Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds

Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature and positive mass. We show that each leaf of the canonical foliation through stable constant mean curvature surfaces of the end of $(M, g)$ is uniquely isoperimetric for the volume it encloses.


Introduction
A complete Riemannian 3-manifold .M; g/ is said to be asymptotically flat if there is a nonempty compact subset K & M and a diffeomorphism M n K fx P R 3 jxj > 1=2g (1.1) with g ij h ij g ij where @ I ij h O.jxj jIj / as jxj 3 I we require that it be minimal. We also require that there be no closed minimal surfaces in the interior of M . Given > 1, we use S to denote the surface in M that corresponds to the centered coordinate sphere fx P R 3 jxj h g in the chart at infinity (1.1). We let B denote the bounded open region in M that is enclosed by S .
The ADM-mass (after R. Arnowitt, S. Deser, and C. W. Misner [1]) of such an asymptotically flat manifold .M; g/ is given by .@ i g ij @ j g ii /x j where integration is with respect to the Euclidean metric. R. Bartnik [2] has shown that this quantity is independent of the particular choice of chart at infinity (1.1). The fundamental positive mass theorem, proven by R. Schoen and S.-T. Yau [39] using minimal surface techniques and then by E. Witten [44] using spinors, asserts that for .M; g/ asymptotically flat with nonnegative scalar curvature, we have that A.V / h area.@M / g inffarea.@/ P R V g: (1.3) When the scalar curvature of .M; g/ is nonnegative, a result of the third-named author [40] combined with an observation in appendix K of [7] shows that there is a region V P R V that achieves the infimum in (1.3). The proof that such isoperimetric regions exist in .M; g/ is indirect and offers no clue as to the position of these regions. The main result of this paper is to show that if .M; g/ is not Euclidean space and if V > 0 is sufficiently large, then V is bounded by the horizon @M and a stable constant mean curvature surface that belongs to the canonical foliation (see Appendix D) of the end of M through stable constant mean curvature spheres. THEOREM 1.1. Let .M; g/ be a complete Riemannian 3-manifold that is asymptotically flat at rate > 1=2 and has nonnegative scalar curvature and positive mass. There is V 0 > 0 with the following property. Let V ! V 0 . There is a region V P R V such that area.@ V / area.@/ for all P R V , with equality only when h V . The boundary of V consists of the horizon @M and a leaf of the canonical foliation of the end of M .
In particular, the solution of the isoperimetric problem in .M; g/ is unique for large volumes.
We have included the assumption that there are are no closed minimal surfaces in the interior of .M; g/ in the definition of asymptotically flat. One could omit this assumption first by replacing .M; g/ with the region outside of all such closed minimal surfaces as in [25, lemma 4.1] and observing that the centering mechanisms obtained in the proof of Theorem 1.1 imply that .M; g/ satisfies the conclusion of Theorem 1.1 as well.
Our proof of Theorem 1.1 shows that the outer boundary of a large isoperimetric region V is close to a centered coordinate sphere S where V $ 4 3 =3. From this, uniqueness of V follows from characterization results for the leaves of the canonical foliation that we discuss in Appendix D.
The special case of Theorem 1.1 where .M; g/ is asymptotic to Schwarzschild with positive mass, i.e., where instead of (1.2) we have for some m > 0 was conjectured by H. Bray [5, p. 44] and G. Huisken and finally settled in joint work [16] by J. Metzger and the second-named author. Their proof develops an ingenious idea of H. Bray [5] for the exact Schwarzschild metric and uses the spherical symmetry in the asymptotic expansion (1.4) in a crucial way. It carries over to higher dimensions [17] and makes no assumption on the scalar curvature. We recall from, e.g., [32] that the value of the scalar curvature at a given point can be characterized by the isoperimetric deficit of small geodesic balls. Qualitatively, in order to enclose a small given amount of volume by a geodesic sphere, less area is needed when the sphere is centered at a point of larger scalar curvature. In Appendix C, we discuss how the isoperimetric deficit of large solutions of the isoperimetric problem detects the mass of .M; g/. Theorem 1.1 expresses the positive mass theorem as a local to global transfer of isoperimetry in the small to isoperimetry in the large in a precise way. More importantly, it adds to the short list of geometries and the even shorter list of geometries with no exact symmetries (see appendix H in [17] for an overview), where we can describe the solutions of the isoperimetric problem exactly.
The uniqueness of isoperimetric regions of a given volume in Theorem 1.1 is in stark contrast to the nonuniqueness of stable constant mean curvature spheres of a given area. This nonuniqueness is particularly dramatic in the following example constructed by A. Carlotto and R. Schoen in [8, p. 561].
Example 1.2 ( [8]). There is an asymptotically flat Riemannian metric g h g ij dx i dx j on R 3 with nonnegative scalar curvature and positive mass so that g ij h ij on R 2 ¢ .0; I/.
We mention that there are examples of .M; g/ asymptotic to Schwarzschild where there are other large stable constant mean curvature spheres than the leaves of the canonical foliation; cf. [6,12].
We emphasize that the examples constructed in [8] are asymptotically flat of rate < 1. They clearly contain an abundance of sequences of stable constant mean spheres that drift off in .M; g/ while their area diverges. On the way of proving Theorem 1.1, we show in Section 2 that, quite generally, the isoperimetric defect along any such sequence tends to 0. THEOREM 1.3. Let .M; g/ be a Riemannian 3-manifold with nonnegative scalar curvature that is asymptotically flat of rate > 3=4. There is 0 > 1 with the following property. For k h 1; 2; : : : , let k h @ k be connected stable constant mean curvature spheres such that k B 0 h ¿ and area. k / 3 I as k 3 I. The case where jj < 1 is covered by the uniqueness of the leaves of the canonical foliation. The proof of Theorem 1.1 outlined above is carried out in Sections 2 and 3. However, it only works when we impose the stronger decay assumptions (3.1) and (3.2) on .M; g/. Incidentally, the decay assumptions stated in Theorem 1.1 are optimal for the positive mass theorem. We obtain Theorem 1.1 in the stated generality from a completely different line of argument that we develop in Section 4.
In this second proof of Theorem 1.1, we study the mean curvature flow of large isoperimetric surfaces. We prove that, upon appropriate rescaling, the level set flow of such large isoperimetric surfaces converges to the Euclidean flow fS p 1 4t ./g tP0; 1 4 / of S 1 ./ in R 3 . When ¤ 0, part of this flow will be in a shell-like region that avoids the center of the manifold, which corresponds to the origin in the rescaled picture. We show that the Hawking mass of the surfaces forming this shell is close to 0. Using this, we apply the monotonicity of the isoperimetric defect from Schwarzschild discovered by G. Huisken in two steps to obtain a contradiction. First, we compare with Schwarzschild of mass m ADM until the time when the surfaces have jumped across the center of .M; g/. Then, we compare with Schwarzschild of mass o.1/m ADM until the surfaces have all but disappeared.
In this argument, we only need a very weak characterization of the leaves of the canonical foliation to conclude uniqueness in Theorem 1.1.
This second proof of Theorem 1.1 is effective in that it provides an explicit estimate for the isoperimetric deficit of general large outward area-minimizing regions that are close to balls B 1 ./ on the scale of their volume. The more analytic, first proof is delicately tuned to large stable constant mean curvature spheres for which it provides very precise information. Furthermore, it allows us to prove Theorem 1.3 on the isoperimetric defect of large, outlying stable constant mean curvature spheres.
After his work was finished, a third proof of Theorem 1.1 was discovered by the fourth-named author [45]. In fact, it is shown in [45] that Theorem 1.1 also holds when the assumption of nonnegative scalar curvature is replaced by a decay condition.

Isoperimetric Deficit of Large Outlying Stable CMC Spheres
Throughout this section, we consider a complete Riemannian 3-manifold .M; g/ that is asymptotically flat at rate h 1 and has nonnegative scalar curvature and positive mass m ADM > 0. We also require the additional decay assumption @ I ij h O.jxj 1 jIj / as jxj 3 I for all multi-indices I of order jIj 3. The control of the third-order derivatives is used in the proof of Proposition 2.3. This could be avoided there by adapting an important observation of J. Metzger from [31]-the derivatives of the ambient curvature enter Simons' identity in divergence form. We also mention that the results in this section could be generalized to the slower decay > 3=4. (This is the threshold for the proof of the key estimate (2.10).) However, as a step in the proof of Theorem 1.1, we require the full strength of the results by S. Ma in [30], which in turn needs the even stronger assumptions (3.1) and (3.2). This is why we restrict the exposition to the present case.
The results proven here will be applied in Section 3 to the study of large isoperimetric regions. Since the methods apply equally well to large stable constant mean curvature surfaces, we consider this more general setting here.
Let h @ & M be a connected stable constant mean curvature surface so that B 0 h ¿ where 0 > 1 is large. The error terms in this section are all with respect to area./ 3 I.
The following result is proven by the first-and second-named authors in section 2 of [11]. 1 LEMMA 2.1. When 0 > 1 and area./ are sufficiently large, then is homeomorphic to a sphere.
Let r > 0 denote the area radius of defined by area./ h 4r 2 : We use H > 0 to denote the mean curvature of . The second fundamental form of and its trace-free part are denoted by h and h, respectively. Any quantity computed with respect to the reference Euclidean metric as opposed to with respect to g will have a bar over it.
pdf, the analysis in this section used an a priori assumption that was spherical. This assumption was subsequently justified by combining the results for spherical isoperimetric regions proven here with a continuity argument involving the isoperimetric profile (inspired by ideas from [10]). This continuity argument, while more complicated than the argument from [ for all x P such that 2jxj ! r 3=4 . PROOF. Assume that the assertion fails with c h k along a sequence of regions k with area radius r k 3 I and at points x k P k h @ k where r 3=4 k 2jx k j.
We work in the chart at infinity fx P R 3 jxj > 1=2g. If we rescale by r 3=4 k and pass to a subsequence, curvature estimates as in [11, app. B] show that the rescaled regions converge in C 2; loc to a half-space in R 3 n f0g. Upon  We can choose a regular value with r 3=4 =2 r 3=4 such that the curve fx P jxj h g has length O.r 3=4 /. A standard variation of the argument leading to the Bonnet-Myers diameter estimate shows that any two points p; q P H are connected by a curve in H whose length is O.r/.
Integrating (2.6) along such curves, we see that there is a P R 3 so that (2.7) holds. Assume now that jaj r g r 3=4 . It follows that there is x 0 P H with jx 0 j h r 3=4 . Using (2.7), it follows that jaj ! ja x 0 j jx 0 j ! r cr 3=4 where c > 0 is independent of . Replacing a by a H h .1 g .c g 2/r 1=4 /a completes the proof. Note that it also follows that n B r 3=4 is connected (so H h n B r 3=4 ).
In the remainder of this section, integrals will be with respect to the measure induced by the Euclidean background unless otherwise indicated. PROOF. We may assume that r g r 3=4 < jaj < 2r. Using  . ij x i x j / ij g o.r/: where we have used (2.9) in the third inequality. This completes the proof.
We arrive at the main results of this section, asserting that the isoperimetric deficit of large outlying stable constant mean curvature spheres is very close to Euclidean. The strategy of the approximation argument we have used here is illustrated in Figure 2 We show that the boundary of the large component of an isoperimetric region, is very close to a sphere S r .a/ outside of B r 3=4 ; note that "¢" represents the origin here. This allows us to approximate the isoperimetric deficit of by that of S r .a/. We show that in the scenario depicted here, the isoperimetric deficit of S r .a/ (and thus of ) is too close to Euclidean.

Proof of Theorem 1.1 Assuming (3.1) and (3.2)
Throughout this section, we assume that .M; g/ is a complete (nonflat) asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature that satisfies the decay assumptions @ I ij h O.jxj 1 jIj / as jxj 3 I

Consider a sequence of large isoperimetric regions
Consider the two alternatives (a) and (b) in Lemma B.2. In (a), V k eventually contains any compact set C . In particular, by the work of S. Ma [30] mentioned above, we find that @ V k is an element of the canonical foliation; cf. the discussion at the end of Appendix D.
On the other hand, if (b) from Lemma B.2 applies, then V k h res k k where res k is contained in a small neighborhood of the horizon and for k sufficiently large, k B 0 h ¿ for any 0 > 1.
In this case, the analysis from the previous section applies. In particular, Corollary 2.7 shows that the isoperimetric deficit of k , and thus V k , is very close to Euclidean. By comparing, via Lemma C.1, with centered coordinate balls, this contradicts the isoperimetric property of V k .

Mean Curvature Flow of Large Isoperimetric Regions
Let .M; g/ be a complete Riemannian 3-manifold that is asymptotically flat at rate > 1=2 and has nonnegative scalar curvature and positive mass m ADM > 0.
The analysis in this and the subsequent sections will establish the proof of Theorem 1.1 in full generality.
Let V k be isoperimetric regions of volumes V k 3 I. Let k be the unique large component of V k . We recall from Appendix B that k is connected with connected outer boundary @ k n @M , and that k is outer area-minimizing in .M; g/.
PROOF. By [42, theorem 3.5], k .t/ is outward area-minimizing in k . Combined with the fact that k is outward area-minimizing, we see that k .t/ is outward area-minimizing in .M; g/. The claim follows from comparison with coordinate spheres.
Define the volume radius k > 0 by the expression vol. k / h 4 We may view k .t/ .M n K/ as a measure on fx P R 3 jxj > 1=2g using the loc .R 3 n f0g/ as k 3 I. Our goal will be to show that h 0.
There is an integral Brakke flow fz .t/g t!0 on R 3 n f0g with the following three properties.
as Radon measures on R 3 n f0g.
(2) For almost every t ! 0, there is a further subsequence f`.k; t/g I kh1 of as varifolds on R 3 n f0g. PROOF. The first two claims follow from T. Ilmanen's compactness theorem for integral Brakke flows, theorem 7.1 in [27]. This result is only stated for sequences of Brakke flows with respect to a fixed complete Riemannian metric in [27]. However, the same proof as in [27] applies in the present setting. The quadratic area bounds carry over from Lemma 4.1.
In view of Proposition J.5, it is now clear that f.t/g t!0 extends to an integral Brakke flow in R 3 with initial condition z .0/ h H 2 S 1 ./: Proposition J.6 shows that such a Brakke motion follows classical mean curvature flow except possibly for sudden extinction: for all t P 0; T where T P 0; 1 4 /. The particular flow at hand is constructed as the limit of level set flows. We use spherical barriers to show that the limiting flow cannot disappear suddenly; i.e., we will show that T h 1 4 .
PROOF. If not, then there is T P 0; 1 4 / so that z .t/ h H 2 S p 1 4t ./ for t P 0; T and z .t/ h 0 for t > T . We will prove the result for jj ! 1 and leave the straightforward modification to the case of jj < 1 to the reader.
Assume that T h 0. Let " > 0 be small. Upper semicontinuity of density for surfaces with bounded mean curvature (see [41, cor. 17.8]) implies that B p 1 4" ./ & z k .0/ h z k for all sufficiently large k. Using that z g k converges to the standard Euclidean inner product in C 2 loc .R 3 n f0g/ and the avoidance principle for the level set flow, we see that B p 1 9" We obtain a contradiction with the assumption that z k ."/ * z ."/ h 0. This is a contradiction for the same reason as before.
Using B. White's version [43] of K. Brakke's regularity theorem [4] for mean curvature flow, we obtain the following consequence.      for some T h 0. We have already seen that the original level set flow f k .t/g t!0 with initial condition k .0/ h k has the property that, for t P t k ; T k , there is a unique large component k . We continue with the notation of Section 4. The strategy of the proof is illustrated in Figure 5 PROOF. Assume that ¤ 0. We continue with the notation set forth above.
Note that f k .t/g tPt k ;T k is a smooth mean curvature flow where k .t/ h @ k .t/. In Corollary 4. 5   Using that x 3=2 g y 3=2 .x g y/ 3=2 for all x; y ! 0, we arrive again at the contradictory estimate (5.2). PROOF OF THEOREM 1.1. Combining Lemma B.2 and Proposition 5.1, we see that every sufficiently large isoperimetric region is connected and close to the centered coordinate ball B 1 .0/ when put to scale of its volume in the chart at infinity (1.1). By the uniqueness of large stable constant mean curvature spheres described in Appendix D, the outer boundary of such an isoperimetric region is a leaf of the canonical foliation.

Appendix A General Properties of the Isoperimetric Profile
Let .M; g/ be an asymptotically flat Riemannian 3-manifold as defined in Section 1. We recall below several properties of the isoperimetric profile A .0; I/ 3 .0; I/ of .M; g/ as defined by (1.3) that we use throughout this paper. The results on the regularity of the isoperimetric profile are given in or follow easily from, e.g., [3,5,19,38].
Locally, the isoperimetric profile can be written as the sum of a concave and a smooth function. In particular, the isoperimetric profile is absolutely continuous.
The left derivate A .V / and the right derivative A g .V / exist at every V > 0. They agree at all but possibly countably many V > 0. Moreover Assume that, for some V > 0, there is V P R V with A.V / h area.@ V / area.@M /: The proof of theorem 1.2 in [17] shows that such isoperimetric regions exist for every sufficiently large volume V > 0 when the mass of .M; g/ is positive. They exist for every volume V > 0 when the scalar curvature of .M; g/ is nonnegative by proposition K.1 in [7]. The outer boundary @ V n@M is a stable constant mean curvature surface. Its mean curvature H is positive when computed with respect to the outward unit normal. Moreover,

A Hg .V / H A H .V /:
In particular, the isoperimetric profile is a strictly increasing function. Moreover, at volumes V > 0 where the isoperimetric profile is differentiable, the outer boundaries of all isoperimetric regions of volume V have the same constant mean curvature.  When jj h 1, a similar situation occurs, except the flow disconnects from the origin after a short time (in the rescaled picture). We must wait this short time before arguing as in (c), so there will be a thin region as in (b) in this case. If jj > 1, the flow is completely disconnected, so we do not need to consider the shaded region as in (b).

Appendix B Divergent Sequences of Isoperimetric Regions
The following result of the first-and the second-named authors is included as corollary 1.13 in [7]. It is a consequence of the solution of the following conjecture by R. Schoen: The only asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature that admits a noncompact area-minimizing boundary is flat Euclidean space.
is a thin smooth region that is bounded by @M and a nearby stable constant mean curvature surface.
The conclusion of the lemma clearly fails in Euclidean space. Under the additional assumption that the scalar curvature of .M; g/ is everywhere positive, this result was observed by the second-named author and J. Metzger as corollary 6.2 in [16]. Together with elementary observations on the number of components of large isoperimetric regions as in Section 5 of [15] and the proof of theorem 1.12 in [7], we obtain the following dichotomy for sequences of isoperimetric regions with divergent volumes. LEMMA B.2. Let .M; g/ be a complete Riemannian 3-manifold that is asymptotically flat with nonnegative scalar curvature and positive mass. Let V k be an isoperimetric region of volume V k where V k 3 I. After passing to a subsequence, exactly one of the following alternatives occurs: (a) Each V k is connected, @ V k n @M is connected, and the sequence is increasing to M . (b) Each V k splits into a union res V k and I V k where the I V k are connected with connected boundary and divergent in M as k 3 I, and where each res V k is contained in an " k -neighborhood of @M where " k 3 0 as k 3 I.
In particular, every isoperimetric region V in .M; g/ of sufficiently large volume V > 0 has exactly one large connected component-either V in alternative (a) or I V in alternative (b). Note that Theorem 1.1 implies that alternative (b) in Lemma B.2 doesn't occur.
We include several additional observations-extracted from the proofs of Theorem 1.2 in [17] and Theorem 1.12 in [7]-about the sequences in Lemma B.2. Let Then, possibly after passing to a further subsequence, loc .R 3 n f0g/ for some P R 3 . In particular,

Appendix C Sharp Isoperimetric Inequality
The characterization of the ADM-mass through the isoperimetric deficit of large centered coordinate spheres in Lemma C.1 below was proposed by G. Huisken [23] and proved by X. The next result was also proposed by G. Huisken [23,24]. A detailed proof, following the ideas of G. Huisken, was given by J. Jauregui and D. Lee as theorem 3 in [29]. below. We also refer to the recording of the Marston Morse lecture given by G. Huisken [24] for the original argument.

Appendix D Canonical Foliation
Here we collect results on the existence and uniqueness of a canonical foliation through stable constant mean curvature spheres of the end of an asymptotically flat Riemannian 3-manifold .M; g/ with positive mass. The asymptotic assumptions in the discussion are tailored to our applications. We refer to [33] and [34, sec. 5] for more general results.
The results discussed below depart from the pioneering work of G. Huisken and S.-T. Yau [26] and of J. Qing and G. Tian [36] for initial data asymptotic to Schwarzschild. We also mention the crucial intermediate results by L.-H. Huang [21] for asymptotically even data. We refer to the recent articles [9,22,30,33] for an overview of the literature on this subject.
The following uniqueness and existence results are, in the stated generality, due to C. Nerz [34, sec. 5]. Let .M; g/ be a Riemannian 3-manifold that is asymptotically flat at rate > 1=2 and has nonnegative scalar curvature and m ADM > 0.

Appendix E A Priori Estimates for the Hawking Mass
The next lemma due to G. Huisken and T. Ilmanen is extracted from section 6 in [25]. Recall from section 4 in [25] that M is diffeomorphic to the complement in in [25] that the boundary y of is C 1;1 and smooth away from the coincidence set y . Assume that ¤ . It follows that the volume of .M / res is strictly larger than that of the isoperimetric region res so that by the monotonicity of the isoperimetric profile of .M; g/ its boundary area is less.
A cut-and-paste argument using that the area of y is less than that of shows otherwise-a contradiction. PROOF. This follows exactly as before when is outer-minimizing. To handle the case when is not outer-minimizing, we can apply the previous argument to the outer-minimizing hull of and then use the fact that after rescaling to unit size, is close to a round sphere (and thus the difference in area between and its outer-minimizing hull is small). The details are given in [11, prop  : When is a sphere, then H 2 area./ g 2 3 .R g j hj 2 /d 16: Here, R denotes the ambient scalar curvature and H and h denote, respectively, the constant scalar mean curvature and the trace-free part of the second fundamental form of with respect to a choice of unit normal, and d is the area element of with respect to the induced metric.

Appendix F Elementary Growth Estimates
The elementary and well-known fact stated in the lemma below follows from an explicit "cut and paste" argument by comparison with balls B for > 1 large.  [25]. We use the positivity of a term dropped in [25] in conjunction with estimates of C. De Lellis and S. Müller [14] to handle an additional technical difficulty brought about by our weaker decay assumptions > 1=2. All integrals below are with respect to the Euclidean background metric unless explicitly noted otherwise.
Let r > 0 so that area x g ./ h 4r 2 : Note that r and are comparable. Following G. Huisken and T. Ilmanen [25, (7.11 This map restricts to an injection of integer n-rectifiable Radon measures M.R ng1 n f0g/ IM n .R ng1 n f0g/ 3 IM n .R ng1 /; which in turn lifts to an injection of integer n-rectifiable varifolds fV P IV n .R ng1 / V .B 1 .0/ n f0g/ < Ig 3 IV n .R ng1 /; which we denote by V U 3 V : The extension of a stationary varifold across a point is not necessarily again stationary as shown by the following well-known example.
The varifold V h m kh1 j`kj is stationary as an element of IM 1 .R 2 n f0g/. It is stationary as an element of IM 1 .R 2 / if and only if e i 1 g ¡ ¡ ¡ g e i m h 0.
However, the phenomenon in the previous example is particular to dimension n h 1. LEMMA J.2. Let n ! 2. There are radial functions k P C I c .B 1 .0// with 0 k 1 such that k .x/ h 1 when jxj < 1=.2k 2 / and k .x/ h 0 when jxj > 1=k and constants c k 8 0 with the following property: Let be a measure on B 1 .0/ n f0g such that, for some c > 0, .B .0/ n f0g/ c n (J.1) for all 0 < 1. Then 1 c jr k j 2 d c k : Below we will often work with the functions ' k h 1 k P C I .R ng1 n f0g/: Note that 0 ' k 3 1 locally uniformly on R ng1 n f0g and that, under the assumptions of the previous lemma, lim k3I jr' k j 2 d h 0: We include a proof of the following, well-known result as preparation for Proposition J.5. LEMMA J.3 (Extending stationary varifolds across a point). Let n ! 2. Let V be a stationary n-rectifiable varifold on R ng1 n f0g such that V .B 1 .0/ n f0g/ < I.

The extension
V of V across the origin is stationary as an n-rectifiable varifold on R ng1 .
PROOF. Let ' k P C I .R ng1 n f0g/ be cutoff functions as in (J.2). Note that (J.1) holds by the monotonicity formula for stationary varifolds as stated in (17.5) of [41]. Let X P C 1 c .R ng1 s R ng1 /. Then ; / is stationary in R n n f0g. As k 3 I, the first term on the right tends to . V /.X/, while the second term tends to 0.
A similar argument gives the following result.
LEMMA J.4. Let n ! 2. Let V be an n-rectifiable varifold on R ng1 n f0g such that, for some c > 0, V .B .0/ n f0g/ c n for all 0 < 1. Let P C I c .R ng1 / be a nonnegative function such that V is n-rectifiable on fx P R ng1 n f0g .x/ > 0g with absolutely continuous first variation such that jHj 2 d V < I. The first variation of the extension V of V across the origin is absolutely continuous on fx P R ng1 .x/ > 0g.
PROOF. Let X P C 1 c .fx P R ng1 .x/ > 0g; R ng1 /. Let ' k P C I .R ng1 n f0g/ be cutoff functions as in (J.2). We compute that ; div X d V g p .proj T r' k / ¡ X d V g ' k .proj T r p / ¡ X d V where V h v.; /. In the last expression, the second term tends to 0 by Hölder's inequality and the construction of ' k , while the first term tends to . V /.X/.
Finally, by Hölder's inequality, we may bound the third term by kjrj=k L 2 . V / kXk L 2 . V / : The first quantity here can be bounded using the estimate in lemma 6.6 of [27].
Putting these facts together, we find that j.
V /. p X/j C kXk L 2 . V / h C kXk L 2 .
V / : This completes the proof.
We now turn to the situation for Brakke flows.
PROPOSITION J.5 (Extending Brakke flows across a point). Let n ! 2. Let f t g t!0 be a codimension 1 integral Brakke flow on R ng1 n f0g such that, for some constant c > 0, t .B .0/ n f0g/ c n for all t ! 0 and 0 < 1. Then fy t g t!0 is a codimension 1 integral Brakke flow on R ng1 .
Recall from lemma 6.6 in [27] that, on fx P R ng1 .x/ > 0g, jrj 2 2 max jr 2 j: In a first step, we verify that, for all t ! 0, Together with (J.6) this finishes the proof.
Example J.1 shows that there is no analogue of Proposition J.5 when n h 1.
The following result and its proof should be compared with the constancy theorem for stationary varifolds [41, §41]. PROPOSITION J.6 (Constancy theorem). Let f t g t!0 be an integral Brakke flow in R 3 such that .0/ h H 2 S 1 ./. There is T P 0; 1 4 such that t h H 2 S p 1 4t ./ for all t P 0; T and t h 0 for all t > T : PROOF. The avoidance principle for Brakke flows as stated in §10.7 of [27] shows that supp t & S p 1 4t ./ for all t P 0; 1 4 . The entropy of S 1 ./ is less than 3=2. The entropy decreases along the Brakke flow by lemma 7 of [28]. Using that f t g t!0 is an integral Brakke flow, we see that for almost every t ! 0 the measure t has an approximate tangent plane with multiplicity 1 at x for t -almost every x. Thus, for almost every t ! 0, there is a measurable subset t & S p 1 4t ./ with t h H 2 t .
We claim that t as a varifold with multiplicity 1 has absolutely continuous first variation for almost every t ! 0. Indeed, by §7.2(ii) in [27], given P C I c .R 3 / we have that I < x D t t ./ for almost every t ! 0. Let such that .x/ h 1 for all x P B 2 ./. Using that x D t t ./ B. t ; / for a Brakke motion, the claim follows.
For such t ! 0 and every X P C 1 c .R 3 ; R 3 /, we have that It follows that the perimeter of t as a subset of S p 1 4t./ vanishes. The Poincaré inequality (as in lemma 6.4 of [41]) shows that either t or its complement in S p 1 4t ./ is a set of 2-dimensional measure zero. We have thus shown that for almost every t ! 0, either t h H 2 S p 1 4t ./ or .t/ h 0.