arXiv:2009.00986v1 [math.DG] 1 Sep 2020 aueo yesrae of hypersurfaces of vature (1.1) curvature 2000 Date .Peevdcrauecniin 2 Given models References Singularity estimate convexity 6. A estimates 5. Derivative estimates 4. Cylindrical conditions curvature 3. Preserved 2. Introduction 1. etme ,22,12:38am. 2020, 3, September : HR ICIGETMTSFRMEAN FOR ESTIMATES PINCHING SHARP ahmtc ujc Classification. Subject Mathematics eas bannwrgdt eut o nin solutions. ancient for results models. rigidity cylin- singularity new of and obtain classification convexity also partial We the to fun- a Together, argument, yield second estimates compactness the estimate. drical a convexity for via a utilize, estimates obtain we derivative which obtain hypersur- form minimal then damental We for theorem rigidity faces. spher the Simons’ in generalize flow curvature which mean for estimates pinching curvature Abstract. n K          UV TR LWI H SPHERE THE IN FLOW CURVATURE ≥ aifigteqartcpnhn condition pinching quadratic the satisfying , | | | | A A A A and 2 A AGODADHYTENGUYEN THE HUY AND LANGFORD MAT | | | | 2 2 2 2 < < < < epoeasieo smttclysapquadratic sharp asymptotically of suite a prove We 3 n n 5 4 3 2 − H H m H 1 m 2 2 2 H ⌈ ≤ + + + 1. 2 S nK 3 3 8 4 2 + K n K K n 2 Introduction +1 Contents ⌉ esuyteeouinb encur- mean by evolution the study we , mK h on peein sphere round the , 1 if if if if rmr 53C44. Primary n n n n and 3 = and 2 = ≥ ≥ and 3 and 4 m m m m 2 = 1 = ⌊ ≤ = R n ⌈ +2 n n 2 2 , , ⌉ ⌋ fsectional of , , e 17 15 14 10 4 1 2 MATLANGFORDANDHUYTHENGUYEN where A is the second fundamental form and H, its trace, is the mean curvature. The first case and the third case with m = 1 were treated in [8] (cf. [1]), while the second case and the third case with m = 2 were treated in [16]. In the former, it was shown that the flow preserves the condition and drives solutions either to a “small Sn” in finite time or to a “large Sn” in infinite time. In the latter, it was shown that the flow preserves the condition and decomposes the solution, via surgery on “necks” (following Huisken and Sinestrari [12]), into a finite number of components, each of which is either a small Sn, a large Sn, or S1 times a small Sn−1. Here, we extend some of the methods of [8, 16] to treat the entire class of conditions (1.1). Namely, we show that (1.1) is preserved and improves, becoming sharp when either the curvature or the time of existence becomes large. These estimates are analogues of [13, Theorem 5.2] and [14, Corollary 1.2] (cf. [15, Theorem 1.1]).

Acknowledgements. M. Langford was supported by an Australian Research Council DECRA fellowship. H. T. Nguyen was supported by the EPSRC grant EP/S012907/1.

2. Preserved curvature conditions 2.1. Quadratic curvature condition. If the strict quadratic curva- Sn+1 ture inequality (1.1) holds on a hypersurface of K , then we can find some α> 0 such that 1 (2.1) A 2 H2 + 2(m α)K. | | ≤ n m + α − − Without loss of generality, α (0, 1). When m min n , 2(n−1) , this ∈ ≤ 2 3 inequality is preserved under mean curvature flow forn any α o(0, 1). ∈ When m min n , 2(n−1) , the inequality is preserved if α > m ≥ 2 3 − min n , 2(n−1) . n o { 2 3 } n Sn+1 Theorem 2.1 (Cf. [7, 1.4 Lemma]). Let X : [0, T ) K be a solution to mean curvature flow such that (2.1)M holds× on → n 0 n 2(n−1) M ×{ } for some α (0, 1). If m<α + min 2 , 3 , then (2.1) holds on n t for∈ all t [0, T ). { } M ×{ } ∈ Proof. Suppose that (2.1) holds on the initial hypersurface for some m 1,...,n and some α (0, 1). If m<α + min n , 2(n−1) , then ∈{ } ∈ { 2 3 } 1 3 n < and < 1 . n m + α n +2 2n 2(m α) − − − SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 3

Set 1 a := and b := 2(m α) . m n m + α m − − Using the evolution equations for H2 and A 2 (see, for example, [16, Equations (2.11) and (2.12)]), we obtain | | (∂ ∆) A 2 a H2 = 2 A 2 a H 2 +2b K( A 2 + nK) t − | | − m − |∇ | − m|∇ | m | | + 2( A 2 a H2 b K)( A 2 + nK)
| | − m − m
| | 4nK A 2 1 H2 . − | | − n 2 n Since = am and 1, we can estimate
2n−bm 2n−bm ≤ 2b K( A 2 + nK) 4nK A 2 1 H2 m | | − | | − n =2K (b 2n) A 2 +2H2 + b nK m −
| | m 2 2 2 nbm = 2K(2n bm) A H
K − − | | − 2n−bm − 2n−bm 2K(2n b )  A 2 a H2 b K .  ≤ − − m | | − m − m Estimating a 3 and applying the Kato-type inequality
A 2 m ≤ n+2 |∇ | ≥ 3 H 2, we arrive at n+2 |∇ | (∂ ∆) A 2 a H2 b K t − | | − m − m 2 A 2 +(b 2n)K A 2 a H2 b K . ≤ |
| m − | | − m − m The claim now follows from the maximum principle.

Note that

2(n−1) 2 4 n 2(n−1) 3 = 3 , 3 when n =2, 3 min 2 , 3 = n ( 2 when n =4,.... n o The non-vacuous cases are therefore: 1 – n = 2: m = 1 and α> 3 . 2 – n = 3: m = 1 and α> 0, or m = 2 and α> 3 . n – n =2k 4: m =1,..., 2 and α> 0. – n = 2k≥+1 5: m = 1,...,k and α > 0, or m = k + 1 and 1 ≥ α> 2 .

2.2. Rigidity. The quadratic curvature condition (1.1) is optimal for m-cylindrical estimates when n 4 and m n . Indeed, consider the ≥ ≤ 2 hypersurfaces k,n−m(r, s)= Sm(r) Sn−m(s) with r2 + s2 = 1, where Sm(r) is the mMdimensional sphere of× radius r. The second fundamental 4 MATLANGFORDANDHUYTHENGUYEN forms have eigenvalues λ with multiplicity m and µ with multiplicity n m and λµ = 1. Observe that − − ms4 +(n m)r4 A 2 = − , | | r2s2 and (n m)r2 ms2 H = − − , rs so that (n m)2r4 + m2s4 2m(n m)r2s2 H2 = − − − , r2s2 which then yields 1 m(n 2m) s2 A 2 H2 2m = − . | | − n m − (n m) r2 − − Thus, for any ε > 0, there is a submanifold of the topological form Sm Sn−m satisfying × 1 A 2 H2 2m ε . | | − n m − ≤ − 2.3. A class of hypersurfaces. Given n 2, m n, K > 0, α ≥ n≤ 2(n−1) ∈ (0, 1), V < and Θ < such that m α< min 2 , 3 , we shall ∞ n ∞ − { } Sn+1 work with the class m(K, α, V, Θ) of hypersurfaces X : K satisfying C M → 2 1 2 2 (1) max A n m α H 2(m α)K , Mn×{0} | | − − + ≤ − n n − 2 (2) µ0( ) VK , and
(3) maxM H≤2 ΘK, Mn×{0} ≤ where µt is the measure induced by X( , t). Every properly immersed Sn+1 · hypersurface of K which satisfies the strict quadratic pinching condi- n tion (1.1) lies in the class m(m, α, V, Θ) for some choice of parameters α, V , and Θ. C

3. Cylindrical estimates Theorem 3.1 (Cylindrical estimates (Cf. [6, 8, 12, 16, 18])). Let X : n Sn+1 [0, T ) K be a solution to mean curvature flow with initial conditionM × in the→ class n (K, α, V, Θ). There exist δ = δ(n, m, α) > 0, Cm η0 = η0(n, m, α) > 0 and, for every η (0, η0), Cη = Cη(n, α, V, Θ, η) < such that ∈ ∞ 1 (3.1) A 2 H2 ηH2 + C K e−2δKt in n [0, T ) . | | − n m +1 ≤ η M × − SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 5

Before proving Theorem 3.1, let us mention some immediate implica- n tions. First observe that, when n 4 and m = 2 , we recover Simons’ ≥ Sn+1⌈ ⌉ 2 theorem [19]: any minimal hypersurface of K satisfying A < nK is totally geodesic. | | Sn+1 Next observe that any ancient solution X : ( , 0) K satisfying the uniform quadratic pinching conditionM × (2.1)−∞ (for suitab→ le m and α) and uniform area and curvature bounds as t will satisfy → −∞ 1 A 2 H2 0 < 2(m 1) | | − n m +1 ≤ − unless m = 1. In particular,− it satisfies the uniform quadratic pinching condition (2.1) with m m 1. Iterating the argument, we find that Sn7→+1 − X : ( , 0) K is umbilic, and hence, assuming is con- nected,M × either−∞ a stationary→ hyperequator or a shrinking hyperpaM rallel (cf. [13, Theorem 6.1]). In fact, the uniform area and curvature bounds are superfluous since we may instead apply Corollary 3.4 below. We thus obtain the following parabolic analogue of Simons’ theorem, which generalizes [13, Theorem 6.1 (2)]. Sn+1 Theorem 3.2. Let X : ( ,ω) K be an ancient solution to mean curvature flow.M If × −∞ → 2 3 2 4 – n =2 and lim sup max ( A 4 H 3 K) < 0, or t→−∞ M×{t} | | − − 2 3 2 8 – n =3 and lim sup max ( A 5 H 3 K) < 0, or t→−∞ M×{t} | | − − 2 2 2 – n 4 and lim sup max ( A n H nK) < 0, ≥ t→−∞ M×{t} | | − − Sn+1 then X : ( ,ω) K is either a stationary hyperequator or a shrinkingM hyperparallel. × −∞ → Finally, we note that, by a similar argument, any immortal solution Sn+1 X : [0, ) K to mean curvature flow with initial datum satisfyingM × the∞ uniform→ quadratic pinching condition converges (assum- ing is connected) to a stationary hyperequator. In particular, is diffeomorphicM to Sn. M Returning to the proof of Theorem (3.1), set 1 1 a := + η η and b := 2(m α) , n m + α − n m +1 0 − − − − where η0 = η0(n, m, α) is the largest number satisfying n + α m nη 2 − 0 ≤ 1 n+2 1 + η . ( − 3 n−m+α 0 Note that, by hypothesis, η0 > 0.
6 MATLANGFORDANDHUYTHENGUYEN

We will prove the estimate (5.1), with δ = nη0, by obtaining a bound for the function 1 f := A 2 + η H2 W σ−1 σ,η | | − n m +1   −   for some σ (0, 1) and any η (0, η ), where W is defined by ∈ ∈ 0 W := aH2 + bK . σ Observe that W > 0 and fσ,η W . The choice of the constants a and b is determined by the need to≤ obtain the good gradient and reaction terms in the following lemma. Lemma 3.3. There exists δ = δ(n, α) > 0 such that (∂ ∆)f 2σ( A 2 + nK)f 4δKf t − σ,η ≤ | | σ,η − σ,η A 2 W (3.2) 2δf |∇ | + 2(1 σ) f , ∇ − σ,η W − ∇ σ,η W   wherever fσ,η > 0. Proof. Set f := A 2 ( 1 + η)H2. Basic manipulations (indepen- η | | − n−m+1 dent of the precise form of fη and W ) yield (∂ ∆)f = W σ−1(∂ ∆)f (1 σ)f W −1(∂ ∆)W t − σ,η t − η − − σ,η t − W W 2 + 2(1 σ) f , ∇ σ(1 σ)f |∇ | . − ∇ σ,η W − − σ,η W 2   The final term will be discarded. Applying the evolution equations for H2 and A 2 [16, Equations (2.11) and (2.12)], we compute | | (∂ ∆)W =2 A 2 + nK (W bK) 2a H 2 t − | | − − |∇ | and
(∂ ∆)f =2 A 2 + nK f 4nK A 2 1 H2 t − η | | η − | | − n (3.3) 2 A 2 1 + η H 2 . − |∇ | −
n−m+1 |∇ |
Combining these preceding three identities and
estimating
fσ,η W σ, H 2 n+2 A 2 and σ < 1 yields ≤ |∇ | ≤ 3 |∇ | W (∂ ∆)f 2σ( A 2 + nK)f + 2(1 σ) f , ∇ t − σ,η ≤ | | σ,η − ∇ σ,η W   f +2W σ−1 bK( A 2 + nK) η 2nK( A 2 1 H2) | | W − | | − n   n +2 1 A 2 2f 1 + η |∇ | . − σ,η − 3 n m + α 0 W   −  SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 7

By the definition of η0, n +2 1 A 2 A 2 2f 1 + η |∇ | 2nη f |∇ | . σ,η − 3 n m + α 0 W ≥ 0 σ,η W   −  So consider the term f Z := (m α) A 2 + nK η + H2 n A 2 − | | W − | | 1
fη (m α) H2 +(b + n)K + H2 n A 2 . ≤ − n m + α W − | |  −  Noting that m α n b − = na + + n(η η ) 1 and m α = , n m + α n m +1 − 0 − − 2 − − and estimating f W , we find η ≤ Z nf + H2 n A 2 ≤ η − | | n b f + + n(η η ) 1 H2 + (b + n)K nbK η n m +1 − 0 − 2 − W  −   n = + nη 1 H2 − n m +1 −  −  n n f + + n(η η ) 1 H2 + m α bK η n m +1 − 0 − − − 2 W  −   n f h i nη H2 + + α m bK η ≤ − 0 2 − W nη f . h i  ≤ − 0 η The claim follows.  Setting σ = η = 0, the maximum principle yields the following. Sn+1 Corollary 3.4. Every solution X : [0, T ) K to mean curva- ture flow with initial datum satisfyingM× the uniform→ quadratic pinching condition (2.1) with α > m min n , 2(n−1) satisfies − { 2 3 } A 2 1 H2 | | − n−m+1 e−4δKt , aH2 + bK ≤ where δ = δ(n, m, α). 2δKt We wish to bound e fσ,η from above. It will suffice to consider points where fσ,η > 0. To that end, consider the function f := max e2δKtf , 0 . + { σ,η } p We first derive an L -estimate for f+ using Lemma 3.3 and the fol- lowing proposition. 8 MATLANGFORDANDHUYTHENGUYEN

Proposition 3.5. Given n 3, α (0, 1) and η (0, 1 1 ) ≥ ∈ ∈ n−m+α − n−m+1 there exists γ = γ(n, α, η) > 0 with the following property: Let X : n Sn+1 2,2 K be a smoothly immersed hypersurface and u W ( ) a Mfunction→ satisfying spt u U , where, introducing the functions∈ M ⊂ α,η 1 f := A 2 H2 ηH2 m,η | | − n m − − and 1 g := A 2 H2 2(m α)K, m,α | | − n m + α − − the set U is defined by− α,η ⊂M U := x : f (x) 0 g (x) . α,η { ∈M m−1,η ≥ ≥ m,α } For any r 1, ≥ u 2 A 2 γ u2W dµ u2 r−1 |∇ | + r |∇ | + K dµ . ≤ u2 W Z Z   Proof. We proceed as in [16, Proposition 2.2]. By a straightforward scaling argument, it suffices to prove the claim when K = 1. Recall Simons’ identity A A = C , ∇(i∇j) kl −∇(k∇l) ij ijkl where the brackets denote symmetrization and C := A A2 A2 A +(g A A g) . ⊗ − ⊗ ⊗ − ⊗ We claim that (3.4) γ(n, α, η)W 3 C 2 + 1 in U ≤| | α,η on any immersed hypersurface X : n Sn. Indeed, if this is not the M → case then there is a sequence of vectors ~λk Rn, k N, satisfying ∈ ∈ 1 f (~λk) := ~λk 2 tr(~λk)2 η tr(~λk)2 0 m−1,η | | − n m +1 − ≥ − and 1 g (~λk) := ~λk 2 tr(~λk)2 2(m α) 0 , m,k | | − n m + α − − ≤ − ~ n where tr(λ) := i=1 λi, but P C(~λk) 2 +1 | | 0 W 3(~λk) → as k , where →∞ n C(~λ) 2 := (λ λ )2(λ λ + 1)2 | | j − i i j i,j=1 X SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 9 and 1 1 W (~λ) := + η η tr(~λ)2 + 2(2 α) . n m + α − n m +1 0 − −  − −  Set r2 := W (~λk)−1 0 and λˆk := r ~λk. Observe that k → k λˆk c(n, α) | | ≤ and hence, up to a subsequence, λˆk λˆ Rn. Computing → ∈ 1 λˆk 2 tr(λˆk)2 η tr(λˆk)2 = r2f (~λk) 0 | | − n m +1 − k m−1,η ≥  −  and 1 λˆk 2 tr(λˆk)2 2(m α)r2 = r2g (~λk) 0 , | | − n m + α − − k k m,α ≤  −  we find 1 (3.5) λˆ 2 tr(λˆ)2 η tr(λˆ)2 | | − n m +1 ≥  −  and 1 (3.6) λˆ 2 tr(λˆ)2 0 . | | − n m + α ≤  −  On the other hand, n 2 λˆkλˆk(λˆk λˆk) +2r2λˆkλˆk(λˆk λˆk)2 + r4(λˆk λˆk)2 = r6 C(~λk) 2 i j j − i k i j j − i k j − i k| | i,j=1 X   so that n 2 (3.7) λˆ λˆ (λˆ λˆ ) =0 . i j j − i i,j=1 X   Together, (3.5), (3.6) and (3.7) are in contradiction: (3.7) implies that λˆ has a null component of multiplicity ℓ and a non-zero component, κ say, of multiplicity n ℓ. The inequalities (3.5) and (3.6) then yield − (n ℓ)2 n ℓ − κ2 η(n ℓ)2κ2 > 0 − − n m +1 ≥ −  −  and (n ℓ)2 n ℓ − κ2 0 . − − n m + α ≤  −  which together imply that ℓ (m 1, m α], which is impossible. This proves (3.4). ∈ − − 10 MATLANGFORDANDHUYTHENGUYEN

Using (3.4), we can estimate u2 γ u2Wdµ C 2 +1 dµ ≤ W 2 | | Z Z u2
= C 2A +1 dµ W 2 ∗∇ Z u2 u
W u2 = ∇ C+ ∇ C+ C Adµ + dµ W 2 u ∗ W ∗ ∇ ∗∇ W 2   Z 2 Z u 3 u 1 C W 2 |∇ | + W 2 W + W A A dµ ≤ W 2 u |∇ | |∇ | |∇ | Z   + u2 dµ Z  2 u A A 2 C u |∇ | + |∇ 1 | |∇ 1 | dµ + u dµ , ≤ u W 2 W 2 Z   Z  where C denotes any constant which depends only on n, α and η. The claim now follows from Young’s inequality.  Lemma 3.6. There exist constants ℓ = ℓ(n, α, η) < and C = C(n, K, α, V, Θ,σ,p) such that ∞

(3.8) f p dµ C e−δpKt + ≤ 1 Z so long as σ ℓp− 2 and p>ℓ−1. ≤ Proof. This follows from Lemma 3.3 and Proposition 3.5 exactly as in [16, Lemma 4.6].  p 2 2 ∞ This L -estimate (for v := f+ ) can be bootstrapped to an L - estimate using Stampacchia iteration, exactly as in the proof of [16, Theorem 4.1].

4. Derivative estimates Next, we use the cylindrical estimate to obtain a sharp “gradient estimate” for the second fundamental form. We need the following universal interior estimates for solutions with initial data in the class n (K, α, V, Θ). Cm n Sn+1 Proposition 4.1. Let X : [0, T ) K be a maximal solution to mean curvature flow with initialM × condition→ in the class n (K, α, V, Θ). Cm Defining Λ0 and λ0 by 1 n (4.1) Λ /2 := Θ2 + 2(m α) and e2nλ0 := 1 + , 0 n m + α − n +Λ − 0 SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 11 we have 2n (4.2) e2nKT 1+ , ≥ Λ0 and (4.3) max kA 2 Λ Kk+1 −1 k Mn×{λ0K } |∇ | ≤ for every k N, where Λ depends only on n, k and Λ . ∈ k 0 Proof. The proof is similar to [16, Proposition 4.7].  Theorem 4.2 (Gradient estimate (cf. [12, Theorem 6.1])). Let X : n Sn+1 [0, T ) K be a solution to mean curvature flow with initial conditionM × in the→ class n (K, α, V, Θ). There exist constants δ = δ(n, α), Cm c = c(n, m, α, Θ) < , η0 = η0(n, m, α) > 0 and, for every η (0, η0), C = C (n, α, V, Θ, η∞) < such that ∈ η η ∞ (4.4) A 2 c (η + 1 )H2 A 2 W + C K2e−2δKt |∇ | ≤ n−m+1 −| | η in n [λ K−1, T ), where λ is defined by (4.1) . M × 0 0 Note that the conclusion is not vacuous since, by Proposition 4.1, the maximal existence time of a solution with initial data in the class n (K, α, V, Θ) is at least 1 log 1+ 2n >λ K−1. Cm 2nK Λ0 0   Proof of Theorem 4.2. We proceed as in [16, Theorem 4.8], which is inspired by [12, Theorem 6.1]. First note that (see, for example, [16, Inequality (2.13)]) (∂ ∆) A 2 2 2A 2 + c A 2 + nK A 2 . t − |∇ | ≤ − |∇ | n | | |∇ | We will control the bad term using the good term
in the evolution equation for A 2 and the Kato inequality. | | By the cylindrical estimate, given any η < η0 we can find a constant Cη = Cη(n, α, V, Θ, η) > 2 such that 1 A 2 H2 ηH2 + C Ke−2δKt , | | − n m +1 ≤ η − where δ = nη0, and hence 1 G := 2C Ke−2δKt + η + H2 A 2 C Ke−2δKt > 0 . η η n m +1 −| | ≥ η  −  1 3 Moreover, since n−m+α < n+2 , 3 G := 2C K + H2 A 2 C K > 0 , 0 0 n +2 −| | ≥ 0 12 MATLANGFORDANDHUYTHENGUYEN where C := 2(m α). By (3.3), 0 − (∂ ∆)G = 2( A 2 + nK)(G 2C Ke−2δKt)+4nK A 2 1 H2 t − η | | η − η | | − n +2 A 2 η + 1 H 2 2δC K2e−2δKt . |∇ | − n−1 |∇ | − η

−2δKt
 −2δKt Since Gη CηKe , we can estimate Gη 2CηKe Gη. By the Kato≥ inequality, we can estimate − ≥ −

A 2 1 + η H 2 n+2 3 1 η A 2 |∇ | − n−m+1 |∇ | ≥ 3 n+2 − n−m+1 − |∇ | β 2
A ,  ≥ 2 |∇ | where

1 3 1 (4.5) β := , 2 n +2 − n m +1  −  so long as η 2 3 β. Estimating, finally, ≤ − 2(n+2)   2δC K2e−2δKt 2δKG , η ≤ η we arrive at

(∂ ∆)G 2( A 2 + nK)G + β A 2 2δKG . t − η ≥ − | | η |∇ | − η Similarly,

(∂ ∆)G 2( A 2 + nK)G . t − 0 ≥ − | | 0 We can now proceed exactly as in [16, Theorem 4.8]: We seek a 2 bound for the ratio |∇A| . Note that, at a local spatial maximum of GηG0 2 |∇A| , GηG0

A 2 A, A A 2 G G 0= |∇ | =2h∇k∇ ∇ i |∇ | ∇k η + ∇k 0 . ∇k G G G G − G G G G η 0 η 0 η 0  η 0  In particular,

A 2 G G A 2 G G 2 2A 2 4|∇ | ∇ η , ∇ 0 |∇ | ∇ η + ∇ 0 4|∇ | . G G G G ≤ G G G G ≤ G G η 0  η 0  η 0 η 0 η 0

SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 13

2 Suppose that |∇A| attains a parabolic interior local maximum at (p , t ). GηG0 0 0 Then, at (p0, t0), A 2 0 (∂t ∆)|∇ | ≤ − GηG0 (∂ ∆) A 2 A 2 (∂ ∆)G (∂ ∆)G = t − |∇ | |∇ | t − η + t − 0 G G − G G G G η 0 η 0  η 0  2 A 2 A 2 G G + |∇ | , (G G ) +2|∇ | ∇ η , ∇ 0 G G ∇ G G ∇ η 0 G G G G η 0  η 0  η 0  η 0  A 2 A 2 |∇ | (c + 4)( A 2 + nK)+2δK β |∇ | ≤ G G n | | − G η 0  η  and hence A 2 (c + 4)( A 2 + nK)+2δK n . |∇ | | | 3 2 2 GηG ≤ 2C K + H A 0 0 n+2 −| | Since 1 A 2 H2 + 2(m α)K, | | ≤ n m + α − − we obtain, at (p0, t0), A 2 |∇ | C, GηG0 ≤ where C depends only on n and α. −2δKt On the other hand, since G0 > C0K and Gη > CηKe , if no interior local parabolic maxima are attained, then, by Proposition 4.1, we have for any t λ K−1 ≥ 0 A 2 A 2 max |∇ | max |∇ | n −1 M ×{t} G0Gη ≤ Mn×{λ0K } G0Gη A 2 max −1 |∇ 2 |−2δλ0 ≤ Mn×{λ0K } C0CηK e λ0 Λ1e

≤ C0Cη Λ eλ0 . ≤ 1 The theorem now follows from Young’s inequality.  Remark 4.3. Fixing η in (4.4) yields the cruder estimate (4.6) A 2 C(H4 + K2) , |∇ | ≤ where C = C(n, α, V, Θ). Indeed, an estimate of this form may actually be obtained without the aid of the cylindrical estimate (simply take 14 MATLANGFORDANDHUYTHENGUYEN

1 1 η = n−m+α n−m+1 , δ = 0, and Cη = 2(m α) in the proof). A non- quantitative− version of the cylindrical estimate− without the exponential decay term may then be obtained via a blow-up argument as in the proof of the convexity estimate (Theorem 5.1) below (cf. [17]). The gradient estimate can be used to bound the first order terms which arise in the evolution equation for 2A. A straightforward max- imum principle argument exploiting this∇ observation yields an analo- gous estimate for 2A. ∇ Theorem 4.4 (Hessian estimate (cf. [6, 12])). Let X : n [0, T ) Sn+1 M × → K be a solution to mean curvature flow with initial condition in the class n (K, α, V, Θ). There exists C = C(n, α, V, Θ) such that Cm (4.7) 2A 2 C(H6 + K3) in n [λ K−1, T ) . |∇ | ≤ M × 0 Proof. The proof is the same as that of [16, Theorem 4.11].  An inductive argument, exploiting estimates for lower order terms in the evolution equations for higher derivatives of A as in Theorem 4.4, can be applied to obtain estimates for spatial derivatives of A to all orders. The evolution equation for A then yields bounds for the mixed space-time derivatives (cf. [12, Theorem 6.3 and Corollary 6.4] and [16, Theorem 4.14]). 5. A convexity estimate Following White [20], the gradient estimates allow us to obtain a convexity estimate via a blow-up argument. n Sn+1 Theorem 5.1. Let X : [0, T ) K be a solution to mean M × → n curvature flow with initial condition in the class m(K, α, V, Θ). For every η > 0 there exists h = h (n, α, V, Θ, η) < Csuch that η η ∞ (5.1) H (p, t) h √K = λ (p, t) η H (p, t) . | | ≥ η ⇒ 1 ≥ − | | Proof. Let η0 be the infimum over all η > 0 such that the conclu- sion holds. By the quadratic pinching hypothesis, η < . Suppose, 0 ∞ contrary to the claim, that η0 > 0. Then we can find a sequence Sn+1 of mean curvature flows Xj : j [0, Tj) K with initial data Sn+1 M × n → Xj( , 0) : j K in the class m(K, α, V, Θ) and a sequence of points· p M →and times t [0, T )C such that j ∈Mj j ∈ j j λ1 Hj(pj, tj) and (pj, tj) η0 . →∞ Hj → − After translating in time, translating and rotating in Rn+2, and parabol- ically rescaling by H(p , t ), we obtain a sequence of flows Xˆ : j j j Mj × SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 15

1 n+1 − 2 ( Tˆj, 0] (S K en ) such that, at the spacetime origin, Hˆj 1 − → Kj − j +2 ≡ and λˆj /Hˆ η , where K := H−2(p , t )K and Tˆ := H2(p , t )t . 1 j → − 0 j j j j j j j j j By Proposition 4.1, Tˆj is bounded uniformly from below. By the gradi- ent estimates, Aˆj is bounded uniformly in a backward parabolic cylin- der of uniform| radius| about the spacetime origin. By standard boot- strapping arguments, we obtain a limit flow X∞ : ∞ ( T∞, 0] Rn+1 0 on which λ∞/H attains a negative localM parabolic× − mini-→ ×{ } 1 ∞ mum, η0, at the spacetime origin. But this violates the strong maxi- mum principle− for the second fundamental form [20, Appendix]. 

6. Singularity models To illustrate the utility of our pinching estimates, we will obtain a precise description of singular regions after performing certain types of ‘blow-ups’. Sn+1 Let X : M [0, T ) K be a maximal solution to mean curvature flow such that×T < →. Standard bootstrapping estimates imply that ∞ lim sup max A 2 = . t→T M×{t} | | ∞ Following standard nomenclature, we say that the “finite time singu- larity” is type-I if lim sup (T t) max A 2 < t→T − M×{t} | | ∞ and type-II otherwise. Well-known ideas due to Hamilton (see, for example, [5, 11]) yield a classification of certain blow-up sequences. For type-I singularities, we obtain the following. Theorem 6.1 (Finite time type-I singularities). Let X : M [0, T ) Sn+1 × → K be a solution to mean curvature flow which undergoes a finite time type-I singularity. Suppose that X( , 0) satisfies (1.1) for some m. Given sequences of times t [0, T·) and points p such j ∈ j ∈ M that lim supj→∞ A(pj, tj) , the sequence of solutions Xj : −2 −|1 S |→∞ Rn+2 M × [ rj tj, 1) rj ( Kj en+2) obtained by translating the point − → − ⊂ n+2 (X(pj, tj), tj) to the spacetime origin in R R, rotating so that Sn+1 Rn+1 × T0( K X(pj, tj)) becomes 0 and parabolically rescaling by − 1 ×{ } −1 − 2 rj := (T tj) , converge locally uniformly in the smooth topology, after passing− to a subsequence and performing a fixed rotation, to a maximal, locally convex, type-I ancient solution X : (Rk n−k) ×M∞ × 16 MATLANGFORDANDHUYTHENGUYEN

( , 1) Rn+1 to mean curvature flow in Rn+1, k < m, which satis- fies−∞ → A 2 c 1 H2 A 2 H2 , |∇ | ≤ n−m+1 −| | n−k where c depends only on X( , 0) and the restriction
X ∞ is lo- · |{0}×M cally uniformly convex and solves mean curvature flow in 0 Rn−k+1. { }× If k = m 1, then X n−k is totally umbilic, else − |{0}×M∞ A 2 < 1 H2 . | | n−m+1 Proof. Using Theorems 3.1, 4.2 and 5.1, the conclusion is obtained as in [11, Section 4] (strict inequality in the quadratic pinching when k < m 1 follows from the strong maximum principle).  − Remark 6.2. In fact, in Theorem 6.1, the blow-up is a shrinking round orthogonal cylinder for every k. Indeed, using Hamilton’s extension of Huisken’s monotonicity formula to general ambient spaces [4] and his matrix Harnack estimate for the heat equation [3], we may proceed as in [9] to show that the the blow-up is a Euclidean self-shrinking solution. By Theorem 5.1, the limit is weakly convex. The conclu- sion then follows from Huisken’s classification result [10]. We believe that it should be possible to reach this conclusion without using the monotonicity formula, however. For type-II singularities, we obtain the following. Theorem 6.3 (Finite time type-II singularities). Let X : M [0, T ) Sn+1 × → K be a solution to mean curvature flow which undergoes a finite time type-II singularity. Suppose that X( , 0) satisfies (1.1) for some · m. There are sequences of times tj [0, T ), points pj and scales ∈ −∈M2 −2 rj 0 such that the sequence of solutions Xj : [ rj tj,rj (T tj −1→ −1 S Rn+2 M× − − − j )) rj ( Kj en+2) obtained by translating (X(pj, tj), tj) → − ⊂Rn+2 R Sn+1 to the spacetime origin in , rotating so that T0( K X(pj, tj)) Rn+1 × −1 − becomes 0 and parabolically rescaling by rj , converge locally uniformly in the×{ smooth} topology, after passing to a subsequence and performing a fixed rotation, to a locally convex eternal solution X : Rk n−k Rn+1 Rn+1 ( ∞ ) ( , 0) to mean curvature flow in , k < m× M1, which× −∞ satisfies → − A 2 < 1 H2 and A 2 c 1 H2 A 2 H2 , | | n−m+1 |∇ | ≤ n−m+1 −| |

n−k where c depends only on X( , 0) and the restriction X {0
}×M∞ is a locally uniformly convex translating· solution to mean curv| ature flow in n−k+1 0 R . If k = m 2, then X n−k is the bowl soliton. { } × − |{0}×M∞ SHARP PINCHING ESTIMATES FOR MCF IN THE SPHERE 17

Proof. Using Theorems 3.1, 4.2 and 5.1, the conclusion is obtained by proceeding as in [11, Section 4] and applying the classification result from [2] (strict inequality in the quadratic pinching on the limit follows from the strong maximum principle). 

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Department of Mathematics, University of Tennessee Knoxville, Knoxville TN, USA, 37996-1320

School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, NSW, Australia, 2308 E-mail address: [email protected] E-mail address: [email protected] School of Mathematical Sciences, Queen Mary University of Lon- don, Mile End Road, London, UK, E1 4NS E-mail address: [email protected]