Abstract
For a general kdimensional Brakke flow in \({\mathbb {R}}^n\) locally close to a kdimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the endtime. Moreover, at any point in spacetime where the Gaussian density is close to 1, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White’s local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakkelike flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.
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1 Introduction
A family of kdimensional surfaces \(M_t\subset {\mathbb {R}}^n\) parameterized by time t is a mean curvature flow (abbreviated as MCF) if the normal velocity is equal to the mean curvature vector of \(M_t\). Given a smooth kdimensional submanifold \(M_0\), there exists a unique smooth MCF with initial datum \(M_0\) until singularities such as vanishing or neckpinching occur. To extend the flow beyond the time of singularity, numerous notions of generalized solution to MCF have been proposed since the 1970s: we mention, among others, the viscosity solutions produced by the level set method [3, 5], BV solutions [14], and varifold solutions [2, 21].
In the present paper, we focus on the varifold solutions known as Brakke flows, proposed and studied in Brakke’s pioneering work [2]. One of the main results of [2] is the partial regularity theorem of Brakke flows [2, 6.12], which states that any unit density Brakke flow is a smooth MCF for a.e. time almost everywhere. Since a timeindependent Brakke flow is a stationary varifold, and since in that case the unit density hypothesis means that the multiplicity function is equal to 1, the result may be seen as the natural parabolic counterpart of the wellknown result established by Allard in [1] in the context of stationary varifolds. For Brakke’s partial regularity theorem, as in many similar problems, the key ingredient is the proof of a “flatness implies regularity” type result, that is, an \(\varepsilon \)regularity theorem. This is referred to as Brakke’s local regularity theorem [2, 6.11] in this context. It states, roughly speaking, that if \(\{M_t\}_{t\in (\Lambda ,\Lambda )}\) is a Brakke flow in a cylinder
which is close to
in the sense of measure over \(t\in (\Lambda ,\Lambda )\), then, in the smaller cylinder \(\textrm{C}_1\), \(M_t\) coincides with a smooth graph over \(B_{1}^k\) evolving by MCF for \(t\in (\Lambda /2,\Lambda /2)\), with estimates on all the derivatives of such graph in terms of the overall height of \(M_t\). The constant \(\Lambda \) depends on how close \(M_t\) is to \(B_2^k\) in measure. While the original proof of Brakke’s local regularity theorem contained various gaps and errors, a rigorous proof was provided in [11, 20] with a different approach than Brakke’s, and for more general flows, allowing for an additive perturbation in the form of a forcing term in the righthand side of the underlying PDE.
Though this local regularity theorem is useful to prove the partial regularity of Brakke flows, there is a drawback in that it does not provide the regularity of the flow up until the “endtime”. Since the problem is parabolic in nature, one would expect the validity of interior estimates away from the “parabolic boundary” of \(B_2^k \times (\Lambda ,\Lambda )\), and thus that the graphical representation over \(B_1^k\) together with the corresponding estimates on the derivatives hold for \(t \in (\Lambda /2,\Lambda )\) instead of \((\Lambda /2,\Lambda /2)\).
The present paper addresses precisely this problem, and proves that such estimates are possible for Brakke flows, even when the aforementioned forcing term is present. There are many moreorless equivalent ways of stating the main regularity theorem proved here: an illustrative form is the following, where, for convenience, we discuss the simple case of Brakke flows with no forcing term and we change the time interval from \((\Lambda ,\Lambda )\) to \([2,0]\). For the sake of accuracy, the statement uses the varifold notation \(V_t\) (see [1, 11]), but the reader may think of the support of the weight measure \(\textrm{spt}\Vert V_t\Vert \) as \(M_t\).
Theorem 1.1
Corresponding to \(E_0 \in \left( 0,\infty \right) \), there exists \(\varepsilon _0= \varepsilon _0(n,k,E_0)\in (0,1)\) with the following property. Suppose \(\{V_t\}_{t\in (2,0]}\) is a kdimensional unit density Brakke flow in the cylinder \(\textrm{C}_3 = \textrm{C}(\mathbb {R}^k \times \{0\},3) \subset {\mathbb {R}}^n\) satisfying:

(1)
\(\sup _{t\in (2,0]}\Vert V_t\Vert (\textrm{C}_3)\le E_0\);

(2)
\(\Vert V_{4/5}\Vert (\textrm{C}_1)\le \frac{5}{4}\,\omega _k,\) (\(\omega _k= \text{ volume } \text{ of } B_1^k\));

(3)
\(0\in \textrm{spt}\Vert V_0\Vert \);

(4)
\(\cup _{t\in [1,0]}\,\textrm{spt}\Vert V_t\Vert \subset \{(x,y)\in {\mathbb {R}}^k\times {\mathbb {R}}^{nk}:y\le \varepsilon \}\) for some \(\varepsilon \in (0, \varepsilon _0]\).
Then, for every \(t \in \left[ 1/4,0\right) \), \(\textrm{C}_{1/2}\cap \textrm{spt}\Vert V_t\Vert \) is a \(C^\infty \) graph over \(B_{1/2}^k\) evolving by MCF, and the spacetime \(C^\ell \)norm of the graph on \(B_{1/2}^k\times [1/4,0)\) is bounded by \(c(\ell ,n,k,E_0)\varepsilon \) for any \(\ell \ge 1\).
Any Brakke flow locally satisfies the assumption (1) for some \(E_0>0\). The assumption (2) excludes the case of two parallel kdimensional planes, which is not a univalent graph, while (3) excludes the sudden vanishing of Brakke flow before the endtime \(t=0\). Since the definition of Brakke flow allows such irregularity, (3) (or some variant of similar nature) is necessary. The last (4) assumes that the height is kept small for \(t\in [1,0]\). The conclusion is that the Brakke flow is a smooth graph away from the parabolic boundary, and all derivatives can be controlled in terms of the height. Note that \(\textrm{spt}\Vert V_0\Vert \) may not be a smooth surface due to a possible (partial) sudden vanishing at \(t=0\), but we can smoothly extend \(\textrm{spt}\Vert V_t\Vert \) as \(t\rightarrow 0\) in \(\textrm{C}_{1/2}\) due to the estimates. As anticipated, the main result of the present paper is in fact more general. Precisely, the assumptions on the flow can be relaxed in various ways. First, the unit density assumption can be entirely dropped, and the theorem can be stated requiring \(\{V_t\}_{t \in (2,0]}\) to be a kdimensional integral Brakke flow in \(\textrm{C}_3\), instead. The reason is that assumption (2) prevents the presence of higher multiplicity points in a slightly smaller parabolic region, as one can see using Huisken’s monotonicity formula and a compactness argument, so that any kdimensional integral Brakke flow satisfying (1)–(4) for sufficiently small \(\varepsilon _0\) is necessarily unit density in a smaller parabolic region. Second, assumption (4) on the smallness of the height can be phrased in a weaker measuretheoretic sense: for the result to be valid, it is in fact sufficient that the (spacetime) \(L^2\)distance of the flow from the plane \(\mathbb {R}^k\times \{0\}\) in \(\textrm{C}_1 \times [1,0]\) (a quantity typically referred to as (\(L^2\))excess) is sufficiently small. Furthermore, the regularity result proved here is in fact valid for the larger class of Brakke flows with forcing term; more precisely, in this case we obtain \(C^{1,\zeta }\) (\(\zeta =1k/p2/q\)) or \(C^{2,\alpha }\) regularity estimates depending on whether the forcing is in the \(L^{p,q}\)integrability class or in the \(\alpha \)Hölder class, respectively. There are several reasons, stemming both from theoretical considerations and from the applications, leading one to consider Brakkelike flows with additional forcing term. A major one is the study of Brakke flows on a Riemannian manifold M: once M is (isometrically) embedded into some Euclidean space \({\mathbb {R}}^N\), the extrinsic curvatures of the immersion act as a forcing term in the corresponding definition of Brakke flow in M; see Sect. 2 for further details on this, and Theorems 2.2 and 2.3 for the precise statements of the main results.
We next discuss some related works. When the Brakke flow in Theorem 1.1 is a smooth MCF or is obtained as a weak limit of smooth MCF, the result has been known as a part of White’s local regularity theorem from [23], and it has been used widely in the literature of MCF to analyze the nature of singularities. White’s theorem applies, for instance, to Brakke flows obtained by the elliptic regularization method of Ilmanen [9], and, since the class of such MCF is weakly compact (see [23, Section 7]), to their tangent flows. The present paper shows that the same conclusions of White’s theorem in various forms hold true even without the proviso of approximability by smooth MCF, and can be derived solely from the definition of Brakke flow. As an illustration, using the main regularity theorem, we can prove the following.
Theorem 1.2
There exists \(\varepsilon _{1}=\varepsilon _{1}(n,k)\in (0,1)\) with the following property. Let \(\{V_t\}_{t\in (a,b]}\) be a kdimensional Brakke flow in a domain \(U\subset {\mathbb {R}}^n\) (or an ndimensional Riemannian manifold). For any point \((x,t)\in U\times (a,b]\) with the Gaussian density \(\Theta (x,t)\in [1,1+\varepsilon _{1})\) (see Sect. 2.6), there exists \(r>0\) such that \(B_r(x)\cap \textrm{spt}\Vert V_s\Vert \) is a smooth MCF in \(B_r(x)\) for \(s\in (tr^2,t)\) and can be extended smoothly to t in the limit as \(s\rightarrow t\).
We remark that there are, in the literature, existence theorems of Brakke flows for which one cannot tell a priori whether they arise as weak limits of smooth MCF or not. The examples include the limits of solutions to the Allen–Cahn equation [8, 18, 19] as well as the flows obtained by means of timediscrete approximate schemes [2, 12, 16, 17]. In the case of Brakke flows with no forcing term, Lahiri [13] showed an analogous endtime \(C^{1,\zeta }\) regularity result using some height growth estimates, a suitable constancy theorem for integral varifolds, and higher order derivative estimates. The proof is very different from that of the present paper, and it appears difficult to generalize it to flows with forcing term. More recently, Gasparetto [7] showed the validity of a similar endtime \(C^{1,\zeta }\) regularity result for Brakke flows with boundary, with a proof based on viscosity techniques. About six months after the present manuscript was made available as a preprint, De Philippis, Gasparetto, and Schulze provided in [4] an alternative proof—again based on viscosity techniques—of the endtime regularity result in the interior for Brakke flows possibly with forcing term in \(L^\infty \).
Next, we describe the idea of the proof. The proof of Theorem 1.1 is achieved by modifying suitable portions of the proof of the local regularity theorem in [11], so to extend the graphicality and the relevant estimates up to the endtime. Just as in many similar problems of this type, a fundamental step towards regularity is the proof of a Caccioppolitype estimate stating that a certain “Dirichlettype energy” can be controlled in terms of the \(L^2\)height of the solution. In the context of Brakke flows, such Dirichlet type energy corresponds, roughly speaking, to the difference (excess) of surface measure of \(\Vert V_t\Vert \) within the cylinder \(\textrm{C}_1\) and the measure \(\omega _k\) of the unit disk. Such difference is shown to be less than a constant times the \(L^2\)height of the flow by means of an ODE argument, see [11, Section 5]: indeed, one proves, by appropriately testing Brakke’s inequality, that the excess of measure—as a function of time—satisfies an ordinary differential inequality. The ODE argument implemented in [11], though, requires some “waiting time” both near the beginning and the end of the time interval: this is the main reason for the lack of estimates up to the endtime in [11]. A key point of the present paper is the observation that such waiting time becomes shorter when the height of the Brakke flow is smaller. The proof of the regularity then proceeds just like in Allard’s regularity theorem: the Brakke flow is approximated by a (parabolic) Lipschitz function, and one initiates a blowup argument. The approximating Lipschitz functions are rescaled by the height of the Brakke flow, but, thanks to the above mentioned key observation, in the process of passing to the limit as the height goes to 0, also the waiting time becomes infinitesimal. One can then show that the rescaled Lipschitz functions converge strongly in \(L^2\) to a solution of the heat equation as long as small neighborhoods of \(t=1\) and \(t=0\) are removed. The contribution to the \(L^2\)norms of the rescaled functions coming from the neighborhood of \(t=0\) can be made small, so that, in combination with the linear regularity theory of the heat equation, one obtains decay estimates for the linearized problem. By iterating, one concludes \(C^{1,\zeta }\) regularity and graphical representation of \(\Vert V_t\Vert \) on a parabolic region of spacetime which touches the origin. In particular, any point on the boundary of this parabolic region is in the support of \(\Vert V_t\Vert \), so that one can repeat the same argument regarding these points as the origin. This implies that the domain of graphicality with estimates can be extended so that it covers the whole support of \(\Vert V_t\Vert \) in \(\textrm{C}_{1/2}\times [1/4,0)\), proving the \(C^{1,\zeta }\) estimate up to the endtime. Once this is done, \(C^{2,\alpha }\) regularity up to the endtime can be obtained by repeating—with essentially no changes—the proof in [20]. Once the \(C^{2,\alpha }\) endtime regularity is available, the classical parabolic regularity theory gives all the higher derivative estimates for the Brakke flow with no forcing term, while the regularity theory for inhomogeneous linear heat equation implies the result when the forcing term is present.
The paper is organized as follows. In Sect. 2 we set up the notation in use throughout the paper, and we provide the formal statements of the main results in their full generality (see Theorems 2.2 and 2.3) as well as the proofs of Theorems 1.1 and 1.2 as a consequence of the general main results. Section 3 contains the enhanced ODE argument which gives energy estimates with short waiting time at the end of the time interval. In Sect. 4 we produce a parabolic Lipschitz approximation of the flow with good estimates up to the endtime, by suitably modifying the corresponding construction in [11, Section 7]. In Sect. 5, the main modification of the blowup argument is described and the main \(C^{1,\zeta }\) regularity on a parabolic domain touching the origin (a subdomain of \(\{(x,t): x^2<t\}\)) is obtained. In Sect. 6, we complete the proof of Theorems 2.2 and 2.3.
2 Assumptions and main results
2.1 Notation
Since the proof follows [11] very closely, we mostly adopt the same notation (see [11, Section 2]), except for a few symbols of norms. Throughout \(1\le k<n\) are fixed, and the dependence of constants on n and k is often not specified for simplicity. We set \({\mathbb {R}}^+:=\{x \in \mathbb {R}:\,x\ge 0\}\). For \(r\in (0,\infty )\) and \(a\in {\mathbb {R}}^n\) (or \(a\in {\mathbb {R}}^k\)) we set
and we often identify \({\mathbb {R}}^k\) with \({\mathbb {R}}^k\times \{0\}\subset {\mathbb {R}}^n\). When \(a=0\), we may write \(B_r\) and \(B_r^k\). For \(a\in {\mathbb {R}}^n\), \(s\in {\mathbb {R}}\) and \(r>0\) we define two types of parabolic cylinders
the first one was used in [11], whereas in the present paper we will prefer to work with the second one. We denote by \({\mathcal {L}}^n\) the Lebesgue measure on \({\mathbb {R}}^n\) and by \({\mathcal {H}}^k\) the kdimensional Hausdorff measure on \({\mathbb {R}}^n\). The restriction of a measure to a (measurable) set A is expressed by \(\lfloor _A\). For an open set \(U\subset {\mathbb {R}}^n\), \(C_c(U)\) is the set of continuous and compactly supported functions defined on U, and \(C_c^k(U)\) is the set of ktimes continuously differentiable functions with compact support in U. The symbols \(\nabla f\) and \(\nabla ^2 f\) always denote the spatial gradient and Hessian of f, respectively, and \(f_t=\partial _t f\) is the time derivative of f. For a function f defined on a domain in spacetime \(D\subset {\mathbb {R}}^n\times {\mathbb {R}}\) and \(\alpha \in (0,1)\), define the following (semi)norms to ease the notation in [11, 20]:
Let \(\textbf{G}(n,k)\) be the space of kdimensional linear subspaces of \({\mathbb {R}}^n\) and let \(\textbf{A}(n,k)\) be the space of kdimensional affine planes in \({\mathbb {R}}^n\). For \(S\in \textbf{G}(n,k)\), we identify S with the corresponding orthogonal projection matrix of \({\mathbb {R}}^n\) onto S. Let \(S^{\perp }\in \textbf{G}(n,nk)\) be the orthogonal complement of S. For \(A\in \textrm{Hom}({\mathbb {R}}^n;{\mathbb {R}}^n)\), we define the operator norm
and we often use this as a metric on \(\textbf{G}(n,k)\). For \(T\in \textbf{G}(n,k)\), \(a \in T\), and \(r\in (0,\infty )\) we define the cylinder
A general kvarifold on \(U\subset {\mathbb {R}}^n\) is a Radon measure defined on \(G_k(U):=U\times \textbf{G}(n,k)\) (see [1, 15] for a more comprehensive introduction), and the set of all general kvarifolds in U is denoted by \(\textbf{V}_k(U)\). For \(V\in \textbf{V}_k(U)\), let \(\Vert V\Vert \) be the weight measure of V (with no fear of confusion with the operator norm), that is the measure defined on U by
For a proper map \(f\in C^1({\mathbb {R}}^n;{\mathbb {R}}^n)\), the symbol \(f_{\sharp } V\) denotes the pushforward of the varifold V through f. We say that \(V\in \textbf{V}_k(U)\) is a rectifiable varifold if there are some \({\mathcal {H}}^k\)measurable and countably krectifiable set \(M\subset {\mathbb {R}}^n\) as well as a nonnegative function \(\theta \in L^1_{\textrm{loc}}({\mathcal {H}}^k\lfloor _M)\) such that
and in such case we write \(V = \textbf{var}(M,\theta )\). Here, \(\textrm{Tan}_x M\) is the approximate tangent space to M at x, which exists for \({\mathcal {H}}^k\)a.e. \(x\in M\). When \(\theta (x)\) is integervalued for \({\mathcal {H}}^n\)a.e. \(x\in M\), V is said to be an integral varifold. The set of all integral varifolds is denoted by \(\textbf{IV}_k(U)\). When \(\theta =1\) additionally, we say that V is of unit density. For \(V\in \textbf{V}_k(U)\), \(\delta V\) denotes the first variation of V and \(\Vert \delta V\Vert \) denotes the total variation of \(\delta V\). When \(\delta V\) is bounded and absolutely continuous with respect to \(\Vert V\Vert \), the Radon–Nikodym derivative (times \(1\)), \(\delta V/\Vert V\Vert \), is denoted by \(h(V,\cdot )\) and is called the generalized mean curvature vector of V. A fundamental geometric property of integral varifolds, of great importance for the analysis of Brakke flows, is Brakke’s perpendicularity theorem [2, Chapter 5]: if \(V\in \textbf{IV}_k(U)\) and \(h(V,\cdot )\) exists, then \(S(h(V,x))=0\) for Va.e. \((x,S)\in G_k(U)\).
For a oneparameter family of varifolds \(\{V_t\}_{t\in [a,b]}\), we often use \(\Vert V_t\Vert \times dt\) to represent the natural product measure on \(U\times [a,b]\); the latter is also expressed as \(d\Vert V_t\Vert dt\) within integration.
Fix \(\phi \in C^\infty ([0,\infty ))\) such that \(0\le \phi \le 1\),
For \(R\in (0,\infty )\), \(x\in {\mathbb {R}}^n\) and \(T\in \textbf{G}(n,k)\), define
and set
The functions \(\phi _{T,R}\) and \(\phi _T\) will be used as smooth test functions to gauge the measure deviation of \(\Vert V\Vert \) away from T with multiplicity one. Notice that \(\textbf{c}\) is independent of T.
2.2 Definition of Brakke flow
Since in this paper we are mostly interested in the endtime regularity, we consider time intervals of the form \([\Lambda ,0]\) with \(\Lambda >0\) in the following.
Definition 2.1
Suppose that \(U\subset {\mathbb {R}}^n\) is a domain and \(1\le k<n\). A family of varifold \(\{V_t\}_{t\in [\Lambda ,0]}\subset \textbf{V}_k(U)\) is a (kdimensional) Brakke flow if the following conditions are satisfied.

(1)
For a.e. \(t\in [\Lambda ,0]\), \(V_t\in \textbf{IV}_k(U)\).

(2)
For all , we have
$$\begin{aligned} \sup _{t\in [\Lambda , 0]}\Vert V_t\Vert ({\tilde{U}})<\infty . \end{aligned}$$(2.5) 
(3)
For a.e. \(t\in [\Lambda , 0]\), \(\delta V_t\) is locally bounded and absolutely continuous with respect to \(\Vert V_t\Vert \), and thus \(h(V_t,\cdot )\) exists. Furthermore, For all ,
$$\begin{aligned} \int _{\Lambda }^0\int _{{\tilde{U}}} h(V_t,x)^2\, d\Vert V_t\Vert dt<\infty . \end{aligned}$$(2.6) 
(4)
For all \(\varphi \in C^1(U\times [\Lambda ,0];{\mathbb {R}}^+)\) with \(\varphi (\cdot , t)\in C^1_c(U)\) for all \(t\in [\Lambda ,0]\), and for all \(\Lambda \le t_1<t_2\le 0\), we have
$$\begin{aligned} \begin{aligned}&\int _U\varphi (x,t_2)\,d\Vert V_{t_2}\Vert (x)\int _U \varphi (x,t_1)\,d\Vert V_{t_1}\Vert (x) \\&\quad \le \int _{t_1}^{t_2}dt\int _{U} \{(\nabla \varphi (x,t)h(V_t,x)\varphi (x,t))\cdot h(V_t,x)+\varphi _t(x,t)\}\,d\Vert V_t\Vert (x). \end{aligned}\nonumber \\ \end{aligned}$$(2.7)
The condition (4) is a weak formulation of MCF due to Brakke [2]. While Brakke’s original formulation of (2.7) is in the form of a differential inequality, nothing is lost if one works in this integral formulation. In fact, the latter is advantageous, in that it may easily accommodate the setting with additional unbounded forcing term as described in the next subsection.
One may naturally consider a MCF and the corresponding notion of Brakke flow in a general ndimensional Riemannian manifold M. By Nash’s isometric embedding theorem, we may always consider M to be a submanifold in a domain \(U\subset {\mathbb {R}}^N\) for some sufficiently large N. A Brakke flow in M can then be defined by asking \(\textrm{spt}\Vert V_t\Vert \subset M\) for all t, (1)–(3), and by replacing the inequality (2.7) by
Here, \(H_M(x,S)=\sum _{i=1}^k\textbf{B}_x(v_i,v_i)\in (\textrm{Tan}_x M)^\perp \), where \(\textbf{B}_x(\cdot ,\cdot )\) is the second fundamental form of \(M\subset {\mathbb {R}}^N\) at \(x\in M\) and the set \(\{v_1,\cdots ,v_k\}\) is an orthonormal basis of \(S\in \textbf{G}(n,k)\). See [20, Section 7] for a further explanation. The term \(H_M\) is already perpendicular to M and, for all analytical purposes, can be regarded as a locally bounded forcing term u as described in the next subsection.
2.3 Assumptions
The following assumptions are the same as [11], and we list them for the reader’s convenience.
For an open set \(U\subset {\mathbb {R}}^n\), suppose that we have a family of kvarifolds \(\{V_t\}_{t\in [\Lambda ,0]}\subset \textbf{V}_k(U)\) and a family of \((\Vert V_t\Vert \times dt)\)measurable vector fields \(\{u(\cdot ,t)\}_{t\in [\Lambda ,0]}\) defined on U and satisfying the following.

(A1)
For a.e. \(t\in [\Lambda ,0]\), \(V_t\) is a unit density kvarifold.

(A2)
There exists \(E_1\in [1,\infty )\) such that
$$\begin{aligned} \Vert V_t\Vert (B_r(x))\le \omega _k r^k E_1\,\, \text{ for } \text{ all } B_r(x)\subset U \text{ and } t\in [\Lambda ,0]. \end{aligned}$$(2.9) 
(A3)
The numbers \(p\in [2,\infty )\) and \(q\in (2,\infty )\) satisfy
$$\begin{aligned} \zeta :=1\frac{k}{p}\frac{2}{q}>0, \end{aligned}$$(2.10)and u satisfies
$$\begin{aligned} \Vert u\Vert _{L^{p,q}(U \times \left[ \Lambda ,0\right] )}:= \left( \int _{\Lambda }^0\left( \int _U u(x,t)^p\,d\Vert V_t\Vert (x) \right) ^{\frac{q}{p}}\,dt\right) ^{\frac{1}{q}} <\infty .\nonumber \\ \end{aligned}$$(2.11) 
(A4)
For all \(\varphi \in C^1(U\times [\Lambda ,0];{\mathbb {R}}^+)\) with \(\varphi (\cdot ,t)\in C^1_c(U)\) for all \(t\in [\Lambda ,0]\), and for all \(\Lambda \le t_1<t_2\le 0\), we have
$$\begin{aligned} \begin{aligned}&\int _U \varphi (x,t_2)\,d\Vert V_{t_2}\Vert (x)\int _U \varphi (x,t_1)\,d\Vert V_{t_1}\Vert (x)\\&\quad \le \int _{t_1}^{t_2}dt\int _U \{ (\nabla \varphi (x,t)h(V_t,x)\varphi (x,t))\cdot (h(V_t,x)\\&\qquad + u^{\perp }(x,t))+\varphi _t (x,t)\}\, d\Vert V_t\Vert (x). \end{aligned} \end{aligned}$$(2.12)Implicitly in the formulation of (A4), it is assumed that the first variation \(\delta V_t\) of \(V_t\) is locally bounded and it is absolutely continuous with respect to \(\Vert V_t\Vert \) (so that \(h(V_t,x)\) exists) for a.e. \(t\in [\Lambda ,0]\), and that \(h(V_t,x)\in L^2_{\textrm{loc}} (U\times [\Lambda , 0])\). For a.e. \(t\in [\Lambda ,0]\), \(u^\perp (x,t)\) is the projection of u onto the orthogonal complement of the approximate tangent space to \(V_t\) at x, which exists for \(\Vert V_t\Vert \)a.e. x due to the integrality of \(V_t\). The inequality (2.12) characterizes formally that the normal velocity of the flow is equal to the mean curvature vector h plus \(u^\perp \). When \(u \equiv 0\), (2.12) simply becomes (2.7), and thus \(\{V_t\}_{t \in \left[ \Lambda ,0\right] }\) is a Brakke flow. More generally, (2.12) includes the case when \(\{V_t\}_{t \in \left[ \Lambda ,0\right] }\) is a Brakke flow in a Riemannian manifold M, which corresponds to \(u(x,t):= H_M(x,\textrm{Tan}_x \Vert V_t\Vert )\): indeed, as already explained, in this case \(u(x,t) \in (\textrm{Tan}_x M)^\perp \), and thus in particular \(u(x,t) \in (\textrm{Tan}_x \Vert V_t\Vert )^\perp \) given that \(\textrm{spt}\Vert V_t\Vert \subset M\) for all t. One technical point to add is that (A1) may be replaced, for all purposes of the present paper, by
 \(({\mathrm{A1'}})\):

for a.e. \(t\in [\Lambda ,0]\), \(V_t\in \textbf{IV}_k(U)\).
The reason for this is that the assumptions of the main theorems essentially allow only unit density varifolds. We will nonetheless adopt (A1) as our working hypothesis, in order to be consistent with [11]. As already mentioned, there are in the literature various results guaranteeing the existence of (generalized) MCF (possibly with forcing term u) satisfying (A1)–(A4).
2.4 Main results
The first theorem is the basic \(\varepsilon \)regularity theorem, and it corresponds to a parabolic version of Allard’s regularity theorem; the second theorem gives a \(C^{2,\alpha }\) estimate. They are the endtime regularity counterpart of [11] and [20], respectively.
Theorem 2.2
Corresponding to \(\nu \in (0,1)\), \(E_1\in [1,\infty )\), p and q satisfying (2.10), there exist \(\varepsilon _{2}\in (0,1)\) and \(c_{1}\in (1,\infty )\) depending only on \(n,k,p,q,\nu ,E_1\) with the following property. For \(R\in (0,\infty )\), \(T\in \textbf{G}(n,k)\), and \(U=\textrm{C}(T,2R)\), suppose \(\{V_t\}_{t\in [R^2,0]}\) and \(\{u(\cdot ,t)\}_{t\in [R^2,0]}\) satisfy (A1)–(A4). Suppose furthermore that we have
Let \({\tilde{D}}:=\left( B_{R/2} \cap T\right) \times [R^2/4,0)\). Then there are \(C^{1,\zeta }\) functions \(f:{\tilde{D}}\rightarrow T^\perp \) and \(F:{\tilde{D}}\rightarrow {\mathbb {R}}^n\) such that \(T(F(y,t))=y\) and \(T^\perp (F(y,t))=f(y,t)\) for all \((y,t)\in {\tilde{D}}\),
As discussed in the Introduction, (2.13) excludes the possibility that \(V_t\) consists of multiple sheets in \(\textrm{C}(T,R)\), and it can replace the assumption that \(V_t\) be unit density. Notice that (2.13) is stated as a property valid at time \(4R^2/5\); nonetheless, the validity of (2.12) implies that in fact \(\Vert V_t\Vert (\phi _{T,R}^2)\) is an almostdecreasing function of t, even when the forcing term u is present. As a consequence, the mass estimate in (2.13) remains valid when \(\Vert V_{4R^2/5}\Vert \) is replaced by \(\Vert V_t\Vert \) for \(t > 4R^2/5\), modulo replacing \(\nu \) with \(\nu ' \in (0,\nu )\), provided \(\varepsilon _{2}\) is sufficiently small depending on \(\nu '\). The assumption (2.14) prevents sudden vanishing of the flow prior to the endtime. Finally, (2.15) is a smallness requirement on the (spacetime) \(L^2\)height of the flow, namely of the spacetime \(L^2\)distance of the flow from the given kdimensional plane T. We notice explicitly that, as a consequence of (2.17)–(2.18), one can naturally extend f and F to \(t=0\) as \(C^{1,\zeta }\) functions. Nonetheless, \(\textrm{C}(T,R/2)\cap \textrm{spt}\Vert V_0\Vert \subset \textrm{image}\, F\), but equality may not hold in general.
When u is \(\alpha \)Hölder continuous, we have the \(C^{2,\alpha }\)regularity estimate as follows.
Theorem 2.3
Corresponding to \(\nu \in (0,1)\), \(E_1\in [1,\infty )\) and \(\alpha \in (0,1)\), there exist \(\varepsilon _{3}\in (0,\varepsilon _{2})\) and \({c_{2}}\in (1,\infty )\) depending only on \(n,k,\alpha , \nu ,E_1\) with the following property. For \(R\in (0,\infty )\), \(T\in \textbf{G}(n,k)\), and \(U=\textrm{C}(T,2R)\), suppose \(\{V_t\}_{t\in [R^2,0]}\) and \(\{u(\cdot ,t)\}_{t\in [R^2,0]}\) satisfy (A1), (A2), (A4) and in place of (A3), assume \(u\in C^{0,\alpha }(\textrm{C}(T,2R)\times [R^2,0])\). Furthermore, assume (2.13), (2.14), (2.15) with \(\varepsilon _{3}\), and in place of (2.16),
Then the conclusion of Theorem 2.2 holds in the \(C^{2,\alpha }\) class, that is (2.18) can be replaced by
Moreover, \(\textrm{image}\,F\) satisfies in the classical (pointwise) sense the motion law that normal velocity \(=h+u^\perp \).
Here one can extend f and F as \(C^{2,\alpha }\) functions to \(t=0\) on \(B_{R/2} \cap T\). Once the regularity goes up to \(C^{2,\alpha }\) and the surfaces satisfy the PDE pointwise, then the parabolic Schauder estimates can be applied in the case that u is more regular. In particular, we will deduce \(C^{k+2,\alpha }\) estimates if \(u \in C^{k,\alpha }\). In the case of Brakke flow, when \(u=0\), we have all the derivative estimates in terms of \(\mu \).
In the next sections, we prove how the results stated in the Introduction, namely Theorems 1.1 and 1.2 can be deduced from Theorems 2.2 and 2.3.
2.5 Proof of Theorem 1.1
Let \(E_0 \in \left( 0, \infty \right) \), and suppose \(\{V_t\}_{t \in \left( 2,0\right] }\) is a kdimensional Brakke flow satisfying (1)–(4) in Theorem 1.1 with \(\varepsilon \in \left( 0, \varepsilon _0\right] \). We prove that, if \(\varepsilon _0\) is chosen sufficiently small, then \(\{V_t\}_{t \in \left[ 1,0\right] }\) satisfies the hypotheses of Theorem 2.3. We set \(R=1\), \(T=\mathbb {R}^k\times \{0\}\), and \(U=\textrm{C}(T,2)=:\textrm{C}_2\), and we notice that (A1)(A3)(A4) are satisfied by assumption. To check (A2), let \(t \in \left[ 1,0\right] \) and \(B_r(x) \subset U\): it is then a classical consequence (see e.g. [21, Proposition 3.5]) of Huisken’s monotonicity formula that
where c is a universal constant. This proves (A2). Next, using that
we see that (2) implies
that is (2.13) holds with \(\nu = 1/8\). Also, (2.15) with \(\varepsilon _{3}\) is an immediate consequence of (4) as soon as \(\varepsilon _0 \le \varepsilon _{3}\), whereas (2.14) follows from (3) and Huisken’s monotonicity formula (see, for instance, [21, Proposition 3.6]). Hence, Theorem 2.3 applies, and Theorem 1.1 follows from the fact that the forcing field \(u \equiv 0\) is smooth. \(\square \)
2.6 Proof of Theorem 1.2
In order to simplify the presentation, we will work under the assumption that \(U = \mathbb {R}^n\), and that \(\textrm{spt}\Vert V_t\Vert \subset B_R\) for every \(t \in \left( a, b \right] \), for some \(R >0\). The general case can be obtained with simple modifications, but the underlying idea is the same; see Remark 2.4.
Before proceeding with the proof, let us recall the classical definition of Gaussian density in the context of Brakke flows; see for instance [22] for a thorough presentation. Under the above assumptions, and setting \({\mathscr {V}} = \{V_t\}_{t\in \left( a, b \right] }\), for any point \((x_0,t_0) \in \mathbb {R}^n \times \left( a, b \right] \) we define
The existence of the above limit is guaranteed by the fact that the function
is monotone increasing as a consequence of Huisken’s monotonicity formula.
Step one. Assume first that \(\Theta ({\mathscr {V}},(x_0,t_0)) = 1\), and let \({\mathscr {V}}' = \{V_t'\}_{t \in \left( \infty ,0\right) }\) be any tangent flow to \({\mathscr {V}}\) at \((x_0,t_0)\). We then have
so that, in particular, \(\Theta ({\mathscr {V}}', (y,s)) \le 1\) for every \((y,s) \in \mathbb {R}^n \times \left( \infty , 0 \right) \). Since it is a general fact that \(\Theta ({\mathscr {V}},(y,s)) \ge 1\) for an integral Brakke flow \({\mathscr {V}}\) (see Appendix A), for every \((y,s) \in \textrm{spt}(\Vert V_t'\Vert \times dt)\), we have
This immediately implies (see e.g. [22, Theorem 8.1]) that there exist \(a \in \left[ 0,\infty \right] \) and \(T \in \textbf{G}(n,k)\) such that
namely that \({\mathscr {V}}'\) is a static kdimensional plane with unit density. Therefore, there exists \(\rho > 0\) such that the hypotheses of Theorem 2.3 are satisfied with \(R=\rho \) by the flow \(\{(\tau _{x_0})_\sharp V_{t_0+s}\}_{s \in \left[ \rho ^2,0\right] }\), where \(\tau _{x_0}\) is the translation \(\tau _{x_0}(x):= xx_0\). Thus, by Theorem 2.3, for all \(t \in \left[ t_0\rho ^2/4,t_0\right) \), \(\textrm{spt}\Vert V_t\Vert \cap \left( x_0 + \textrm{C}(T,\rho /2)\right) \) coincides with the graph of a \(C^\infty \) function
which satisfies the mean curvature flow in the classical sense and which can be extended smoothly on \(B_{\rho /2}(x_0) \cap (x_0+T)\) up to \(t=t_0\). This completes the proof in case \(\Theta ({\mathscr {V}},(x_0,t_0))=1\).
Step two. The proof that the same result holds when \(\Theta ({\mathscr {V}},(x_0,t_0)) \le 1+\varepsilon _1\) for some sufficiently small \(\varepsilon _1\) is by a standard blowup argument. First, notice that it is sufficient to prove that there exists \(\varepsilon _1>0\) such that if \({\mathscr {V}}\) is a tangent flow^{Footnote 1} and \(\Theta ({\mathscr {V}},(0,0)) \le 1 + \varepsilon _1\) then \({\mathscr {V}}\) is a static kdimensional plane with unit density.
To see this, let \(\{{\mathscr {V}}_j\}_{j \in {\mathbb {N}}}\) be a sequence of tangent flows such that \(\Theta ({\mathscr {V}}_j, (0,0)) \le 1 + {1/j}\), and notice that, for each j, the function
is constant, so that, in particular
Apply next the compactness theorem for Brakke flows, and let \({\mathscr {V}}\) be the limit Brakke flow of a (not relabeled) subsequence of \(\{{\mathscr {V}}_j\}_j\). We have then
and thus \(\Theta ({\mathscr {V}},(0,0))=1\). By Step one, \(\textrm{spt}\Vert V_t\Vert \) is a smooth graph evolving by mean curvature in some \(B_\rho (0)\) for all \(t \in \left[ \rho ^2,0\right) \), and the flow can be extended smoothly in \(B_\rho (0)\) up to \(t=0\). Since \({\mathscr {V}}\) is the limit Brakke flow of \({\mathscr {V}}_j\), then for all sufficiently large j also the flow \({\mathscr {V}}_j\) satisfies all assumptions of Theorem 2.3 (with \(u \equiv 0\)) in a parabolic domain \(\textrm{C}(T,\rho /2) \times [\rho ^2/16,0]\), and thus \({\mathscr {V}}_j\) is a smooth mean curvature flow in a neighborhood of \(x=0\) until the endtime \(t=0\). Since \({\mathscr {V}}_j\) is a tangent flow, it must then be a static kdimensional plane with unit density, and the proof is complete. \(\square \)
Remark 2.4
In case \(\{V_t\}_{t \in (a,b]}\) is a kdimensional Brakke flow in a domain \(U \subset \mathbb {R}^n\), the same proof goes through, except that we need a suitably modified monotonicity formula to make sense of the Gaussian density. More precisely, if \((x_0,t_0) \in U \times (a,b]\) and \(B_{2r}(x_0) \subset U\) then for any function \(\psi :B_{2r}(x_0) \rightarrow [0,1]\) that is smooth, compactly supported, equal to 1 on \(B_r(x_0)\) and satisfying a bound of the form \(r \nabla \psi  + r^2 \Vert D^2\psi \Vert \le b\), the limit
exists and it is independent of \(\psi \). This limit is the Gaussian density \(\Theta ({\mathscr {V}}, (x_0,t_0))\). The same limit also exists in the case when \({\mathscr {V}}=\{V_t\}_{t \in (a,b]}\) is a kdimensional Brakke flow in a domain U of an ndimensional Riemannian manifold M, or, more generally, when \({\mathscr {V}}=\{V_t\}_{t\in (a,b]}\) is a flow with a locally bounded forcing term u, with the only caveat that the proof of existence of the limit involves a more complicated monotonicity formula. Once the existence of the density has been established, tangent flows to such a \({\mathscr {V}}\) at \((x_0,t_0)\) are Brakke flows in \(\mathbb {R}^n\) (in the manifold case, we are identifying \(\mathbb {R}^n\) with \(\textrm{Tan}_{x_0}M\)), and the proof proceeds verbatim. For the proof of the monotonicity formulas needed in these cases, the interested reader can consult [22, Sections 10 and 11].
3 Energy estimates
The main result of this section is the following theorem, which establishes that the deviation of the kdimensional area of surfaces that are \(L^2\)close to a plane T and move by (forced) unitdensity Brakke flow from the area of a single kdimensional disk can be estimated in terms of the \(L^2\)height with respect to T. An analogous result was proved in [11, Theorem 5.7], but the version we are going to present here has an important advantage, which is ultimately the key to unlock the endtime regularity. More precisely, while [11, Theorem 5.7] concludes the validity of the estimate up to some waiting time both at the beginning and at the end of the time interval where the \(L^2\)height is assumed to be small, here we extend the estimate arbitrarily near the endtime as long as we know that the area of the moving surfaces is a sufficiently large portion of the area of the disk (namely, as long as we know that the flow is not vanishing). The price to pay is that the estimate comes with a constant which deteriorates while approaching the endtime. The endtime regularity will result from appropriately balancing the size of this constant with the vanishing of the \(L^2\)height along a blowup sequence.
Theorem 3.1
Corresponding to \(E_1\in [1,\infty )\) and \(\tau \in \left( 0,\frac{1}{2}\right) \), there exist \(\varepsilon _{4}=\varepsilon _{4}(E_1,\tau ) \in (0,1)\) and \(K = K(E_1) \in \left( 1,\infty \right) \) independent of \(\tau \) with the following property. Given \(T \in \textbf{G}(n,k)\), suppose that \(\{V_t\}_{t \in \left[ 1,0\right] }\) and \(\{u(\cdot ,t)\}_{t \in \left[ 1,0\right] }\) satisfy (A1)–(A4) with \(U = \textrm{C}(T,1)\). Assume also that
Then,
Furthermore, if
then
Before coming to the proof of Theorem 3.1, we record here the following result, which is [11, Proposition 5.2].
Proposition 3.2
Corresponding to \(E_1\in [1,\infty )\) and \(\nu \in (0,1)\) there exist \(\alpha _2\in (0,1)\), \(\mu _1\in (0,1)\), and \(P_2\in [1,\infty )\) with the following property. For \(T \in \textbf{G}(n,k)\) and a unit density varifold \(V \in \textbf{IV}_k (\textrm{C}(T,1))\) with finite mass, define
Suppose \(\textrm{spt}\Vert V\Vert \) is bounded and

(A)
If
$$\begin{aligned} \left \Vert V\Vert (\phi _T^2)  \textbf{c}\right \le \frac{\textbf{c}}{8}, \quad \alpha \le \alpha _2, \quad \text{ and } \mu \le \mu _1, \end{aligned}$$(3.11)then we have
$$\begin{aligned} \left \Vert V\Vert (\phi _T^2)  \textbf{c}\right \le {\left\{ \begin{array}{ll} P_2(\alpha ^{\frac{2k}{k2}} + \alpha ^{\frac{3}{2}} \mu ^{\frac{1}{2}} + \mu ^2) &{}\quad \text{ if } k > 2, \\ P_2 (\alpha ^{\frac{3}{2}} \mu ^{\frac{1}{2}} + \mu ^2) &{}\quad \text{ if } k \le 2. \end{array}\right. } \end{aligned}$$(3.12) 
(B)
If, instead
$$\begin{aligned} \frac{\textbf{c}}{8} < \left \Vert V\Vert (\phi _T^2)  \textbf{c}\right \le (1\nu ) \textbf{c}\quad \text{ and } \mu \le \mu _1 \end{aligned}$$(3.13)then \(\alpha \ge \alpha _2\).
The following is an immediate corollary of Proposition 3.2, and it is [11, Corollary 5.3]
Corollary 3.3
Let \(\alpha _2,\mu _1\), and \(P_2\) be as in Proposition 3.2. Set \(\mu _2:= \min \{\mu _1, \left( \frac{\textbf{c}}{32P_2} \right) ^{1/2}\}\). For V and T as in Proposition 3.2, define \(\alpha \) and \(\mu \) as in (3.8) and (3.9). Also define
Assume (3.10) as well as
Then, we have
Proof of Theorem 3.1
The general scheme follows the proof of [11, Theorem 5.7]. We define the function
where
Arguing precisely as in the proof of [11, (5.53)], namely by testing Brakke’s inequality (2.12) with \(\varphi = \phi _T^2\), we conclude that
We first prove (3.5). Towards a contradiction, suppose that there exists \(t_* \in \left[ \frac{1}{2},0\right] \) such that
where \(1< K < \infty \) will be chosen later. In particular, from the definition of E(t) we have for every \(t \in \left[ 1,t_*\right] \) that
if we choose \(K \ge \max \{1, 2 K_2\}\). On the other hand, we also have, due to (3.18), (3.2), (3.3), and (3.4),
for \(\varepsilon _{4}\) suitably small. In particular, if \(P_2\) is the constant from Proposition 3.2 corresponding to \(E_1\) and, for instance, \(\nu =1/2\), then choosing also \(K \ge 4P_2\) we have that
Hence, we can apply Corollary 3.3 with \(V=V_t\) for all \(t \in \left[ 1,t_*\right] \), and conclude that for a.e. \(t\in \left[ 1,t_*\right] \) it holds
where
and \(\alpha _2 \in \left( 0,1\right) \) is the same constant as in Proposition 3.2 corresponding to \(E_1\) and \(\nu = 1/2\). Let us consider the case \(k > 2\), as the case \(k \le 2\) is easier and can be treated similarly. Note that, since \(\varepsilon _{4} < 1\),
On the other hand, for \(t \in \left[ 1,t_*\right] \) we have
so that the first alternative does not occur. Let \({\bar{t}}\) be the supremum of \(s \in [1,t_*]\) such that \(\mu _*^{\frac{2k}{k+6}} \le E(t) \le 1\) for \(t \in \left[ 1, s \right] \). Then, (3.18) and (3.22) imply that the differential inequality \(E'(t) \le  P E(t)^{\frac{k2}{k}}\) is satisfied a.e. on \(\left[ 1, {\bar{t}} \right] \). Integrating and using (3.3), we find then that
In particular, for \(\varepsilon _{4}\) suitably small it is \({\bar{t}} < \frac{3}{4}\). By the monotonicity of E(t), we then have that the differential inequality \(E'(t) \le  P \mu _*^{\frac{2}{3}} E(t)^{\frac{4}{3}}\) is satisfied a.e. on \(\left[ {\bar{t}}, t_* \right] \), so that, integrating, we find
Since \(t_*{\bar{t}} \ge 1/4\), (3.23) is in contradiction with (3.20) as soon as we choose \(K \ge \frac{4}{P^3} 12^3\). This completes the proof of (3.5). Assume now that (3.6) holds, and let \({\bar{t}} \in \left[ \tau ,0\right] \) be such that
Towards a contradiction, assume that (3.7) is violated: due to (3.5), this means that there exists \(t_* \in \left[ \frac{1}{2},2\tau \right] \) such that
We then have
by monotonicity, and thus
On the other hand, again for \(t \in \left[ t_*,{\bar{t}} \right] \) we have
for \(\varepsilon _{4}\) sufficiently small, where we have used (3.24) together with (3.2) and (3.4). We can then apply again Corollary 3.3 with \(V=V_t\), \(t \in \left[ t_*,{\bar{t}}\right] \), and conclude that for a.e. \(t \in \left[ t_*,{\bar{t}} \right] \)
On the other hand, as a consequence of (3.26) we have that for every \(t \in \left[ t_*,{\bar{t}} \right] \)
and thus
Arguing as above, we only treat the case \(k>2\), and we notice that \(E(t)=E(t) < 1\). Assume that \({\hat{t}}\) is the infimum of \(s \in \left[ t_*, {\bar{t}} \right] \) such that \(E(t) \ge \mu _*^{\frac{2k}{k+6}}\) for all \(t \in \left[ s, {\bar{t}}\right] \). Then, (3.18) and (3.28) imply that the differential inequality \(E'(t) \le  P \left(  E(t) \right) ^{\frac{k2}{k}}\) is satisfied a.e. on \(\left[ {\hat{t}}, {\bar{t}} \right] \). Integrating we find that
In particular, for \(\varepsilon _{4}\) sufficiently small (depending on \(\tau \)) we have \({\hat{t}} \in \left[ \frac{3}{2}\tau ,{\bar{t}}\right] \). Now, by monotonicity of E(t), it holds \(E(t) \le \mu _*^{\frac{2k}{k+6}}\) on \(\left[ t_*,{\hat{t}} \right] \), and thus the differential inequality \(E'(t) \le  P \mu _*^{\frac{2}{3}} \left( E(t)\right) ^{\frac{4}{3}}\) holds a.e. on \(\left[ t_*,{\hat{t}} \right] \). We integrate to find that
which contradicts (3.25) if \(K \ge 2 (6/P)^3\) and completes the proof of (3.7). \(\square \)
4 Lipschitz approximation
The following proposition states the existence of a Lipschitz approximation of the flow in spacetime, with good estimates up to the endtime. The result is similar to [11, Theorem 7.5], the only difference being that the Lipschitz approximation is obtained up to the endtime. In the next Sect. 5, \(t=0\) in Proposition 4.1 will correspond to a time slightly before the endtime, up to which we have a good excess estimate.
Proposition 4.1
Corresponding to \(E_1\in [1,\infty )\), p and q, there exist \(\varepsilon _{5}\in (0,1)\), \(r_1\in (0,1)\) and \({c_{3}}\in [1,\infty )\) with the following property. For \(U=\textrm{C}(T,1)\), suppose that \(\{V_t\}_{t\in [3/5,0]}\) and \(\{u(\cdot ,t)\}_{t\in [3/5,0]}\) satisfy (A1)–(A4). Write \(V_t=\textbf{var}(M_t,1)\) for a.e. t and identify T with \({\mathbb {R}}^k\times \{0\}\). Suppose that we have
Set
and
Then there exist maps \(f\,:\, B_{1/3}^k\times [1/2,0]\rightarrow {\mathbb {R}}^{nk}\) and \(F\,:\,B_{1/3}^k\times [1/2,0]\rightarrow {\mathbb {R}}^n\times [1/2,0]\) such that for all \((x,s),\,(y,t)\in B_{1/3}^k \times [1/2,0]\),
and with the following property. Define
Then
Proof
To be consistent with the notation in [11, Section 7], we change the time intervals \([3/5,0]\) and \([1/2,0]\) in the statement above to [0, 1] and [1/4, 1] respectively in the following, which does not change the proof in any essential way. With this replacement, we discuss the proof. We simply describe the exact locations where we need to change in [11, Section 7] and the equation numbers are those of [11] in the following for the rest of the proof. For [11, Proposition 7.1], one replaces the parabolic cylinder \(P_r(a,s)\) in (7.3) and (7.4) by \({\tilde{P}}_r(a,s)\) defined in Sect. 2 and the same conclusion (7.6) follows by the same proof. Next, no change is required in [11, Lemma 7.3], where one obtains a small constant \(r_1\in (0,1)\) depending only on \(E_1,\,p\) and q. In the proof of [11, Theorem 7.5], one replaces (1/4, 3/4) by (1/4, 1) and P by \({\tilde{P}}\) in (7.58), (7.59), (7.62), (7.65) and (7.66). The only essential modification is the part following (7.66) on the covering argument. The modified statement (7.66) is the following: For each \((x,s)\in B\), there exists some \(r(x,s)\in (0,r_1)\) such that
This follows from the definition of A, (7.58). Thus \(\{\overline{{\tilde{P}}_{r(x,s)}(x,s)}\}_{(x,s)\in B}\) is a covering of B. Here, unlike \(P_r(x,s)\), since \({\tilde{P}}_{r}(x,s)\) is not a metric ball with respect to the metric \(d((x_1,s_1),(x_2,s_2)):=\max \{x_1x_2,s_1s_2^{1/2}\}\), we cannot invoke the standard Vitali covering lemma as given. On the other hand, by following the same proof of the Vitali lemma applied to \(\{\overline{{\tilde{P}}_{r(x,s)}(x,s)}\}_{(x,s)\in B}\) (see for example [15, Theorem 3.3]), one can prove that there exists a countable subset \(\{\overline{{\tilde{P}}_{r(x_j,s_j)}(x_j,s_j)}\}\subset \{\overline{{\tilde{P}}_{r(x,s)}(x,s)}\}_{(x,s)\in B}\) such that it is pairwise disjoint and
Note that the righthand side are the closed metric balls with respect to the parabolic distance. Then, using the above inequality and the property of the covering,
The rest of the proof is the same. \(\square \)
Remark 4.2
In [11], the generalized Besicovitch covering theorem in [6, 2.8.14] was invoked for parabolic cylinders at the bottom of page 40. After the publication of [11], Ulrich Menne communicated the secondnamed author that the parabolic cylinders do not satisfy the assumption in [6, 2.8.14] (called directionally \(\xi ,\,\eta ,\,\zeta \) limited), so that the theorem is not applicable. However, one can fix the proof in [11] by using the Vitali covering lemma, which holds true for any metric balls, instead of using Besicovich. Later it was proved that, even though the precise assumption in [6] is not satisfied, the Besicovich covering theorem still holds true for parabolic cylinders of type P (not \({\tilde{P}}\)), see [10] for the proof.
5 Blowup argument
We first state the regularity result for a domain which is at positive distance away from the endtime \(t=0\). This is a direct consequence of [11, Theorem 8.7] with modifications to shorten the waiting time near the endtime.
Proposition 5.1
Corresponding to \(E_1\in [1,\infty )\), \(\nu \in (0,1)\), p, q and \(\iota \in (0,1/4)\), there exist \(\varepsilon _{6}\in (0,1)\), \(c_{4}\in (1,\infty )\) with the following property. For \(T\in \textbf{G}(n,k)\), \(R\in (0,\infty )\), \(U=\textrm{C}(T,2R)\), suppose \(\{V_t\}_{t\in [R^2,0]}\) and \(\{u(\cdot ,t)\}_{t\in [R^2,0]}\) satisfy (A1)–(A4) and (2.13)–(2.16) with \(\varepsilon _{6}\) in place of \(\varepsilon _{2}\). Write \({\tilde{D}}:=(B_R \cap T)\times [R^2/2,\iota R^2]\). Then there are \(f\,:\,{\tilde{D}}\rightarrow T^\perp \) and \(F:\,{\tilde{D}}\rightarrow {\mathbb {R}}^n\) such that \(T(F(y,t))=y\) and \(T^{\perp }(F(y,t))=f(y,t)\) for all \((y,t)\in {\tilde{D}}\) and
where the norms are measured on \((B_R \cap T)\times [R^2/2,\iota R^2]\).
Proof
We may assume that \(R=1\) by the parabolic change of variables. We first use the \(L^2L^\infty \) height estimate [11, Proposition 6.4] with \(R=1\), \(\Lambda =1\), \(U=B_1(a)\) with \(a\in T\cap B_1\) (and the timeinterval [0, 1] translated to \([1,0]\)), so that there exist \(c_{5}=c_{5}(k,p,q)\) and \(c_{6}=c_{6}(n,k)\) such that, for all \(t\in [4/5,0]\), we have
where
In particular, by moving a within \(T\cap B_1\), (5.3) shows
for all \(t\in [4/5,0]\). Using the lower density ratio bound (see [11, Corollary 6.3]), for all sufficiently small \(\varepsilon _{6}\) depending only on \(E_1\), p and q, one can show that
for all \(t\in [4/5,0]\). Thus, (5.5) and (5.6) show
for all \(t\in [4/5,0]\). Next, we use [11, Theorem 8.7]. Corresponding to \(E_1\), p and q with \(\nu =1/2\), there exist \(\varepsilon _{7}\in (0,1)\) (\(\varepsilon _6\) in [11]), \(\sigma \in (0,1/2)\), \(\Lambda _{1}\in (2,\infty )\) (\(\Lambda _3\) in [11]) and \(c_{7}\in (1,\infty )\) (\(c_{16}\) in [11]) with the properties stated there. We identify T with \({\mathbb {R}}^k\times \{0\}\) in the following. We fix a small \(0<{\tilde{R}}\le 1/6\) depending only on \(\iota \) and \(\Lambda _{1}\) (for example, \({\tilde{R}}=\sqrt{\iota /(4\Lambda _{1})}\)) so that, for any \((x,t)\in B^k_1\times [1/2,\iota ]\), we have
The choice of such \({\tilde{R}}\) depends ultimately only on \(\iota \), \(E_1\), p and q. We use [11, Theorem 8.7] with \(R={\tilde{R}}\) and \((x,t)\in B_1^k\times [1/2,\iota ]\) as the origin. There are four assumptions in [11, Theorem 8.7], the smallness of height [11, (8.83)] and \(\Vert u\Vert \) [11, (8.84)], and the existence of \(t_1\) and \(t_2\) in [11, (8.85)] and [11, (8.86)] with respect to \(B^k_{3{\tilde{R}}}(x)\times (t\Lambda _{1}{\tilde{R}}^2,t+\Lambda _{1}{\tilde{R}}^2)\) and \(\nu =1/2\). The first two conditions are fulfilled if we restrict \(\varepsilon _{6}\) so that \(\varepsilon _{6} {\tilde{R}}^{(k+4)/2}<\varepsilon _{7}\). In the following, we prove that the latter two are satisfied by using a compactness argument. Let \(\phi _{T,{\tilde{R}},x}\) be defined by \(\phi _{T,{\tilde{R}},x}(y):=\phi _{T,{\tilde{R}}}(yx)\). We claim that, given any \(\delta >0\), for all sufficiently small \(\varepsilon _{6}>0\) depending only on \(\iota ,\,E_1,\,\nu ,\,p,\,q\) and \(\delta \), we have
for all \((x,t)\in B^k_1\times [3/5,0]\). Note that, by using the monotone decreasing property of E(t) corresponding to \(\phi _{T,{\tilde{R}},x}\) in place of \(\phi _T\) in (3.18), the increase of \(\Vert V_t\Vert (\phi _{T,{\tilde{R}},x}^2)\) in time can be made small by restricting \(\mu \) and \(\Vert u\Vert \) appropriately depending on \(\delta \) and \({\tilde{R}}\) (in the following, we may refer to this fact as “almost monotone property”), so we need to prove \({\tilde{R}}^{k}\Vert V_{3/5}\Vert (\phi _{T,{\tilde{R}},x}^2)\le \textbf{c}+\delta \) for all \(x\in B^k_1\). Assume for a contradiction that there exist \(\{V_t^{(m)}\}_{t\in [1,0]}\) and \(\{u^{(m)}(\cdot ,t)\}_{t\in [1,0]}\) satisfying the assumptions of the present theorem with \(\varepsilon =1/m\), and \(x_m\in B_1^k\) such that \({\tilde{R}}^{k}\Vert V_{3/5}^{(m)}\Vert (\phi _{T,{\tilde{R}},x_m}^2)>\textbf{c}+\delta \). Again by the almost monotone property, we have
for all large m. Since
is uniformly bounded by (3.18) and (A2), using Fatou’s lemma and (A1) we conclude that for almost all \(t_0\in [4/5,3/5]\), there exists a subsequence \(V_{t_0}^{(m_j)}\in \textbf{IV}_k(\textrm{C}(T,2))\) such that the \(L^2(\Vert V_{t_0}^{(m_j)}\Vert )\)norms of \(\{h(V_{t_0}^{(m_j)})\}_j\) are bounded uniformly in \(\textrm{C}(T,3/2)\). Then, by Allard’s compactness theorem of integral varifolds, a further subsequence converges to \(V\in \textbf{IV}_k(\textrm{C}(T,3/2))\), and due to (5.7), it is supported on T. Since the \(L^2\)norm of the generalized mean curvature is lowersemicontinuous under varifold convergence, V has \(h(V,\cdot )\in L^2(\Vert V\Vert )\) in \(\textrm{C}(T,3/2)\) and the multiplicity of V on T has to be a constant function with integer value, and by (5.10), the integer has to be \(\ge 2\). But this implies that \(\liminf _{j\rightarrow \infty }\Vert V_{t_0}^{(m_j)}\Vert (\phi _T^2)\ge \Vert V\Vert (\phi _T^2)\ge 2\textbf{c}\). Since \(t_0\ge 4/5\) and by the almost monotone property, one can obtain a contradiction to (2.13) for all large \(m_j\). This proves (5.9). Similarly, we claim that, given \(\delta >0\), for small \(\varepsilon _{6}>0\),
for all \((x,t)\in B_1^k\times [3/5,\iota /2]\). Again by the almost monotone property, we need to prove the claim at \(t=\iota /2\). The similar contradiction argument applied to the time interval \([\iota /2,\iota /4]\) in place of \([4/5,3/5]\) (with the same notation) shows that, for almost all \(t_0\in [\iota /2,\iota /4]\), there exists a subsequence such that \(\lim _{j\rightarrow \infty } \Vert V_{t_0}^{(m_j)}\Vert =0\) on \(\textrm{C}(T,3/2)\). But then, with the clearingout lemma (see [11, Corollary 6.3]), one can show that \((\Vert V_t^{(m_j)}\Vert \times dt)(\textrm{C}(T,1)\times (\iota /8,0))=0\) for all large j (where \(\iota \) needs to be smaller than a constant depending only on k, n, p, q and \(E_1\) for the clearingout lemma). This is a contradiction to (2.14). This proves (5.11). Now we are ready to apply [11, Theorem 8.7]: we choose a small \(\delta >0\) so that \(\textbf{c}\delta >\textbf{c}/2\) and \(\textbf{c}+\delta <3\textbf{c}/2\) and let \(\varepsilon _{6}\) be restricted so that we have (5.9) and (5.11). Then for each \(T^{1}(B^k_{3{\tilde{R}}}(x))\times (t\Lambda _{1}{\tilde{R}}^2,t+\Lambda _{1}{\tilde{R}}^2)\) with \((x,t)\in B_1^k\times [1/2,\iota ]\), all the assumptions for [11, Theorem 8.7] are satisfied. Thus the support of \(\Vert V_t\Vert \) can be represented as the graph of a \(C^{1,\zeta }\) function in \(T^{1}(B^k_{\sigma {\tilde{R}}}(x)) \times (t{\tilde{R}}^2/4,t+{\tilde{R}}^2/4)\) with estimate in terms of \(\mu \) and \(\Vert u\Vert \). Since \(\textrm{C}(T,1)\times [1/2,\iota ]\) can be covered by a finite number of such domains, the support of the flow is represented as a \(C^{1,\zeta }\) graph over \(B^k_1\times [1/2,\iota ]\) with estimates in terms of \(\mu \) and \(\Vert u\Vert \). The resulting constant \(c_{4}\) depends only on \(E_1,\,\nu ,\,p,\,q,\,\iota \). This concludes the proof. \(\square \)
The constants in the claim of Proposition 5.1 deteriorate as \(\iota \) approaches to 0, and we will use it with a fixed \(\iota \) depending only on \(E_1\), \(\nu \) and \(\zeta \) in Proposition 5.3. We next prove the main decay estimate under the parabolic dilation centered at the endtime, which will be iterated to obtain the desired \(C^{1,\zeta }\) estimate.
Proposition 5.2
Corresponding to \(E_1\in [1,\infty )\), \(\nu \in (0,1)\), p and q there exist \(\varepsilon _{8}\in (0,1)\), \(\theta \in (0,1/4)\) and \(c_{8}\in (1,\infty )\) with the following property. For \(W\in \textbf{G}(n,k)\), \(0<R<\infty \) and \(U=\textrm{C}(W,2R)\), suppose that \(\{V_t\}_{t\in [R^2,0]}\) and \(\{u(\cdot ,t)\}_{t\in [R^2,0]}\) satisfy (A1)–(A4). Suppose
Then there are \({\tilde{T}}\in \textbf{G}(n,k)\) and \({\tilde{A}}\in \textbf{A}(n,k)\) such that
Moreover, if \(\Vert u\Vert <\varepsilon _{8}\), we have
Proof
We may assume that \(R=1\) after a parabolic change of variables. The outline of proof is similar to [11, Proposition 8.1], with the crucial difference that we work with (5.16) and that the result is for a domain centered at the endtime point \((x,t)=(0,0)\). We give a description on the different points on the proof for this result. The proof proceeds by contradiction. We will fix \(\theta \in (0,1/4)\) later depending only on \(E_1\) and \(\zeta \). If the claim were false, then, for each \(m\in {\mathbb {N}}\) there exist \(\{V_t^{(m)}\}_{t\in [1,0]}\), \(\{u^{(m)}(\cdot ,t)\}_{t\in [1,0]}\) satisfying (A1)–(A4) on \(\textrm{C}(W^{(m)},2)\times [1,0]\) for \(W^{(m)}\in \textbf{G}(n,k)\) such that, by assuming \(T={\mathbb {R}}^k\times \{0\}\) after suitable rotation,
(5.16) and (5.17), but for any \({\tilde{T}}\in \textbf{G}(n,k)\) with \(\Vert T{\tilde{T}}\Vert \le m\mu ^{(m)}\) and \({\tilde{A}}\in \textbf{A}(n,k)\) which is parallel to \({\tilde{T}}\), we have
By taking \({\tilde{A}}={\tilde{T}}=T\) in (5.24), we obtain
which shows in particular that
By (5.25), (5.4) and (5.7), we have
for all \(t\in [4/5,0]\), where \(\sqrt{c_{6}}=c(n,k)\). The same argument used to prove (5.9) combined with (5.17) shows
Using (5.22) and the similar argument leading to (5.10), one can prove that
Then, with (5.26)–(5.28), for all sufficiently large m, we may apply Theorem 3.1 with \(\tau =\theta ^6/2\). Thus there exists a constant \(c_{9}=c_{9}(\theta ,\nu ,p,q,E_1)\) such that
We now apply Proposition 4.1 with the time interval shifted from \([3/5,0]\) to \([3/5\theta ^6,\theta ^6]\). For all sufficiently large m, note that (4.2)–(4.4) are all satisfied due to (5.29), (5.26) and (5.25). The smallness condition of (4.1) can be proved by (A4) and (5.29) as it was done for (3.18). Thus we have Lipschitz functions \(f^{(m)}\) and \(F^{(m)}\) defined on \(B^k_{1/3}\times [1/2\theta ^6,\theta ^6]\) with quantities (4.5) and (4.6) defined in terms of \(V^{(m)}\) and where \(f^{(m)}\) and \(F^{(m)}\) satisfy (4.7)–(4.9). Once we achieve this, arguing exactly as in [11, p.45], one can prove that the righthand side of (4.9) corresponding to \(V^{(m)}\) can be bounded by \(c(\mu ^{(m)})^2\) with c depending only on \(\theta ,\,\nu ,\,E_1,\, p,\,q\). We define the blowup sequence by
for all sufficiently large m on \(B^k_{1/3}\times [1/2\theta ^6,\theta ^6]\). Writing \(\Omega ':=B^k_{1/3}\times (1/2\theta ^6,\theta ^6]\), the verbatim proof for [11, Lemma 8.3, 8.4] gives the existence of a subsequence \(\{{\tilde{f}}^{(m_j)}\}\) and \({\tilde{f}}\in C^{\infty }(\Omega ')\) such that
At this point, it is important to note that (5.26) gives
where \(c_{6}=c(n,k)\). We then define \(T^{(m)}\in \textbf{G}(n,k)\) as the graph of the map
which is the tangent space to the graph \(\{(x,\mu ^{(m)}{\tilde{f}}(x,\theta ^6)) \, :\, x \in B^k_{1/3}\}\) at \(x=0\), and also define the affine plane \(A^{(m)}\in \textbf{A}(n,k)\) by \(A^{(m)}=T^{(m)}+(0,\mu ^{(m)}{\tilde{f}}(0,\theta ^6))\). By the standard estimates for parabolic PDE, all the partial derivatives of \({\tilde{f}}\) on \(B^k_{2\theta }\times [\theta ^2,\theta ^6]\) are bounded in terms of constant multiple of \(\sqrt{c_{6}}\). In particular, there exists a constant \(c_{10}=c(n,k)\) such that
Following the verbatim proof in [11], this leads to
Thus, for all large m, we have
On the integral over the time interval \((\theta ^6,0)\), since \(\textrm{dist}\,(x,A^{(m)})\le c(c_{10})\mu ^{(m)}\) on the support of \(\Vert V_t^{(m)}\Vert \), combined with (A2), we have
where \(c_{11}\) depends only on \(c_{10}\) and \(E_1\). Then (5.35) and (5.36) show
Now, choosing \(\theta \) small enough depending only on \(n,\,k,\,E_1,\zeta \), we may assume that \((c_{10}+ c_{11})\theta ^2 <\theta ^{2\zeta }/2\). Since T can be replaced by \(W^{(m)}\) for the limit (see [11]) in (5.37), we have a contradiction to (5.24). This completes the proof of claims (5.18)–(5.20). For (5.21), since \(\theta \) is fixed, we may argue as for the proof of (5.9) and restrict \(\varepsilon _{8}\) to make sure that (5.21) holds. This completes the proof. \(\square \)
It is possible to apply Proposition 5.2 iteratively; in combination with Proposition 5.1, we have then the following.
Proposition 5.3
Corresponding to \(E_1\in [1,\infty )\), \(\nu \in (0,1)\), p and q, there exist \(\varepsilon _{9}\in (0,1)\) and \(c_{12} \in (1,\infty )\) with the following property. For \(T\in \textbf{G}(n,k)\), \(R\in (0,\infty )\) and \(U=\textrm{C}(T,2R)\), suppose that \(\{V_t\}_{t\in [R^2,0]}\) and \(\{u(\cdot ,t)\}_{t\in [R^2,0]}\) satisfy (A1)–(A4). Suppose
Identifying T as \({\mathbb {R}}^k\cong {\mathbb {R}}^k\times \{0\}\subset {\mathbb {R}}^n\), let \({\tilde{D}}:=\{(x,t)\in {\mathbb {R}}^k\times [R^2/2,0)\,:\, x^2< t\}\). Then there exist \(f\,:\,{\tilde{D}} \rightarrow T^{\perp }\) and \(F\,:\,{\tilde{D}}\rightarrow {\mathbb {R}}^n\) such that \(F(x,t)=(x,f(x,t))\) for \((x,t)\in {\tilde{D}}\) and

(1)
\(\textrm{spt}\Vert V_t\Vert \cap \textrm{C}(T,\sqrt{t})=\textrm{Image}\, F(\cdot ,t)\,\,\text{ for } \text{ all } \, t\in [R^2/2,0)\),

(2)
\(R^{1}\Vert f\Vert _0+\Vert \nabla f\Vert _0+R^\zeta [ f]_{1+\zeta }\le c_{4}c_{12}\max \{\mu ,c_{8}\Vert u\Vert \}\).
Proof
We may set \(R=1\) without loss of generality. With \(E_1\), \(\nu \), p and q given, we use Proposition 5.2 to obtain \(\varepsilon _{8}\), \(\theta \) and \(c_{8}\). Setting \(\iota =\theta ^2/2\), we use Proposition 5.1 to obtain \(\varepsilon _{6}\) and \(c_{4}\). We choose \(\varepsilon _{9}\) so that
We first use Proposition 5.2 with \(W=A=T\), and note that (5.12)–(5.17) are satisfied due to (5.38)–(5.41) and (5.42). Thus there exist \(T_1\in \textbf{G}(n,k)\) and \(A_1\in \textbf{A}(n,k)\) such that (5.18)–(5.20) are satisfied with \(R=1\), \(W=T\), \({\tilde{A}}=A_1\) and \({\tilde{T}}=T_1\). Similarly, we may use Proposition 5.1 since (2.13)–(2.16) are satisfied with \(R=1\) and \(\varepsilon _{6}\), so that we have \(f_1\) and \(F_1\) defined on \(B_1^k\times [1/2,\theta ^2/2]\) satisfying (5.1) and (5.2). We next claim that Proposition 5.2 can be inductively used for \(R=\theta ^j\), \(j\in {\mathbb {N}}\), where we obtain \(T_j\in \textbf{G}(n,k)\) and \(A_j\in \textbf{A}(n,k)\) satisfying
where \(T_0:=T\), and writing \(\mu _j\) as \(\mu \) in (5.14) corresponding to \(A_j\) and \(R=\theta ^j\),
The case \(j=1\) follows from Proposition 5.2. Assume that it is true until \(j\ge 1\). Then we check that (5.12)–(5.17) are true for \(W=T\), \(T=T_j\), \(A=A_j\) and \(R=\theta ^j\). We have
where we used (5.45), (5.38), (5.39) and (5.44). Thus (5.12) is satisfied. Since \(A_j\) and \(T_j\) are parallel, (5.13) is fine. By (5.42), (5.43) and (5.46), we have \(\mu _j<\varepsilon _{8}\), so that (5.14) is satisfied. The condition (5.16) follows from (5.40), and (5.42), (5.39) and (5.21) give (5.17) for j. Thus, we may apply Proposition 5.2 with \(R=\theta ^j\), and obtain \(T_{j+1}\) and \(A_{j+1}\) which are parallel and
where we used (5.46), and
by (5.20) and (5.46). This closes the inductive step and proves (5.45) and (5.46) for all j. We next prove that we can apply Proposition 5.1 on each domain \(\textrm{C}(T_j,2\theta ^j)\times [\theta ^{2j},\theta ^{2(j+1)}/2]\) for all \(j\ge 1\). Note that for each \(j\ge 0\), by the same argument leading up to (5.7), we have
for all \(t\in [4\theta ^{2j}/5,0)\). To apply Proposition 5.1, we need to have T there replaced by \(A_j\), so we need to tilt the plane whose tilt is estimated by (5.47). For this reason, we may actually need to use a slightly smaller cylinder than \(C(T_j,2\theta ^j)\) so that the smallness of corresponding \(\mu _j\) (with respect to the distance function to \(A_j\)) can be assured from (5.46). Inductively, we know that the support of \(\Vert V_t\Vert \) in \(\textrm{C}(T_{j1},\theta ^{j1})\times [\theta ^{2(j1)}/2,\theta ^{2j}/2]\) is a \(C^{1,\zeta }\) graph, so that the condition (2.13) is satisfied. Condition (2.14) follows from (5.40), and (2.15) follows from (5.46), (5.42) and (5.43). Thus we may apply Proposition 5.1 and obtain a graph representation \({\tilde{f}}_j\) over \(A_j\) with the \(C^{1,\zeta }\) estimate of the form \(c_{4}\theta ^{j\zeta } \max \{\mu ,c_{8}\Vert u\Vert \}\). Note that, by the implicit function theorem, one can equally represent the same set as a graph \(f_j\) over T. The norm \(\Vert \nabla f_j\Vert _0\) over \(B_{\theta ^j}^k\times [\theta ^{2j}/2,\theta ^{2(j+1)}/2]\) can be different by a constant multiple of \(\Vert T_jT\Vert \) which is bounded as in (5.47). The Hölder seminorm \([f]_{1+\zeta }\) has two terms, \([\nabla f]_\zeta \) and the \((1+\zeta )/2\)Hölder seminorm in time. The first is seen as the variation of the tangent space and one can see that it is bounded by a multiple of constant (which is close to 1) under the small rotation. The estimate for the latter is obtained by applying [11, Proposition 6.4] with the gradient Hölder norm, and the small rotation affects little. Hence we can obtain the desired \(C^{1,\zeta }\) estimate for \(f_j\) representing \(\textrm{spt}\Vert V_t\Vert \) over the domain \(B_{\theta ^j}^k\times [\theta ^{2j}/2,\theta ^{2(j+1)}/2]\), by \(2c_{4}\theta ^{j\zeta }\max \{\mu ,c_{8}\Vert u\Vert \}\). We next observe that
so that we have a representation of \(\textrm{spt}\Vert V_t\Vert \) as the graph of a single function f over \({\tilde{D}}\). The estimate \(\Vert f\Vert _0+\Vert \nabla f\Vert _0 \le 2 c_{4} \max \{\mu , c_{8} \Vert u\Vert \}\) is immediate. For the Hölder seminorm \([f]_{1+\zeta }\), we proceed as follows. Let \((y_1,s_1)\), \((y_2,s_2)\) be points in \({\tilde{D}}\) with \((y_1,s_1) \ne (y_2,s_2)\), assume without loss of generality that \(s_1 \le s_2\), and let \(h,l \ge 0\) be such that \((y_1,s_1) \in B^k_{\theta ^h} \times [ \theta ^{2\,h}/2,\theta ^{2(h+1)}/2]\) and \((y_2,s_2) \in B^k_{\theta ^{h+l}} \times [ \theta ^{2(h+l)}/2,\theta ^{2(h+l+1)}/2]\). By the triangle inequality, we estimate
where \(c_{12}=c_{12}(k,p,q)\). The estimate for the second summand in \([f]_{1+\zeta }\) is analogous. The proof is now complete. \(\square \)
6 Proof of the main results
We are now ready to prove Theorems 2.2 and 2.3.
Proof of Theorem 2.2
By scaling, we may assume \(R=1\). Given \(\nu \in \left( 0,1\right) \), \(E_1 \in \left[ 1, \infty \right) \), p and q, let \(\varepsilon _{9}\), \(c_{4}\), \(c_{12}\) and \(c_{8}\) be as in Proposition 5.3. Let now \(\varepsilon _{2} \in \left( 0,1\right) \) and \(c_1 \in \left( 1, \infty \right) \) be such that the following conditions are satisfied:
For \(T \in \textbf{G}(n,k)\), and \(U = \textrm{C}(T,2)\), suppose that \(\{V_t\}_{t \in \left[ 1,0\right] }\) and \(\{u(\cdot ,t)\}_{t \in \left[ 1,0\right] }\) satisfy (A1)–(A4) as well as (2.13)–(2.16). We identify, as usual, T with \(\mathbb {R}^k \cong \mathbb {R}^k \times \{0\} \subset \mathbb {R}^n\), and we claim the following: for every \(j \ge 1\), setting
there exist \(f_j :D_j \rightarrow T^\perp \) and \(F_j :D_j \rightarrow \mathbb {R}^n\) such that \(F_j(x,t)=(x,f_j(x,t))\), and

(1)
\(\textrm{spt}\Vert V_t\Vert \cap \textrm{C}(T,\sqrt{\sigma _jt}) = \textrm{Image}\, F_j(\cdot ,t) \text{ for } \text{ all } t \in \left[ \tau _j,0\right) \),

(2)
\(\Vert f_j\Vert _0 + \Vert \nabla f_j\Vert _0 \le c_1 \max \{\mu ,\Vert u\Vert _{p,q}\}\).
Assume the claim for the moment. It is then an immediate consequence of (6.2) that
which implies that
Since \(\lim _{j \rightarrow \infty } \tau _j = 0\), (6.4) and (1)–(2) imply that one can define a function \(f :B^k_{\frac{1}{2}} \times \left[ \frac{1}{4},0\right) \rightarrow T^\perp \) such that, setting \(F(x,t)=(x,f(x,t))\) for \((x,t) \in B^k_{\frac{1}{2}} \times \left[ \frac{1}{4},0\right) \) one has
that is (2.17) and part of the estimate in (2.18). In what follows, we will first prove the claim; then, we will show that the resulting function f also satisfies \([f]_{1+\zeta } \le c_1 \max \{\mu ,\Vert u\Vert _{p,q}\}\).
The proof of the claim is by induction on \(j\ge 1\). The induction base, \(j=1\), is Proposition 5.3. We then assume that the claim is true for j, and prove it for \(j+1\). Fix any point \((x_0,t_0) \in \partial D_j\), and translate in spacetime so to consider the flow \(\{{\tilde{V}}_s\}_{s\in \left[ 1t_0,0\right] }\), with \({\tilde{V}}_s:= (\tau _{x_0})_\sharp V_{s+t_0}\) where \(\tau _{x_0}(y):= yx_0\). Set \({\tilde{R}}^2 = {\tilde{R}}_{t_0}^2:= \frac{1}{4}+\frac{t_0}{4}\), and notice that \(\textrm{C}(T,x_0,2 {\tilde{R}}) \subset \textrm{C}(T,0,2)\). In particular, \(\{{\tilde{V}}_s\}\) satisfies (A1)–(A4) in \(U=\textrm{C}(T,2{\tilde{R}})\) corresponding to the forcing term \({\tilde{u}}(y,s) = {\tilde{u}}_{(x_0,t_0)}(y,s):= u(y+x_0,s+t_0)\). We next claim that (5.38)–(5.41) are satisfied. We clearly have
by (2.13) and (6.1). Moreover, \((T^{1}(0)\times \{0\}) \cap \textrm{spt}(\Vert {\tilde{V}}_s\Vert \times ds) = (T^{1}(x_0)\times \{t_0\}) \cap \textrm{spt}(\Vert V_t\Vert \times dt) \ne \emptyset \), because for any sequence \((x_h,t_0) \in D_j\) such that \(x_h \rightarrow x_0\) we have \((x_h,f_j(x_h,t_0)) \in T^{1}(x_h) \cap \textrm{spt}\Vert V_{t_0\Vert }\) by (1), and thus \((T^{1}(x_0) \times \{t_0\}) \cap \textrm{spt}(\Vert V_t\Vert \times dt)\) contains all subsequential limits of \((x_h, f_j (x_h,t_0),t_0)\). We also readily estimate
so that (2.16) implies (5.39). Finally, we have
using the same argument leading to (5.9). We can then apply Proposition 5.3 and conclude after translating back the origin to \((x_0,t_0)\) that, setting
there exist functions \(f^{(x_0,t_0)} :{\tilde{D}}^{(x_0,t_0)} \rightarrow T^\perp \) and \(F^{(x_0,t_0)} :{\tilde{D}}^{(x_0,t_0)} \rightarrow \mathbb {R}^n\) such that \(F^{(x_0,t_0)}(x,t)=(x,f^{(x_0,t_0)}(x,t))\) for all \((x,t) \in {\tilde{D}}^{(x_0,t_0)}\) and
 \((1)_\star \):

\(\textrm{spt}\Vert V_t\Vert \cap \textrm{C}(T,x_0,\sqrt{tt_0}) = \textrm{Image}\,F^{(x_0,t_0)}(\cdot ,t)\) for all \(t \in \left[ t_0  \frac{{\tilde{R}}_{t_0}^2}{2}, t_0 \right) \),
 \((2)_\star \):

\({\tilde{R}}_{t_0}^{1} \Vert f^{(x_0,t_0)}\Vert _0 + \Vert \nabla f^{(x_0,t_0)}\Vert _0 + {\tilde{R}}_{t_0}^\zeta [f^{(x_0,t_0)}]_{1+\zeta } \le c_{4}c_{12} \max \{\mu _{(x_0,t_0)}, c_8 \Vert {\tilde{u}}_{(x_0,t_0)}\Vert \}\) .
In particular, there is a well posed extension of the functions \(f_j\) and \(F_{j}\) to the region
We let \(f_{j+1}\) and \(F_{j+1}\) denote such extensions, and we proceed with the proof that conditions (1)–(2) hold true with \(j+1\) in place of j. To this aim, it is sufficient to show the following: for \(t \in \left[ \tau _{j+1},0\right) \) and \(\sigma _j t \le x^2 < \sigma _{j+1} t\), there exists \((x_0,t_0) \in \partial D_j\) such that \((x,t) \in {\tilde{D}}^{(x_0,t_0)}\). Once this is established, indeed, one immediately gains that
see Fig. 1, and (1) at step \(j+1\) follows immediately from (1) at step j and \((1)_\star \), while (2) at step \(j+1\) follows from (2) at step j and \((2)_\star \) thanks to (6.1)
To prove the above claim, let then \((x,t) \in \mathbb {R}^{k} \times \left[  \tau _{j+1},0\right) \) be such that \(\sigma _j t \le x^2 < \sigma _{j+1}t\), and set
for some number \(\alpha =\alpha _j > 1\) to be determined. Notice that \((x_0,t_0) \in \partial D_j\) by construction. We then only need to prove that there exists \(\alpha > 1\) such that \((x,t) \in {\tilde{D}}^{(x_0,t_0)}\). On the other hand, by the definitions of \(t_0\) and \(x_0\) it holds that
so that, recalling the definition of \(\sigma _j\), \((x_0,t_0) \in {\tilde{D}}^{(x_0,t_0)}\) provided \(\alpha > 1\) is chosen so that
We now show that (6.7) has a solution \(\alpha =\alpha _j>1\) for every j. Direct calculation shows that \(\alpha =2\) is a solution to (6.7) when \(j=1\) and \(j=2\). On the other hand, it holds
so that solutions to
also solve (6.7). Changing variable
(6.8) reduces to
which admits \(\xi = \frac{1}{2}\) as a solution for every \(j \ge 3\). Going back to the original variables, we have that the number \(\alpha =\alpha _j > 1\) such that \(\frac{1}{\alpha } = 1\frac{1}{4\sigma _j}\) is a solution to (6.7) for \(j \ge 3\). This concludes the proof of (6.5).
We are only left with the proof of the estimate on the Hölder seminorm \([f]_{1+\zeta }\). Given that \(\textrm{spt}\Vert V_t\Vert \cap \textrm{C}(T,1/2)\) is the graph of a function defined on \(B^k_{1/2}\) for all \(t \in [1/4,0)\), we know now that for every \((x_0,t_0) \in B^k_{1/2} \times [1/4,0)\) the flow \(\{{\tilde{V}}_s\}_{s \in [1t_0,0]}\) with \({\tilde{V}}_s = (\tau _{x_0})_\sharp V_{s+t_0}\) as above satisfies the assumptions of Proposition 5.3 with, say \(R=3/4\). In particular, we have \(C^{1,\zeta }\) estimates for f with \(c_{4} c_{12} \max \{\mu ,c_{8}\Vert u\Vert _{p,q}\}\) in the parabolic region \({\tilde{D}}^{(x_0,t_0)}=\{(x,t) \in \mathbb {R}^k \times [t_0  1/4,t_0) \, :\, xx_0^2 < tt_0\}\). To prove the desired Hölder estimate, let now \((y_1,s_1)\) and \((y_2,s_2)\) be points in \(B^k_{1/2} \times [1/4,0)\) with \((y_1,s_1) \ne (y_2,s_2)\), and assume without loss of generality that \(s_1 \le s_2\). Consider the parabolic region \({\tilde{D}}^{(y_2,s_2)}\) with vertex at \((y_2,s_2)\). If \(y_1y_2^2 < s_1s_2\), then \((y_1,s_1) \in {\tilde{D}}^{(y_2,s_2)}\), and the estimate is a consequence of Proposition 5.3 and (6.1). Otherwise, if \(y_1y_2^2 \ge s_1s_2 = s_2s_1\), we use the triangle inequality to estimate
which yields the estimate for \([\nabla f]_\zeta \) thanks to (6.1). The estimate for the second summand in \([f]_{1+\zeta }\) is analogous, and we omit it. The proof is complete. \(\square \)
Proof of Theorem 2.3
Here we briefly record the outline of the \(C^{2,\alpha }\) regularity of [20] and point out the key estimates. The idea is to look at a graphical distance function from the solution of the heat equation g, denoted by \(Q_g\) ( [20, Definition 4.1]) and one shows a decay estimate of the \(L^2\)norm of \(Q_g\) by the blowup argument. The key identity is Lemma 4.2, which shows certain “subcaloric” property of \(Q_g\), and the resulting \(L^\infty \) estimate Proposition 4.3, both of [20]. Note that the latter is an estimate up to the endtime. Since this is the basis of the blowup argument, if we have already \(C^{1,\zeta }\) graph representation up to the endtime, all the following argument in [20] works verbatim with obvious modifications of changing the domain of integration to the one with center at the endtime point from the center of the spacetime domain. The second order Taylor expansion of the blowup should be changed to the endtime point as well. The end result is the estimate away from the parabolic boundary, as stated in the claim. \(\square \)
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Notes
That is, \({\mathscr {V}}=\{V_t\}_{t \in (\infty ,0)}\) satisfies \(\Vert (\iota _r)_\sharp V_{r^2}\Vert = \Vert V_{1}\Vert \) for every \(r > 0\), where \(\iota _r(x):=r^{1}x\), as well as
$$\begin{aligned} h(V_t,x)=\frac{S^\perp (x)}{2t} \qquad \text{ for } V_t \text{a.e. } (x,S) \in G_k(\mathbb {R}^n), \text{ for } \text{ a.e. } t < 0. \end{aligned}$$
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Acknowledgements
The second author dedicates the present paper to his mother Suzuko Tonegawa. The work was carried out during the sabbatical year of the second author at the University of Milan. The research of S.S. was supported by the project PRIN 2022PJ9EFL “Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations”, funded by the European Union under NextGenerationEU and by the Italian Ministry of University and Research, and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of INdAM. Y.T. was partially supported by JSPS 18H03670, 19H00639.
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Gaussian density lower bound
Gaussian density lower bound
We include the following Lemma for the reader’s convenience. The localized version can be proved similarly.
Lemma A.1
Suppose that \({\mathscr {V}}=\{V_t\}_{t\in (a,b]}\) is a Brakke flow as in Definition 2.1 and that \(\textrm{spt}\Vert V_t\Vert \subset B_R\) for every \(t\in (a,b]\) for some \(R>0\). Then for any \((x_0,t_0)\in \textrm{spt}(\Vert V_t\Vert \times dt)\), we have \(\Theta ({\mathscr {V}},(x_0,t_0))\ge 1\).
Proof
The proof is by a contradiction argument. If \(\Theta ({\mathscr {V}},(x_0,t_0))<1\), by the definition of the Gaussian density and the continuity of the integrand, there exist some \(\tau _0>0\), \(\delta _0>0\) and \(\varepsilon _0>0\) such that \((x_0,t_0)(x',t')<\varepsilon _0\) implies
By the definition of Brakke flow, we can choose an arbitrarily close point \((x',t')\) to \((x_0,t_0)\) such that \(V_{t'}\in \textbf{IV}_k(U)\) and \(V_{t'}\) has, at \(x'\), the approximate tangent space with integermultiplicity, say, \(j'\in {\mathbb {N}}\). Then, by the property of the approximate tangent space, one can prove that
In particular, (A.2) implies that
for all sufficiently small \(\tau >0\). Since \(t'\) may be arbitrarily close to \(t_0\), we may assume that \(t'>t_0\tau _0\), and by the monotonicity and (A.3), we have
Since \(\tau \) is arbitrarily small, we may assume \((x_0,t_0)(x',t'+\tau )<\varepsilon _0\), and (A.4) is a contradiction to (A.1). This proves the claim. \(\square \)
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Stuvard, S., Tonegawa, Y. Endtime regularity theorem for Brakke flows. Math. Ann. (2024). https://doi.org/10.1007/s00208024028268
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DOI: https://doi.org/10.1007/s00208024028268