then or any diagonal matrix B and totally symmetric 3-tensor T, it holds that Ġkl pq,rs kl Gpq,rs B T kpq T lrs 0 Proo. We will show that each o the terms in the decomposition 2.4 in Lemma 2.5 is nonpositive. Note that it suices to compute at matrices having distinct eigenvalues, since the result at a general matrix B may be obtained by taking a limit B k B such that each matrix B k has distinct eigenvalues. We irst compute, ġ k = g k + δ k σi σi k, g pq = g ϕ r σ ϕ r σ + ϕ r σ σi pq δ σi p σi p δ σi q σi q, 2 In act, ϕ need only be twice continuously derentiable.

9 where we are denoting r σ := ġ k pq k g pq = ϕ r σ CYLINDRICAL ESTIMATES 9 σi c 1 m k ϕ r σ δ σi k pq. It ollows that δ σi p σi = k ϕ r σ 0. p δ σi q σi δ σi p σi q. I we ix the index k and set ξ p = T kpp, then, by convexity o ϕ and positivity o k, we have k ϕ r σ δ p σi σi p δ q σi σi q ξ p ξ q 2 p ξ p On the other hand, since ϕ is monotone non-increasing, and is convex, ϕ k r σ δ σi pq ξ p ξ q 0. Since both inequalities hold or all k, we deduce that ġk pq k g pq T kpp T kqq 0. We next consider p ġ q ġ p p q q k,p,q = q ϕ δ σi p p δ σi q r σ p q ϕ p r σ q i m + 1 p > q p q = ϕ p r σ i p > m + 1 q p q 0 i p > q > m + 1. We may suppose that the eigenvalues o B are ordered such that 1 < < n, in which case each o the above terms is non-positive. Thereore, p ġ q ġ p q T pqq 2 + T qpp 2 0. p q Now consider g kpq = g p>q ϕ r σ σi kpq + ϕ r σ δσi k, δ σi p, δ σi q.

10 10 BEN ANDREWS AND MAT LANGORD Then where and g kpq kpq kpq = ϕ r σ = ϕ r σ [ δσi k, δ σi p, δ σi q kpq ] kpq [ δσi p q δ σi q p p q k p k q + δ σi q k k δ σi q k q k p p q + δ ] σi k p p δ σi k k p k q p q S 1 i m + 1 k > p > q = ϕ S 2 i k > m + 1 p > q r σ S 3 i k > p > m + 1 q 0 i k > p > q > m + 1, S 1 := p p q k p k q k k q k p p q + k p k p k q p q, S 2 := p p q k p k q k k q k p p q + k k p k q p q, p p q S 3 := k p k q k k q k p p q. We now demonstrate the non-positivity o each o the terms S 1, S 2 and S 3, completing the proo o the proposition. Modulo the positive actor [ k p k q p q ] 1, we have S 1 p q p q 2 k q k q 2 + k p k p 2 = k p k p 2 k q 2 + p q p q 2 k q 2. This is non-positive, since k p q. This covers the irst case. We next consider S 2 p p q 2 + k k p 2 k q 2 = p p q 2 k p q k q + k p p p q 2 k p q 2, which is again non-positive, since k p q, and k p q by convexity o. Thus the term o the second case is also non-positive. inally, we consider S 3 p p q 2 k k q 2 p k k q 2, which is non-positive since k p. This implies that the term o the third case is also non-positive, completing the proo o the proposition.

11 CYLINDRICAL ESTIMATES 11 In order to obtain the uniorm estimate required by property iii, we modiy G in order to obtain a unction with a strictly positive term in Q. A well-known trick c. [HS99b, Theorem 2.14], [ALM13, Lemma 3.3] then allows us to extract the required uniorm estimate. irst, we relabel the preliminary pinching untion g g 1 G G 1, and consider the new pinching unction g deined by: g := Kg 1, g 2 := g2 1, 3.2 g 2 where g 2 = M n i or some large constant M >> 1, or which g 2 is positive along the low. That there is such a constant ollows rom applying the maximum principle to the evolution equation 2.3 or the unction G 2 x, t := g 2 κx, t as in [ALM13, Lemma 3.1]. Note that K 1 > 0, K 2 < 0 and K > 0 wherever g 1 > 0. Observe that Properties i and ii are not harmed in the transition rom g 1 to g. We now show that the estimates listed in Properties iii and iv are satisied by the curvature unction deined in 3.2. Proposition 3.2. Let g be the pinching unction deined by 3.2 and G its corresponding matrix unction. Then, or every ε > 0 there exists c ε > 0 such that or all diagonal matrices B and totally symmetric 3-tensors T, it holds that Ġkl pq,rs kl Gpq,rs B T kpq T lrs c ε T 2 whenever GB ε B. Proo. irst note that supressing dependence on B Ġkl pq,rs kl Gpq,rs α T kpq T lrs = K Ġkl α pq,rs kl Kαβ Ġ pq α kl Gpq,rs α T kpq T lrs Ġrs β T kpqt lrs 2 K Ġkl 2 pq,rs kl Gpq,rs 2 T kpq T lrs K 2 kl Gpq,rs 2 T kpq T lrs, where we used Lemma 3.1, convexity o K, and the inequalities K 1 0 and 0 in the irst inequality, and the inequalities Ġ2 0 and K 2 0, and convexity o in the second. Since K 2 < 0 whenever G 1 > 0 and G 2 is strictly concave in non-radial directions, the claim ollows rom a well-known trick, exactly as in [ALM13, Lemma 3.3]. The estimate we obtain rom the previous lemma or the term Q W, W in the evolution equation or the pinching unctions G is a crucial component in obtaining the L p -estimates o the ollwing section. This is the starting point or the Stampacchia-De Giorgi iteration argument. The second crucial estimate is the Poincaré-type inequality see sections 4 and 5 o [HS09]; in particular, Lemma 5.5, which we can obtain with the help o property iv. This estimate corresponding to Lemma 5.2 o [HS09] provides an estimate on the ero order term that occurs in contracting the Simons-type identity or pq p q h ij with Ġij c. [ALM13, Proposition 4.4]. Proposition 3.3. Let g be the pinching unction deined by 3.2 and G its corresponding matrix unction. Then, or every δ > 0, ε > 0, and C > 0 there exist γ ε > 0 and γ δ > 0 such that G m kl Ġkl m B Bkl 2 γ ε 2 G m δ B + γ δ 2 B

12 12 BEN ANDREWS AND MAT LANGORD or all B {G m B ε B} { i δ B C or all i}, where i are the eigenvalues o B. Proo. Let B be a symmetric matrix with eigenvalues 1 n. Deine ZB := ĠB2 G B 2. Then ZB = ġ p p 2 g p p 2 = ġp q ġ q p p q p q p>q = ϕ r σ p p q p q + p>q l ϕ r σ p p q p q + q=1 p=l+1 p> q p> q p p q p q p p q p q, where we have deined l m n 2 as the number o non-positive eigenvalues i. Now, observe that the bound on G/ implies an upper bound on ϕr along the low, and thereby an upper bound on ϕ r. Since each o the derivatives k are bounded along the low we have, in particular, upper and lower bounds or p. This yields the ollowing simpliication o the estimate: ZB c p=l+1 q=1 l p q p q + p> q p q p q 3.3 or some c > 0 independent o δ and ε. We shall denote positive constants which depend neither on ε nor δ generically by the letter c. We control the irst sum using the convexity estimate 1 δ C as ollows: p=l+1 q=1 l p q p q m + 1 l 2m + 1 lc 2 2 l q q 3.4 q=1 l q=1 q 2m + 1 lc 2 2 δ + C c 2 δ + C, 3.5 where we estimated c i c or each i with c independent o δ and ε.

13 CYLINDRICAL ESTIMATES 13 Recall m n 2. Then we may decompose the second term in the brackets on the right hand side o 3.3 as p> q p q p q = = n p=m+2 q=1 n p=m+2 q=l+1 l + 2 k + k=1 p q p q p q p q 2 n p=m+2 q=1 where l is again the number o non-positive eigenvalues. irst line, S 1 := n p=m+2 l k=1 k l p q p q, Let s ocus on the term o the q=l+1 p q p q 2 l k=1 k. By assumption, the principal curvatures satisy Γ ε := { Γ : ε g c 0 }. Since each o the summands is non-negative, S 1 can only vanish i k = 0 or all k l and p p q = 0 or all p > q > l. That is, i there are no negative eigenvalues, and the positive ones o which there are at least n m are all equal. But this implies c 1 m k+1 0, which in turn implies g = 0 < ε, a contradiction, whence we obtain a positive lower bound or S 1 / 2 G: S 1 c ε 2 G or some c ε > 0 depending on ε. On the other hand, the arguents leading to 3.5 yield the estimate l n l S 2 := 2 k + p q p q c 2 δ + C k=1 p=m+2 q=1 with c > 0 independent o ε and δ. The claim ollows. We note that the above estimate is only useul in the presence o the convexity estimate, Theorem 1.1, since in that case, or any δ > 0, there is a constant C δ > 0 or which the set Γ δ,cδ := { Γ : i > δ C δ i} is preserved by the low. 4. Proo o Theorem 1.3 In order to prove Theorem 1.3 it suices to obtain or any ε > 0 an upper bound on the unction G G ε,σ := ε σ or some σ > 0. We will use the estimates o Propositions 3.3 and 3.2 to obtain bounds on the space-time L p -norms o the positive part o G ε,σ, so long as p is suiciently large and σ suiciently small, just as in [HS99a, HS99b, HS09] see also [ALM13] where these techniques are applied in the ully non-linear setting. The Stampacchia-De Giorgi iteration procedure introduced in [Hu84] see also [HS99a, ALM13] then allows us to extract a supremum bound on G ε,σ. We recall the ollowing evolution equation rom [ALM13]:

14 14 BEN ANDREWS AND MAT LANGORD Lemma 4.1. The unction G ε,σ satisies the ollowing evolution equation: t LG ε,σ = σ 1 Ġkl pq,rs kl Gpq,rs k h pq l h rs + where u, v := kl u k u l. 21 σ G ε,σ, σ1 σ σg ε,σ W 2, 4.1 Now set E := max{g ε,σ, 0}. We need to obtain space-time L p -estimates or E. Let us irst observe that integration by parts and application o Young s inequality, in conjunction with Lemma 2.2 and Proposition 3.2, yields the estimate c. [ALM13] d dt E p dµ A 1 pp 1 A 2 p 3 2 E p 2 G ε,σ 2 dµ B 1 p B 2 p 1 2 E p W 2 2 dµ + C 1 σp E p W 2 dµ 4.2 or some positive constants A 1, A 2, B 1, B 2, C 1 which are independent o σ and p. To estimate the inal term, we use Proposition 3.3 in a similar manner to [HS09, Section 5]. We obtain: Lemma 4.2. There are positive constants A 3, A 4, A 5, B 3, B 4, C 2 which are independent o p and σ such that: E p ZW dµ A 3 p A4 p A5 E p 2 G ε,σ 2 dµ + B 3 p B4 E p W 2 2 dµ. Proo. As in [ALM13, Section 4], contraction o the commutation ormula or 2 W with and Ġ yields the identity LG ε,σ = σ 1 Q W, W + σ 1 ZW + σ 2 Ġkl G kl k l + σ G ε,σl 2 1 σ σ1 σ, G ε,σ + 2 G ε,σ 2. The claim is now proved using integration by parts 3 and Young s inequality, with the help o Lemma 2.2 and Propostion 3.2 c. [ALM13, Lemma 4.2]. Corollary 4.3. or all ε > 0 there exist constants l > 0 and L > 0 such that or all p > L and 0 < σ < lp 1 2 there is a constant K = K ε,σ,p or which we have the ollowing estimate: G ε,σ p + dµ G ε,σ, 0 p + dµ 0 + tkµ 0 M, where µ 0 is the measure induced on M by the initial immersion. Proo. Recall Proposition 3.3. Setting δ = ε/2 we obtain ZW ε 2 γ 1 2 γ 2 3 We remark that the extra terms coming rom integration o terms involving the non-divergence-orm operator L by parts are rather easily handled, as in [ALM13], using the homogeneity o the derivatives o via Lemma 2.2 and the existence o a preserved cone.

15 CYLINDRICAL ESTIMATES 15 whenever G ε > 0. We now note that or all positive σ, p there is a constant K σ,p such that σp 2 + K σ,p σp, so that ε 2 γ 1 σpγ 2 2 K σ,p σp + ZW. I we are careul to ensure σpγ 2 εγ 1 /4, we obtain εγ K σ,p σp + ZW. Since G ε,σ is bounded by σ, and W 2 is bounded by 2, we obtain E p W 2 K ε,σ,p + c ε E p ZW, or some constants K ε,σ,p > 0 depending on ε, σ and p, and c ε > 0 depending on ε but independent o σ and p. Combining Lemma 4.2 and inequality 4.2 now yields d E p dµ K ε,σ,p µ 0 M dt α 0 p 2 α 1 σp 5 2 α2 p 3 2 α3 p E p 2 G ε,σ 2 dµ β 0 p β 1 σp 3 2 β2 σp β 3 p 1 2 E p W 2 2 dµ. or some positive constants α i and β i, which depend on ε but not on σ or p, and K ε,σ,p, which depends on ε, σ and p. It is clear that L > 0 and l > 0 may be chosen such that α 0 p 2 α 1 σp 5 2 α2 p 3 2 α3 p 0 and β 0 p β 1 σp 3 2 β2 σp β 3 p or all p > L and 0 < σ < lp 1 2. The claim then ollows by integrating with respect to the time variable. The proo o Theorem 1.3 is completed by proceeding with Huisken s Stampacchia-De Giorgi iteration scheme. We omit these details as the arguments required already appear in [ALM13, Section 5] with no signiicant changes necessary. Reerences [An94a] B. Andrews, Contraction o convex hypersuraces in Euclidean space. Calc. Var. Partial Dierential Equations , [An94b] B. Andrews, Harnack inequalities or evolving hypersuraces. Math. Z , [An07] B. Andrews, Pinching estimates and motion o hypersuraces by curvature unctions. J. Reine Angew. Math, , [ALM] B. Andrews, M. Langord, J. McCoy, Convexity estimates or ully non-linear surace lows. Submitted to J. Dierential Geom. Preprint available at maths-people.anu.edu.au/~langord/ ConvexityEstimatesSuraces.pd.

16 16 BEN ANDREWS AND MAT LANGORD [ALM13] B. Andrews, M. Langord, J. McCoy Convexity estimates or hypersuraces moving by convex curvature unctions To appear in Analysis & PDE. Preprint available at maths-people.anu.edu.au/~langord/ ConvexityEstimatesConvex.pd. [AMZ13] B. Andrews, J. McCoy and Y. Zheng, Contracting convex hypersuraces by curvature. Calc. Var. Partial Dierential Equations , [Ba] R. C. Baker, The mean curvature low o submaniolds o high codimension, PhD thesis, Australian National University, [Ev82] L. C. Evans. Classical solutions o ully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math., , [Ge90] C. Gerhardt, low o nonconvex hypersuraces into spheres, J. Dierential Geometry , [Gl63] G. Glaeser, onctions composées diérentiables. Ann. Math , [Ha84] R. Hamilton, our maniolds with positive curvature operator. J. Dierential Geom , [Hu84] G. Huisken, low by mean curvature o convex suraces into spheres. J. Dierential Geometry , [HS99a] G. Huisken and C. Sinestrari, Mean curvature low singularities or mean convex suraces. Calc. Var. Partial Dierential Equations, , [HS99b] G. Huisken and C. Sinestrari, Convexity estimates or mean curvature low and singularities o mean convex suraces. Act. Math , [HS09] G. Huisken and C. Sinestrari, Mean curvature low with surgeries o two-convex hypersuraces, Inventiones Mathematicae , [Kr82] N. V. Krylov. Boundedly inhomogeneous elliptic and parabolic equations. Iv. Akad. Nauk SSSR Ser. Mat., , , 670. Mathematical Sciences Institute, Australian National University, ACT 0200 Australia Mathematical Sciences Center, Tsinghua University, Beijing , China Morningside Center or Mathematics, Chinese Academy o Sciences, Beijing , China address: ben.andrews@anu.edu.au Mathematical Sciences Institute, Australian National University, ACT 0200 Australia address: mathew.langord@anu.edu.au