A SHORT PROOF OF ALLARD’S AND BRAKKE’S REGULARITY THEOREMS

. We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.


Introduction
Allard's and Brakke's ε-regularity theorems are key tools in the study of, respectively, minimal surfaces and mean curvature flows, and they can be roughly stated as follows: If a m-dimensional minimal surface (resp.the space-time track of a mean curvature flow) is sufficiently flat in the ball of radius 1 (resp.parabolic cylinder of radius 1) and its area is roughly the one of the unit m-dimensional disk (resp.the weighted Gaussian density of a unit disk) then in a smaller ball (resp.parabolic cylinder) it can be written as the graph of a smooth function which enjoys suitable a-priori estimates.
See Theorems 2.1 and 3.8 below for the rigorous statement.Note these results are also relevant in the smooth category, since the scale and the regularity of the graphical parametrization of the surface only depend on the a priori assumption of (weak) closeness to the m-dimensional unit disk.
The original proofs by Allard and Brakke are modeled on the pioneering ideas introduced by De Giorgi in the regularity theory for co-dimension 1 area minimizing surfaces, [6] and on their implementation done by Almgren in [2].In particular the proof is, roughly speaking, divided into the following steps: (a) Under the desired assumption it is possible to show that most of the surface can be covered by the graph of a Lipschitz function, whose W 1,2 norm can be estimated by the difference in area between the surface and the plane.(b) Since the minimal surface equation (respectively the mean curvature flow) linearizes on the Laplacian equation (resp.the heat equation), this function is close to a harmonic (resp.caloric) function which enjoys strong a priori estimates.(c) These estimates can be pulled back to the minimal surface (resp.space-time track of the mean curvature flow) to show that the initial assumptions are satisfied also in the ball of radius 1/2.A suitable iteration provides then the conclusion.In this approach the two main difficulties lie in the approximation procedure in step (a) and in proving that the closeness in step (b) is in a strong enough topology to be able to pull-back the estimates from the linearized equation.Among various references, we refer the reader to [7] for a very clear account of Allard's theorem and to [11] for a simplified proof of Brakke theorem.Inspired by the work of Caffarelli and Cordoba [5], in [12] Savin provided a viscosity type approach to the above proofs, in which step (a) is completely avoided and step (b) is replaced by a partial Harnack inequality, obtained via Aleksandrov-Bakelmann-Pucci (ABP) type estimates, see [14] and [19] for the extension to the minimal surface system and to parabolic equations, respectively.This approach has been recently exploited by the second named author to prove a boundary version of Brakke's regularity theorem, [8].We also mention that Savin's partial Harnack inequality has been a crucial ingredient in the proof of the De Giorgi conjecture on solutions of the Allen Cahn equation, [13].
In this note we show how combining both the "variational" and the "viscosity" approach, it is possible to obtain a very short and self-sufficient proof of both Allard's and Brakke's theorems.The key observation is that while viscosity techniques are very robust in allowing to pass to the limit in the equation under L ∞ convergence (a key step in Savin's approach) the ABP estimate can be replaced by a simple variational argument based on the fact that coordinates are harmonic (resp.caloric) when restricted to the minimal surface (resp.mean curvature flow) and that for harmonic functions the mean value inequality can be easily obtained by testing the weak formulation of the equation with a suitable truncation of the fundamental solution, see for instance [4].
This short note is organized as follows: in Section 2 we prove Allard's theorem and in Section 3 we prove Brakke's theorem.In order to make the note self-contained, we conclude with an appendix where we record the proof of the maximum principle for varifolds and Brakke flows.Although the proofs are done in the natural context of varifolds and Brakke flows, we invite the reader to take in mind the simple case of a smooth surface with zero mean curvature and a smooth mean curvature flow.
For the purpose of open access, the authors have applied a Creative Commons Attribution (CC-BY) license to any Author Accepted Manuscript version arising from this submission.

Allard's regularity theorem
In the following, we denote by M m (U ) the space of m-dimensional rectifiable Radon measures on U , namely those Radon measures M on U for which there is a m- ω m r m wherever the limit exists.For M -almost every x, the approximate tangent plane to M at x is well defined and we denote it by T x M ∈ Gr(m, d).For S ∈ Gr(m, d) and where {η i } m i=1 is any orthonormal basis of S. Next, for every p ∈ (1, +∞], we let M p m (U ) be the set of those measures M ∈ M m (U ) such that, for every , where p ′ is the conjugate exponent of p.To each M ∈ M p m (U ) we associate a vector field We call H M the generalized mean curvature vector of M .Whenever the indication of M is unnecessary, we write H in place of H M .Before stating the main result of this section, we introduce the following notation: for which corresponds to the radius of the smallest cylinder of the form {x ∈ R d : The goal of the present section is proving the following version of Allard's theorem: Theorem 2.1 (Allard's regularity theorem).For every α ∈ (0, 1), there are δ 0 > 0 and C > 0 with the following property.Let M ∈ M ∞ m (B 1 ) and assume that 0 If H is more regular, then Schauder estimates entail higher regularity for supp M , as well.We shall prove Theorem 2.1 in Subsection 2.2.The next subsection is dedicated to proving a decay property of the oscillations of supp M , which allows us to prove an improvement of flatness (see Theorem 2.8 below).
2.1.Decay of oscillations.In the present subsection, we assume the following: for some C 0 universal.
Remark 2.4.Although this fact will not be used in the following, we point out that the above result holds true provided f is m-convex, meaning that the sum of the m smallest eigenvalues of D 2 f is non-negative.
Proof.We begin with some preliminary computations.Assume 1 m > 2 and let For any 0 < r ≤ s < R, we then let Note that g r,s ≥ 0 and g r,s ≡ 0 outside B s .Straightforward computations give, using (2.4) and (2.5), for any S ∈ Gr(m, d): Moreover, since f is non-negative and convex div S (f ∇g r,s − g r,s ∇f ) = f div S (∇g r,s ) − g r,s div S (∇f ) (2.6) Let now By (2.2) and (2.6), for any 0 < r < s ≤ R, it holds By Hölder's inequality, we may estimate and where in the second inequality we have used the fact that ∇f ∞ ≤ 1 and Item 2 in Assumption 2.2 above.Direct computations give, using (2.4) and (2.5) together with the convexity of t → t −m , lim sup for some C depending only on m.Thus, letting s ց r in (2.7), together with standard approximation arguments (see for example [15, §17]), gives in the sense of distributions.We now multiply both sides of the latter inequality by the integrating factor F (r) = e CΛr and integrate from some h > 0 to r to obtain where in the second inequality we have used the facts that e t ≥ 1 + t and that, by Item 2 in Assumption 2.2, with the caveat that C 0 in the conclusion of Proposition 2.6 depends on E 0 .
The above result allows us to prove a partial Harnack inequality.We refer the reader to (2.3) for the definition of osc S (M, B r (x 0 )).Whenever x 0 = 0 and the indication of M and of the m-plane S is unnecessary, as it is in the next two results, we write osc(B r ) in place of osc S (M, B r (x 0 )).
Proposition 2.6 (Harnack inequality).Let M satisfy Assumption 2.2 and let 0 ∈ supp M .There exists a universal constant η ∈ (0, 1) such that, if and let Notice that, by assumption, Therefore, by Proposition 2.3, for every θ ∈ [η, 1/2] (to be determined later) it holds Note that supp f 1 ∩ supp f 2 = ∅.Using this fact and the assumptions M (B r ) ≤ 3 2 ω m r m and |y 1 |, |y 2 | ≤ 2εR ≤ 2ηR, we have for some C 1 depending only on m.We now specify our choices of θ and η.We first choose θ much smaller than min{C Fixing η ≤ θ 2 we obtain a contradiction from (2.9) and summing for i ∈ {1, 2}.
As a corollary, we have the following Proposition 2.7.There are positive universal constants β and C with the following property.Let M satisfy Assumption 2.2 and 0 ∈ supp M .Then, for every r ∈ Proof.By rescaling, assume R = 1.For some K large to be determined later, let We claim that, if η is the constant determined in Proposition 2.6, then for any r such that osc(B r ) ≤ ηr it holds (2.10) provided β is chosen small so that η β ≥ (1 − η) and C ≥ 1 1−η .
2 ω m r m ; and, for some ε ≤ ε 0 and S ∈ Gr(m, d), Then there is Before proceeding with the proof, we introduce the following notation.To every ) and H M = 0, we say that V M is stationary.
Proof.By rescaling and translating, we assume R = 1 and x 0 = 0. We argue by contradiction and compactness.Assume there are sequences ε j ց 0, c j ց 0 and ) such that the assumptions of the theorem are satisfied with ε 0 replaced by ε j and c replaced by c j , for which however (2.11) fails for any choice of η and T .
Before proceeding, we remark that, by compactness (see, for instance, [15, Theorem 42.7]), there is Let now Σ j = supp M j ; for every x ∈ R d , define F j (x) = (Sx, ε −1 j S ⊥ x) ∈ R d and let and that they are non-empty, since by assumption 0 ∈ Σ j for every j.Therefore there is a relatively closed set Σ in such that, up to subsequences, Σ j converges to Σ in the Hausdorff distance.For every x ′ ∈ B S 1 , we let u(x ′ ) = {y ∈ S ⊥ : (x ′ , y) ∈ Σ}.
First, we show that u(x ′ ) = ∅ for every x ′ ∈ B S 1 .Indeed, otherwise, there would be r > 0 such that for every j large enough 2 ) = ∅, hence, by compactness and lower semicontinuity of the mass, which is false.Secondly, we prove that u(x ′ ) is a singleton for every x ′ ∈ B S 1/2 .Indeed, by Proposition 2.7, for every j, every x ∈ Σ j ∩ B 1/2 and every r ∈ Cε j , 1  2 , it holds osc S (M j , B r (x)) ≤ Cε j r β .Therefore, by Hausdorff convergence, for every x, y In particular, u(x ′ ) is a singleton; for the rest of the proof, we denote by u(x ′ ) the only element of u(x ′ ).Note further that this implies that u ∈ C 0,β (B S 1/2 ).By Lemma 2.9 below, u is harmonic in B S 1/4 .In particular, since sup B S 1/4 |u| ≤ 2, classical elliptic estimates yield sup for some C universal.Since u(0) = 0, we may choose η small depending only on C and α so that 2η .However, this implies that, for every j large enough and every , which contradicts the assumptions made at the beginning of the proof.Lemma 2.9.u defined in the proof of Theorem 2.8 is harmonic.
Proof.We argue as in [14,Lemma 2.4].Let h : B S 1/4 → S ⊥ be the harmonic function such that (h − u)| ∂B S 1/4 = 0.If u = h, then there is 0 < δ < 1/2 small such that, for all j large enough, the function is such that G j | Σ j achieves its maximum at some point x j with |Sx j | ≤ 1 4 − δ.We claim that j can be chosen so large that, for every T ∈ Gr(m, d), it holds (2.12) div T ∇G j (x j ) > H M j L ∞ |∇G j (x j )|, which would contradict Proposition A.1.In the rest of the proof, C denotes constants (whose value may change from one expression to the other) which depend only on d, m and δ, but they are independent of j.We start by noticing that, by standard elliptic estimates, max{|∇h(Sx Therefore |∇G j (x j )| ≤ C ε j for j large enough and H M j ∞ |∇G j (x j )| ≤ Cc j → 0 as j → ∞.Thus, in order to prove (2.12), it is sufficient to prove that for j sufficiently large (2.13) inf and let ½ S denote the orthogonal projection onto S. Then In particular, since ∆h = 0, where trace S A j = m i=1 A j ξ i •ξ i for any orthonormal basis {ξ i } m i=1 of S. Since |A j | ≤ C, by continuity it holds div T ∇G j (x j ) ≥ δ for every T ∈ Gr(m, d) such that |T − S| ≤ γ for γ > 0 sufficiently small.This proves (2.13) in the case |T − S| ≤ γ.
On the other hand, if |T − S| ≥ γ, then there is a unit vector η ∈ T such that |S ⊥ η| ≥ c 0 γ for some c 0 > 0 depending only on the dimension.Then div T ∇G j (x j ) ≥ trace T A j + |(∇f j (x j )) T η| 2 .

We have |trace
Notice that the conclusion of Theorem 2.8 may be iterated at all scales.In particular, if we let for some C large, then Theorem 2.8 yields the existence of some universal constants η and ε 0 such that

inf
S∈Gr(m,d) We finally prove Theorem 2.1.
Proof of Theorem 2.1.
Step 1.We claim that, if δ 0 is small enough, then there exists S ∈ Gr(m, d) such that the assumptions of Theorem 2.8 are in place for R = 1 4 and any x 0 ∈ supp M ∩ B 3/4 .We argue by compactness and contradiction: consider sequences δ j ց 0 and {M j } ⊂ M ∞ m (B 1 ) that satisfy the assumptions of Theorem 2.1 with δ 0 replaced by δ j .
We first remark that, up to subsequences, there exists M ∈ M m (B 1 ) such that Θ m (M, x) ≥ 1 M -almost everywhere, V M is stationary and M (B 1 ) ≤ ω m .By [1, Theorem 5.3], we have that M = H m S for some S ∈ Gr(m, d).
Given α ∈ (0, 1), let now ε 0 be the constant given in Theorem 2.8.Then, for j large enough, supp M j ⊂ {y : The only thing left to prove is that for every j large and every x ∈ supp M j ∩ B 3/4 .Given the above inequality, by monotonicity ([15, §17]), provided δ 0 is smaller than some universal constant, we obtain M j (B r (x)) ≤ 3 2 ω m r m for every 0 ≤ r ≤ 1 4 , as required in Theorem 2.8.We now prove (2.15).If the result is false, then there exist a subsequence j k and points Up to extracting a further subsequence, x k → x ∈ B 3/4 and, by monotonicity, x ∈ supp M .Therefore, for any ε > 0, contradicting (2.16) and thus proving the claim.
Step 2. We now prove that supp M ∩B 3/4 is the graph of some function u : S(supp M ) → R d−m .For the rest of the proof, we let where S ∈ Gr(m, d) was determined in the previous step, and we set Σ := supp M .By iterating Theorem 2.8, for every x ∈ Σ ∩ B 3/4 we find T x ∈ Gr(m, d) such that (2.17) and, for every r ≤ 1 4 , osc Tx (M, B r (x)) ≤ Cεr 1+α .It then follows that, for any two x, y ∈ Σ ∩ B 3/4 , it holds . together with (2.17), the above inequality shows at once that there is u : and that, for every x ′ ∈ S(Σ), there is a linear function L x ′ : S → S ⊥ such that, for every y ′ ∈ S(Σ), it holds Step 3. We conclude the proof by showing that S(Σ) ⊃ B m 1/2 .Once that is proved, (2.18) gives ||u|| C 1,α ≤ Cε, as desired.We argue by contradiction: if 1/2 and 0 ∈ supp M by assumption, there must be a ball B m r (x ′ ) ⊂ B m 1/2 \ S(Σ) and a point y ′ ∈ ∂B m r (x ′ ) ∩ S(Σ).If ε is smaller than some universal constant, then y = (y ′ , u(y ′ )) ∈ Σ ∩ B 3/4 .Consider a blow-up sequence (2.18) yields that supp M ∞ is included in a m-dimensional half-plane.This contradicts the Constancy Theorem (see, for instance, [15, Theorem 41.1]), concluding the proof.

Brakke's regularity theorem
In this section, we show how the arguments in the previous section may be adapted to the case of mean curvature flows.Inspired by [11], we give the following definition.(1) for almost every t ∈ [0, Ω], M t ∈ M 2 m (U ) and, for M t -almost every x, Θ m (M t , x) is a positive integer.
(2) For every W ⊂⊂ U , (3) For every non-negative test function where H is the generalized mean curvature vector of M t , as defined in (2.2), and v ⊥ (x) = (T x M t ) ⊥ v(x) for M t -almost every x.
For a Brakke flow with transport as above, we define the measure M on U × [0, Ω] by ϕ(x, t) dM (x, t) = ϕ(x, t) dM t (x) dt and the space-time track of M: at M -almost every (x, t).This fact is used in Proposition 3.5 below.

Decay of oscillations.
In the following, we denote by The following assumptions will be used in the present subsection.
Notice that, in this case, div S ∇f ≥ 0 for any S ∈ Gr(m, d).By (3.3) and the above inequality, we have In particular, for almost every t, it holds where we have used the facts that M t ∈ M 2 m for a.e.t and that M t (B 1 ) ≤ E 1 .Let now I(s) = f Ψ(•, s) dM s .We use φ = f Ψ as a test function in Definition 3.1.By (3.5), we obtain, for −1 ≤ t ≤ s < 0: where (3.2) was used in the above equality.We then use Young's inequality to bound and obtain where the latter inequality follows from Item 2 in Assumption 3.3.Therefore, going back to (3.6), dividing both sides by s − t and letting s ց t, we obtain in the sense of distributions, for some C depending also on E 1 .Hence for every −1 ≤ t ≤ s < 0 it follows that where in the second inequality we used the inequality e t ≥ 1 + t and the fact that I(s) ≤ CE 1 ε.With the above inequality at hand, it is fairly standard to prove that, for any two sequences (x j , t j ) → 0 and τ j ր 0, it holds By choosing (x j , t j ) such that M t j has an approximate tangent plane at x j and τ j converging to 0 fast enough, we find lim sup j→∞ f (•) Ψ(• − x j , τ j ) dM t j +τ j ≥ f (0), as desired.
Similarly to what we did in Section 2, for S ∈ Gr(m, d) and X ∈ Σ M we define the quantity Proposition 3.6 (Harnack inequality).For every E 1 , there exists η ∈ (0, 1) with the following property.Let M and v satisfy Assumption 3.3 with (0, 0) ∈ Σ M .Moreover, assume that, for every t Assume by contradiction that there are two points and consider Consider η ≤ θ ≤ 1 and let T = −θR 2 .Since the f i are convex and ∇f i ∞ ≤ 1, by Proposition 3.5 where we have used that f i ∞ ≤ εR and Λ ≤ εR −1 .By choosing θ small so that 3C 0 θ ≤ 1  10 and remarking that f i (y i ) ≥ 1 2 − 2η εR, we obtain Next, we bound from above the left-hand side of (3.7): provided η 2 is much smaller than θ chosen above, it holds T − s i ≤ − θ 2 R 2 , hence there is some constant where we have used the fact that M T (B R ) If η is smaller than some universal constant, then supp f 1 ∩supp f 2 = ∅, thus summing (3.7) and (3.8) for i = 1, 2, we obtain for some C depending on E 1 .Choosing η smaller, if needed, contradicts the assumption that Ψ R/2 (•, T ) dM T ≤ 3 2 , thus concluding the proof.As Proposition 2.6 implies Proposition 2.7, we obtain the following result as a corollary of Proposition 3.6.Proposition 3.7 (Decay of oscillations).For every E 1 , there exist C and β with the following property.Let M be a Brakke flow with transport term v in Q R that satisfies Assumption 3.3 with (0, 0) ∈ Σ M and such that, for every t Then, for every r ≥ C(osc 3.2.Brakke's theorem.The present subsection is dedicated to the proof of the following version of Brakke's regularity theorem: Theorem 3.8 (Interior regularity).For every α ∈ (0, 1) and every E 1 , there are positive and small constants θ and δ 0 with the following property.Let M be a Brakke flow with transport term v in Q 1 .Assume that: In the above statement, by u ∈ C 1,α (Q m θ/8 ; R d−m ) we mean that (x, t) → u(x, t) is differentiable with respect to x and that there is C > 0 such that, for every u C 1,α corresponds(up to a multiplicative constant) to the smallest C for which the above inequality holds.
Remark 3.9.This result is analogous to the "end-time regularity" proved in [16], although our proof requires the forcing term v to be a L ∞ function.We also remark that, differently from the case of minimal varifolds, higher regularity on v does not straightforwardly yield higher regularity for Σ M .In this regard, see [17].
Similarly to Theorem 2.1, Theorem 3.8 follows from an improvement of flatness, which we state and prove next.We first make some preliminary assumptions: convergence established in Step 2 and by classical Schauder estimates for the heat equation.
The following result was used in the proof of Theorem 3.11.
Proposition 3.12 (Compactness).Let {M j }, {v j } be two sequences such that, for each j, M j is a Brakke flow with transport term v j in Q R .Assume that, for every Then there exist a subsequence {j ℓ } ⊂ N and a Brakke flow Proof.The proof, in the case of Brakke flows without transport term, can be found in [10, §7]: the more general case can be proved by straightforward modifications.See also [9,Section 4.2.2].We conclude the present section by sketching the proof of Theorem 3.8.
Proof of Theorem 3.8.The proof is analogous to that of Theorem 2.1, thus we will only highlight the relevant differences.
Step 1 .We show that the assumptions of Theorem 3.11 are in place for every X 0 ∈ Σ M ∩ Q 3θ/4 and R = θ/4, provided θ and δ 0 in the statement of Theorem 3.8 are chosen small enough.In order to do so, we argue by contradiction.For E 1 and α fixed, assume there is a sequence {M j } of Brakke flows with transport term v j that satisfy the assumptions of Theorem 3.8 with δ 0 and θ replaced by some δ j ց 0 and θ j ց 0, respectively.By Proposition 3.12 above, the rescalings (where µ r (y) = y r ) converge to a Brakke flow M without transport term in R d × (−∞, 0].By Huisken's monotonicity formula, M must be a stationary unit-density plane.Therefore the assumptions of Theorem 3.11 are satisfied for j large enough.
letting h ց 0 in the above inequality gives the desired result.Otherwise, if 0 ∈ supp M then there is a sequence x j → 0 such that Θ m (M, x j ) ≥ 1.The result follows by continuity of f .Remark 2.5.The role of the factor 3/2 in Assumption 2.2 (2), as will be clear from the proof of Proposition 2.6 below, is to rule out the possibility that M consists of two separated sheets, which would clearly violate the conclusion of a Harnack inequality.On the other hand, for what concerns Proposition 2.3, by virtue of the classical monotonicity formula (see, for instance, [15, §17]), (2) in Assumption 2.2 may be replaced by the weaker assumptions