Convexity Estimates for High Codimension Mean Curvature Flow

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.


INTRODUCTION
Let us consider a compact smooth n-manifold M , and a family of immersions which move by mean curvature flow, that is where H is the mean curvature vector. The mean curvature flow constitutes a system of quasilinear weakly parabolic partial differential equations for F , and since M is compact the flow must form a singularity in finite time. Singularity formation may be characterised analytically as follows: if we let A denote the second fundamental form and take T to be the maximal time then there holds lim sup tրT sup M |A|(·, t) = ∞.
A profound and challenging problem is characterising and classifying the geometry of singularities, whose formation depends on the initial submanifold M 0 where M t := F (M, t). In the seminal work of Huisken-Sinestrari [HS99], the singularities formed by mean convex codimension one solutions were shown to be weakly convex (White obtained a similar result for embedded mean convex solutions in [Whi03]). It is natural to seek a corresponding theorem for solutions of higher codimension. However, we encounter a number of new difficulties, the foremost being that the second fundamental form and the mean curvature are vector-valued and consequently there is no direct corresponding notion of mean convexity. We instead use a different but related condition introduced by Andrews-Baker [AB10]. They showed that when n ≥ 2, the quadratic pinching condition is preserved for each c < 4 3n and a > 0. That is, if the condition is satisfied at the initial time, then it is satisfied by M t for every t ∈ [0, T ). We note that for compact hypersurfaces, positive mean curvature implies this condition but for all sufficiently large c. We will refer to submanifolds satisfying (1.1) as being quadratically pinched. Andrews-Baker showed that if c < { 4 3n , 1 n−1 } the flow contracts quadratically pinched solutions to round points. This result is a high codimension generalisation of Huisken's work on convex solutions of codimension one [Hui84]. Recently the second-named author has constructed a flow with surgeries for solutions of dimension n ≥ 5 which are quadratically pinched with c < c n := 3(n+1) 2n(n+2) n = 5, 6, 7, 4 3n n ≥ 8.
This generalises the surgery construction for two-convex hypersurface flows due to Huisken-Sinestrari [HS09]. An important ingredient in [Ngu20] (and in the present work) is the codimension estimate due to Naff, which implies the singularities formed by a quadratically pinched solution are codimension one if c < c n . In this paper we show that a quadratically pinched mean curvature flow with c < c n is asymptotically convex in a quantifiable manner. As the second fundamental form is vector-valued, we denote by λ 1 ≤ · · · ≤ λ n the eigenvalues of the second fundamental form in the principal normal direction, that is ν 1 = H |H| (see Section 2 for precise definitions).
Theorem 1.1. Let F : M × [0, T ) → R n+k be a mean curvature flow of dimension n ≥ 5 which is quadratically pinched with c < c n . Then for any ε > 0 there exists a constant C ε > 0 which depends only on n, ε and M 0 such that As ε > 0 is arbitrary, this shows that the negative part of the first eigenvalue in the principal normal direction does not grow as fast as |H|. We then use this estimate to characterise type II singularities of the quadratically pinched mean curvature flow near the maximum of the curvature: Theorem 1.2. Suppose a type II singularity forms at time T , that is Then there exists a sequence of rescalings of F that subconverges smoothly to a codimension one limiting flow which is either: a strictly convex translating solution; or the isometric product of R m with a strictly convex translating solution of dimension n − m.
The paper is set out as follows. In Section 2 we gather together the necessary evolution equations and technical tools. In particular, λ 1 is not smooth but is locally Lipschitz and semiconvex, and its evolution equation must be understood in a distributional sense. In Section 3 we obtain a Poincarétype inequality which requires the Simons' identity for high codimension submanifolds. In Section 4, we complete the proof of the convexity estimate by applying Huisken's Stampacchia iteration. Finally, in Section 5 we study singularity formation and prove Theorem 1.2. Acknowledgements. The second named author was supported by the EPSRC grant EP/S012907/1.

EVOLUTION EQUATIONS
Let F : M × [0, T ) → R n+k solve mean curvature flow and write M t := F (M, t). We recall from the work of Andrews-Baker [AB10] the following evolution equations for the second fundamental form and mean curvature vector. With respect to local orthonormal frames {e i } and {ν α } for the tangent and normal bundles, From these equations we can compute that We use R ⊥ to denote the normal curvature, which is given by Under the quadratic pinching assumption we have |H| > 0, so at any point in M t we can choose a local orthonormal frame for the normal bundle which is such that We also use the notationÂ to denote the components of the second fundamental form orthogonal to the mean curvature vector, and write for the scalar part of the mean curvature component of A. Hence A admits the decomposition A = hν 1 +Â.

2.1.
Pinching is preserved. With this notation in place we can state the estimate proven by Andrews-Baker showing that quadratic pinching is preserved by the flow.

At every point in
Note that by Proposition 6 in [AB10] we have so the gradient term on the right-hand side is nonpositive. At points where Q ≤ 0, each of the zeroth-order reaction terms is also nonpositive. From now on we suppose the initial submanifold M 0 , and hence M t for all t ∈ [0, T ), is quadratically pinched with For ease of notation let us define and observe that by the quadratic pinching W ≥ ε0 2 |H| 2 on M t . and where δ 0 > 0 depends only on n and ε 0 .
Proof. From Lemma 2.1 we obtain Using (2.2) we estimate which gives the desired inequality for W . It follows that and since W ≤ 4 3n |H| 2 we have Thus it suffices to take 2.2. The evolution of h. From the equations for A and H, we readily compute the projection A, H satisfies The first of the reaction terms can be split into a hypersurface and a codimension component, as follows: Similarly, the remaining reaction terms can be written as Therefore, For a positive function f , we have hence the quantity |H| satisfies For a tensor B ij divided by a positive scalar function f there holds Therefore, dividing A, H by |H|, we obtain Let us introduce the abbreviation We simplify the gradient terms by decomposing and so obtain: where T ij is the quantity defined in (2.4).
Since h is a symmetric bilinear form it has n real eigenvalues, which we denote by The smallest eigenvalue can be written as and is therefore a Lipschitz continuous function on M × [0, T ). We will use the evolution equation for h to estimate (∂ t − ∆)λ 1 , interpreted in an appropriate weak sense (cf. [Whi03] and [Lan17]).
with equality at (x 0 , t 0 ). If the inequality is reversed then ϕ is an upper support for f at (x 0 , t 0 ).
With this definition in place we have the following estimate: and extend the {e i } to a spatial neighbourhood of x 0 by parallel transport with respect to g(t 0 ). We then extend to an orthonormal frame on a backward spacetime neighbourhood of (x 0 , t 0 ) by parallel transport with respect to the connection ∇ ∂t . On this neighbourhood we can define a smooth function Observe that by the definition of λ 1 there holds On the other hand, at (x 0 , t 0 ) there holds which shows that ∆e 1 is orthogonal to e 1 , so since h is diagonal at (x 0 , t 0 ) we obtain ∆h 11 = ∆η, and consequently It follows then Eventually we will want to prove integral estimates for the function λ 1 . To do so we appeal to Alexandrov's theorem, following Brendle [Bre15] (see also [Lan17]). We call a function f : there is a small open neighbourhood on which f can be expressed as the sum of a smooth and a convex (resp. concave) function.
Here µ t is the measure induced by the immersion F (·, t).
Proof. Choosing local coordinates and applying Alexandrov's theorem [EG15, Section 6.4], we see that f has two derivatives at a.e. point in By approximation, the same inequality also holds if ϕ is only Lipschitz continuous.
Since h is smooth, on every small enough set in spacetime, λ 1 can be expressed as the minimum over a set of smooth functions which is compact in C 2 . This is sufficient to ensure that λ 1 is locally semiconcave on M × [0, T ), so by the lemma we conclude that there is a set of full measure Then λ 1 admits a lower support ϕ at (x 0 , t 0 ), to which we can apply Lemma 2.5. Since ϕ(x 0 , t 0 ) = λ 1 (x 0 , t 0 ), this gives the desired inequality.
Remark 2.8. Notice that the first of the gradient terms is nonnegative whenever λ 1 ≤ 0, whereas the remaining gradient terms both contain ∇Â as a factor. It is this structure of the gradient terms which allows us to prove the convexity estimate.
Lemma 2.9. There is a positive constant C = C(n) such that Proof. We use the formula where in the last line we have used |A| 2 ≤ 4 3n |H| 2 . By (2.3) we have and estimate Then since and use this to bound Substituting this inequality into (2.5) and using the quadratic pinching gives the desired estimate.
2.4. Modifying λ 1 . We now form the quantity where ε and Λ are positive constants to be chosen later. Combining the evolution equations for the three components we obtain: Lemma 2.10. At each point in Q there holds where C = C(n).
Proof. At any point in Q we compute so by Lemma 2.7, Inserting the estimates from Lemma 2.2 and Lemma 2.9 we find Using the definition of f and rearranging we obtain Next we estimate in order to obtain Finally, by bounding we find

A POINCARÉ INEQUALITY
In this section we establish a Poincaré-type inequality for the high codimension solution M t . The proof loosely follows [Hui84, Lemma 5.4], in that we combine Simons' identity with an integration by parts argument. We also incorporate an idea from [BH17, Proposition 3.1], where the authors symmetrise and then take the square of Simons' identity to fully exploit the structure of the cubic zeroth-order terms. Simons' identity for high codimension submanifolds states that We symmetrise to get Using the relation Breaking U into components parallel and orthogonal to the mean curvature vector we obtain Substituting this back in we arrive at There is a purely dimensional constant C such that |V | ≤ C|A||R ⊥ |, so for a larger constant C there holds |V | ≤ C|A| 2 |Â|.
Since |U | ≤ C|A| 3 we have We are now ready to prove the Poincaré inequality. The proof does not actually use the fact that M t moves by mean curvature, so this result can be viewed as a general statement about high codimension submanifolds.
Proposition 3.2. Fix t ∈ [0, T ) and let u : M → R be a nonnegative Lipschitz function which is supported in supp(f (·, t)). Then there is a positive constant C = C(n, ε 0 , ε, Λ) such that Proof. For a symmetric matrix B with eigenvalues µ i there holds Observe that the right-hand side vanishes if and only if B is the second fundamental form of a codimension-one cylinder. Let us define which has as eigenvalues µ i = λ i − Λv. In particular, the computation above shows that which is to say that µ 1 (x, t) ≤ −εw(x, t). Furthermore, since Λv(x, t).
If the right-hand side is nonnegative then we can square both sides to get an estimate of the form Λv(x, t) < 0 then trivially there holds so in either case we can bound Putting these estimates together we find on the support of f there holds where C = C(n, ε 0 , ε, Λ). Combining this inequality with the result of the last lemma we find on the support of f there holds Let u be a nonnegative Lipschitz function supported in supp(f ). Then we can multiply this inequality by |A| −4 u 2 and integrate over M to get We are going to estimate each of the four Hessian terms on the right. Since each of these is handled in the same way, we only give the argument for the first one. Defining we can write The divergence term vanishes upon integration, and there is a purely dimensional constant C such that so making C a bit larger, we have Estimating the remaining Hessian terms in the same way and substituting back in we arrive at

STAMPACCHIA ITERATION
In this section we establish the convexity estimate by proving an a priori supremum estimate for the function where σ ∈ (0, 1) is chosen small depending on n and M 0 . Recall from Lemma 2.10 that at each point in Q there holds where C = C(n). Let us fix Λ = C 2εδ 0 so that the last term vanishes. Then using Hence at points in Q ∩ supp(f σ ) we have where C = C(n). All of the computations until now were for a quadratically pinched solution with From here on we assume n ≥ 5 and the more restrictive condition c ≤ c n − ε 0 where This is the range of pinching constants for which Naff's codimension estimate is valid.
4.1. L p -estimates. For each k > 0 let us define Using the Poincaré inequality we now establish an L p -estimate for f σ,k . In the codimension one case similar estimates have appeared in [Hui84] and [HS99]. and by Young's inequality for almost every t ∈ [0, T ). Using that f σ,k = f σ − k on supp(f σ,k ) and rearranging slightly, this Hence the function for almost every t ∈ [0, T ). Since ϕ is Lipschitz continuous in time it follows that In particular, ϕ can be bounded from above in terms of its value at the initial time, and the constants C 4 , η, σ, p and T . Recall that C 4 depends only on n, η, Λ and L. Also, f p σ,k ≤ C p 0 |H| σp , so ϕ(0) can be bounded purely in terms of n, Λ, σ, p, L and µ 0 (M ). This completes the proof.
4.2. The supremum estimate. Combining the L p -estimates just established with the Michael-Simon Sobolev inequality [MS73] we obtain the following iteration inequality. The proof is very similar to that of Theorem 5.1 in [Hui84], so we omit the details.
Recall that η depends only on n and ε 0 , and we chose Λ depending only on n, ε 0 and ε. Therefore, by the corollary we have an estimate of the form where C = C(n, ε 0 , ε, L, µ 0 (M ), T ). Appealing to the codimension estimate of Theorem 4.1, we finally obtain where C has the same dependencies as before. From here, since ε can be made arbitrarily small, an application of Young's inequality to the two lower-order terms on the right-hand side gives the convexity estimate of Theorem 1.1 (note that T can be bounded in terms of M 0 by applying the maximum principle to the evolution equation of W ).
Remark 4.5. There is another way to prove Theorem 1.1 using compactness and the strong maximum principle, which we now sketch. Let F : M × [0, T ) → R n+k be a quadratically pinched mean curvature flow with n ≥ 5 and c < c n , fix a constant ε > 0, and set Then the δ j form a bounded nondecreasing sequence and therefore converge to some δ. Choose a sequence (x j , t j ) ∈ P j such that f ε (x j , t j ) → δ and form a sequence of solutions by shifting F (x j , t j ) to the spacetime origin in R n+k × R and parabolically rescaling by |H|(x j , t j ). Then the gradient estimates established in Section 3 of [Ngu18] ensure that this sequence converges smoothly in a small spacetime neighbourhood about the origin. The limit lies in R n+1 by Naff's codimension estimate, and the quantity f ε attains its minimum δ at the spacetime origin. The strong maximum principle applied to the evolution equation for f ε shows that |∇h| ≡ 0 on the limiting solution, which is consequently either a piece of shrinking sphere or cylinder. In either case we have λ 1 ≥ 0, so δ > 0. In other words, on the original solution, there exists a threshold C ε depending on ε and M 0 such that whenever |H|(x, t) ≥ C ε there holds Note that this argument does not yield a quantitative blow-up rate for the negative part of the second fundamental form, in contrast to the estimate (4.4) proven above. Moreover, the gradient estimates in [Ngu18] are difficult to establish and will not be available in other situations where the Stampacchia iteration goes through. Indeed, an estimate showing asymptotic positivity of curvature is often needed to establish gradient estimates; this is the case for the fully nonlinear flow studied in [BH17] and for three-dimensional Ricci flow [Per02].

SINGULARITY FORMATION
In the study of parabolic evolution equations it is natural to distinguish between singularities which form at different rates. For a solution of mean curvature flow F : M × [0, T ) → R n+k where T is the maximal time we say that a type I singularity forms as t → T if there is a positive constant C such that max Note that this is the blow-up rate for solutions which shrink homothetically (such as shrinking spheres and cylinders). If on the other hand then the singularity forming at time T is said to be of type II. For certain type I singularities, Baker used Huisken's monotonicity formula to show that appropriate rescalings about the singularity converge to a homothetically shrinking solution [Bak11]. Moreover, Baker could show that the only such solutions satisfying the quadratic pinching condition are shrinking spheres and (generalised) cylinders. The analogous result for mean-convex solutions of codimension one was proven earlier by Huisken [Hui90]. In [HS99] Huisken and Sinestrari used their convexity estimate to show that at a type II singularity, appropriate rescalings about the maximum of the curvature converge to a convex translating solution. In this section we use our convexity estimate to generalise their result to higher codimensions. Fix a smooth mean curvature flow F : M × [0, T ) → R n+k of dimension n ≥ 5 which is quadratically pinched with c < c n , and suppose a type II singularity is forming as t → T . Consider a sequence of timest j → T and let (x j , t j ) be such that Then we have By the type II assumption, for each K > 0 there is a point (y, τ ) ∈ M × [0, T ) such that If j is large enough so thatt j > τ then we have Hence if j is sufficiently large there holds and since K can be made arbitrarily large this shows that It follows that t j → T . Let L 2 j := |H| 2 (x j , t j ) and consider the sequence of rescaled solutions defined by where H j is the mean curvature vector of F j . More generally, for t ≤ L 2 Therefore, for times t ≤ δL 2 Passing to a subsequence in j, we can guarantee that there is a sequence τ j → ∞ such that It is well known that for a compact solution of mean curvature flow, a global upper bound for |A| implies bounds on all of the higher derivatives of A. This follows from the estimates in [EH91] in the codimension-one case, and similar arguments work in higher codimensions (the details can be found in Section 4.3 of [Bak11]). Standard compactness theorems therefore imply that there is a smooth solutionF :M × (−∞, ∞) → R n+k such that the sequence F j subconverges smoothly toF in the following local sense. There is a sequence of nested open sets U l ⊂M such that M = l∈N U l and local diffeomorphisms ϕ l : U l → M such that the sequence converges smoothly to (x, t) →F (x, t), (x, t) ∈ U l × [−l, l] as j → ∞ for every l ∈ N. This follows for example from Hamilton's compactness theorem [Ham95a], as is illustrated in Section 6.1 of [Bak11].
Theorem 5.1. The smooth limiting solutionF :M × (−∞, ∞) → R n+k obtained by the above rescaling procedure lies in an (n + 1)-dimensional affine subspace and is either: a strictly convex translating solution; or the isometric product of R m with a strictly convex translating solution of dimension n − m.
Proof. We denote the second fundamental form ofF byÃ, and use this convention for other curvature quantities as well. We know that the mean curvature vector ofF satisfies |H|(0, 0) = 1, but a priori, there could be a point on the limiting solution where |H| = 0. However the evolution equation is still valid, so by the strong maximum principle we either haveW > 0 orW ≡ 0. In the latter situation |H| 2 (0, 0) = 0, which is a contradiction, so appealing to the pinching we conclude that 4 3n |H| 2 >W > 0 onM × (−∞, ∞). Hence Naff's codimension estimate impliesh ≡Ã, and consequently, the image ofF lies in an (n + 1)-dimensional subspace of R n+1 . By the convexity estimate,h ≥ 0 oñ M × (−∞, ∞).
To recap, the blow-up limitF is codimension one, has nonnegative second fundamental form, its scalar mean curvature |H| is globally bounded from above by one, and this global upper bound is attained at the spacetime origin. This is exactly the situation considered in Section 4 of [HS99]. Applying Hamilton's strong maximum principle for tensors to the evolution of the second fundamental form, (∂ t − ∆)h i j = |h| 2hi j , we conclude that the solutionM t :=F (M, t) splits as an isometric product R m × N t , where N t is a strictly convex solution of dimension n − m which exists for all t ∈ (−∞, ∞). Since the spacetime maximum of the mean curvature of N t is attained at the spacetime origin, the rigidity case of Hamilton's Harnack inequality [Ham95b] implies the family N t moves by translation.
Remark 5.2. By the gradient estimate in [Ngu18], the limiting flowF cannot be the product of R n−1 with a grim reaper. On the other hand, the grim reaper is the only strictly convex translator in R 2 , so we conclude that m ≤ n − 2.
Remark 5.3. If N t is uniformly two-convex in the sense that the smallest two principal curvatures satisfy λ 1 + λ 2 ≥ αH globally for some α > 0, then by the gradient estimate in [Ngu18] and work of Bourni-Langford [BL16], N t must be rotationally symmetric and hence a bowl soliton of dimension m. See also the paper [Naf19b].