Uniqueness of twoconvex closed ancient solutions to the mean curvature flow
Abstract.
In this paper we consider closed noncollapsed ancient solutions to the mean curvature flow () which are uniformly twoconvex. We prove such an ancient solution is up to translations and scaling the unique rotationally symmetric closed ancient noncollapsed solution constructed in [19] and [11].
Contents
1. Introduction
In this paper we consider closed noncollapsed ancient solutions to the mean curvature flow ()
(1.1) 
for , where is the mean curvature of and is the outward unit normal vector. We know by Huisken’s result [14] that the surfaces will contract to a point in finite time.
The main focus of the paper is the classification of twoconvex closed ancient solutions to mean curvature flow, i.e. solutions that are defined for , for some . Ancient solutions play an important role in understanding the singularity formation in geometric flows, as such solutions are usually obtained after performing a blow up near points where the curvature is very large. In fact, Perelman’s famous work on the Ricci flow [16] shows that the high curvature regions are modeled on ancient solutions which have nonnegative curvature and are noncollapsed. Similar results for mean curvature flow were obtained in [12], [18], [19] assuming mean convexity and embeddedness.
Daskalopoulos, Hamilton and Sesum previously established the complete classification of ancient compact convex solutions to the curve shortening flow in [8], and ancient compact solutions the Ricci flow on in [9]. The higher dimensional cases have remained open for both the mean curvature flow and the Ricci flow.
In an important work by XuJia Wang [17] the author introduced the following notion of noncollapsed solutions to the MCF which is the analogue to the noncollapsing condition for the Ricci flow discussed above. In the same work XuJia Wang provided a number of results regarding the asymptotic behavior of ancient solutions, as , and he also constructed new examples of ancient MCF solutions.
Definition 1.1.
Let be a smooth domain whose boundary is a mean convex hypersurface . We say that is noncollapsed if for every there are balls and of radius at least such that and , and such that and are tangent to at the point , from the interior and exterior of , respectively (in the limiting case , this means that is a halfspace). A smooth mean curvature flow is noncollapsed if is noncollapsed for every .
In [1] Andrews showed that the noncollapsedness property is preserved along mean curvature flow, namely, if the initial hypersurface is noncollapsed at time , then evolving hypersurfaces are noncollapsed for all later times for which the solution exists. Haslhofer and Kleiner [12] showed that every closed, ancient, and noncollapsed solution is necessarily convex.
In recent breakthrough works, Brendle and Choi [6, 7] gave the complete classification of noncompact ancient solutions to the mean curvature flow that are both strictly convex and uniformly twoconvex. More precisely, they show that any noncompact and complete ancient solution to mean curvature flow (1.1) that is strictly convex, uniformly twoconvex, and noncollapsed is the Bowl soliton, up to scaling and ambient isometries. Recall that the Bowl soliton is the unique rotationallysymmetric, strictly convex solution to mean curvature flow that translates with unit speed. It has the approximate shape of a paraboloid and its mean curvature is largest at the tip. The uniqueness of the Bowl soliton among convex and uniformly twoconvex translating solitons has been proved by Haslhofer in [10].
While the noncollapsedness property for mean curvature flow is preserved forward in time, it is not necessarily preserved going back in time. Indeed, XuJia Wang ([17]) exhibited examples of ancient compact convex mean curvature flow solutions
Ancient selfsimilar solutions to MCF are of the form for some fixed surface and some “blowup time” . We rewrite a general ancient solution as
(1.2) 
Haslhofer and Kleiner [12] proved that every closed ancient noncollapsed mean curvature flow with strictly positive mean curvature sweeps out the whole space. By XuJia Wang’s result [17], it follows that in this case the backward limit as of the typeI rescaling of the original solution , defined by (1.2), is either a sphere or a generalized cylinder of radius . In [3] we showed that if the backward limit is a sphere then the ancient solution has to be a family of shrinking spheres itself.
Definition 1.2.
We say an ancient mean curvature flow is an Ancient Oval if it is compact, smooth, noncollapsed, and not selfsimilar.
Definition 1.3.
We say that an ancient solution is uniformly 2convex if there exists a uniform constant so that
(1.3) 
Throughout the paper we will be using the following observation: if an Ancient Oval is uniformly 2convex, then by results in [17], the backward limit of its typeI parabolic blowup must be a shrinking round cylinder , with radius .
Based on formal matched asymptotics, Angenent [2] conjectured the existence of an Ancient Oval, that is, of an ancient solution that for collapses to a round point, but for becomes more and more oval in the sense that it looks like a round cylinder in the middle region, and like a rotationally symmetric translating soliton (the Bowl soliton) near the tips. A variant of this conjecture was proved already by White in [19]. By considering convex regions of increasing eccentricity and using a limiting argument, he proved the existence of ancient flows of compact, convex sets that are not selfsimilar. Haslhofer and Hershkovits [11] carried out White’s construction in more detail, including, in particular, the study of the geometry at the tips. As a result they gave a rigorous and simple proof for the existence of an Ancient Oval.
Our main result in this paper is as follows.
Theorem 1.4.
The proof of this theorem will follow from the results stated below.
Theorem 1.5.
If is an Ancient Oval which is uniformly 2convex, then it is rotationally symmetric.
Our proof of Theorem 1.5 closely follows the arguments by Brendle and Choi in [6, 7] on the uniqueness of strictly convex, noncompact, uniformly 2convex, and noncollapsed ancient mean curvature flow. It was shown in [6] that such solutions are rotationally symmetric. Then, by analyzing the rotationally symmetric solutions, Brendle and Choi showed that such solutions agree with the Bowl soliton.
Given Theorem 1.5, we may assume in our proof of Theorem 1.4 that any Ancient Oval is rotationally symmetric. After applying a suitable Euclidean motion we may assume that its axis of symmetry is the axis. Then, can be represented as
(1.4) 
for some function , and from now on we will set and . We call the points and the tips of the surface. The function , which we call the profile of the hypersurface , is only defined for . Any surface defined by (1.4) is automatically invariant under acting on . Convexity of the surface is equivalent to concavity of the profile , i.e. is convex if and only if .
A family of surfaces defined by evolves by mean curvature flow if and only if the profile satisfies
(1.5) 
If satisfies MCF, then its parabolic rescaling defined by (1.2) evolves by the rescaled MCF
where is the parametrization of , and is the corresponding unit normal. Also,
for a profile function , which is related to by
The points and are referred to as the tips of rescaled surface . Equation (1.5) for is equivalent to the following equation for
(1.6) 
It follows from the discussion above, that our most general result 1.4 reduces to the following classification under the presence of rotational symmetry.
Theorem 1.6.
Let and , be two invariant Ancient Ovals with the same axis of symmetry (which is assumed to be the axis) whose profile functions and satisfy equation (1.5) and rescaled profile functions and satisfy equation (1.6). Then, they are the same up to translations along the axis of symmetry (translations in ), translations in time and parabolic rescaling.
Since the asymptotics result from [3] will play a significant role in this work, we state it below for the reader’s convenience.
Theorem 1.7 (Angenent, Daskalopoulos, Sesum in [3]).
Let be any invariant Ancient Oval (see Definition 1.2) . Then the solution to (1.6), defined on , has the following asymptotic expansions:

For every ,
as .

Define and . Then,
uniformly on compact subsets in .

Denote by the tip of , and define for any the rescaled flow at the tip
where
Then, as , the family of mean curvature flows converges to the unique unit speed Bowl soliton, i.e. the unique convex rotationally symmetric translating soliton with velocity one.
Before we conclude our introduction we give a short description of our proof for Theorem 1.6. A more detailed outline of this proof is given in Section 3.
Discussion on the proof of Theorem 1.6. The proof of Theorem 1.6 makes extensive use of our previous work [3] where the detailed asymptotic behavior of Ancient Ovals, as , was given under the assumption of symmetry (see Theorem 1.7 below). Note that our symmetry result, Theorem 1.5, which will be shown in Section 2, only shows the symmetry of solutions and not the symmetry assumed in Theorem 1.7. However, as we will demonstrate in the Appendix of this work (see Theorem 8.1), the estimates in Theorem 1.7 simply extend to the symmetric case. Since the proof of Theorem 1.6 is quite involved, in Section 3 we will give an outline of the different steps of our proof. The main idea is simple: given and any two solutions of (1.5), we will find parameters , corresponding to translations along the xaxis, translations in time and parabolic rescaling respectively, such that , where denotes the image of under these transformations (see (3.3)). To achieve this uniqueness, we will consider the corresponding rescaled profiles and show that . It will mainly follow from analyzing the equation for in the cylindrical region (the region , for some and small). We restrict to the cylindrical region by introducing an appropriate cut off function and setting . The difference in this region satisfies the equation
(1.7) 
for a nonlinear error term . The operator is simply the linearized operator for equation (1.6) on the cylinder which we see in the middle, i.e. constant the . This operator is well studied and it is known to have two unstable modes (corresponding to two positive eigenvalues) and one neutral mode (corresponding to the zero eigenvalue). The uniqueness at the end follows by a coercive estimate on (1.7) with the right norm (we call it ), which roughly implies that if , then
(1.8) 
thus leading to a contradiction. It is apparent that to obtain such a coercive estimate one needs to adjust the parameters in such a way that the projections and onto the positive and zero eigenspaces of are all simultaneously zero at some time . The main challenge in showing (1.8) comes from the error terms which are introduced by the cutoff function and supported at the transition region between the cylindrical and tip regions (the latter is defined to be the region ). To estimate these errors one needs to consider our equation in the tip region and show a suitable coercive estimate there which allows us to bound back in the tip region back in terms of . To achieve this, one heavily uses the a priori estimates and Theorem 1.7 from [3]. We also need to introduce an appropriate weighted norm in the tip region which lets us show the Poincaré type estimate we need to proceed. Unfortunately, numerous technical difficulties arise from various facts including the noncompactness of the limit as and the fact that at the tips.
In previous classifications of ancient solutions to mean curvature flow and Ricci flow, [8], [9], [6, 7], an essential role in the proofs was played by the fact that all such solutions were given in closed form or they were solitons. One of the significance of our techniques in our current work is that they overcome such a requirement and potentially can be used in many other parabolic equations and particularly in other geometric flows. To our knowledge, our work and the recent work by Bourni, Langford and Tinaglia [5] are the first classification results of geometric ancient solutions where the solutions are not given in closed form and they are not solitons. Let us also point out that our current techniques are reminiscent of the significant work by Merle and Zaag in [15] which has provided an inspiration for us.
Acknowledgements: The authors are indebted to S. Brendle for many useful discussions regarding the rotational symmetry of ancient solutions.
2. Rotational symmetry
The main goal in this section is to prove Theorem 1.5. Our proof of Theorem 1.5 follows closely the arguments of the recent work by Brendle and Choi [6, 7] on the uniqueness of strictly convex, uniformly 2convex, noncompact and noncollapsed ancient solutions of mean curvature flow in . It was shown in [6] that such solutions are rotationally symmetric. Then by analyzing the rotationally symmetric solutions, Brendle and Choi showed that such solutions agree with the Bowl soliton. For the reader’s convenience we state their result next.
Theorem 2.1 (Brendle and Choi [6]).
Let be a noncompact ancient mean curvature flow in which is strictly convex, noncollapsed, and uniformly 2convex. Then agrees with the Bowl soliton, up to scaling and ambient isometries.
In the proof of Theorem 1.5 we will use both the key results that led to the proof of the main theorem in [6] (see Propositions 2.5 and 2.6 below), and the uniqueness result as stated in Theorem 2.1.
Before we proceed with the proof of Theorem 1.5, let us recall some standard notation. Our solution is embedded in , for all and in the mean curvature flow, time scales like distance squared. We denote by the parabolic cylinder centered at of radius , namely the set
where denotes the closed Euclidean ball of radius in .
Also, following the notation in [13] and [6], we denote by the rescaled by the mean curvature parabolic cylinder centered at of radius , namely the set
Note that in [13, §7] Huisken and Sinestrari consider parabolic cylinders with respect to the intrinsic metric on the solution , which in our case is equivalent to the extrinsic metric on spacetime that we are considering here.
We recall Brendle and Choi’s [6] definition of a mean curvature flow being symmetric, in terms of the normal components of rotation vector fields. In what follows we identify with the subalgebra of consisting of skew symmetric matrices of the form
Thus acts on the second factor in the splitting . Any generates a vector field on by . If is a Euclidean motion, with and , then the pushforward of the vector field under is given by
Any vector field of this form is a rotation vector field.
Definition 2.2.
A collection of vector fields on is a normalized set of rotation vector fields if there exist an orthonormal basis of , a matrix , and a point such that
Definition 2.3.
Let be a solution of mean curvature flow. We say that a point is symmetric if there exists a normalized set of rotation vector fields such that in the parabolic neighborhood .
Lemma 4.3 in [6] allows us to control how the axis of rotation of a normalized set of rotation vector fields varies as we vary the point .
The proof of Theorem 1.5 relies on the following two key propositions which were both shown in [6]. The first proposition is directly taken from [6] (see Theorem 4.4 in [6]). The second proposition required some modifications of arguments in [6] and hence we present parts of its proof below (see 2.6).
Definition 2.4.
A point of a mean curvature flow lies on an neck if there is a Euclidean transformation , and a scale such that

maps to

for all the hypersurface is close in to the cylinder of length , of radius , and with the axis as symmetry axis.
Proposition 2.5 (Neck Improvement  Theorem 4.4 in [6]).
There exists a large constant and a small constant with the following property. Suppose that is a mean curvature flow, and suppose that is a point in spacetime with the property that every point in is symmetric and lies on an neck, where . Then is symmetric.
Proof.
The proof is given in Theorem 4.4 in [6]. ∎
The next result will be shown by slight modification of arguments in the proof of Theorem 5.2 in [6]. The proof of Proposition 2.6 below follows closely arguments in [6].
Proposition 2.6 (Cap Improvement [6]).
Let and be chosen as in the Neck Improvement Proposition 2.5. Then there exist a large constant and a small constant with the following property. Suppose that is a mean curvature flow solution defined on . Moreover, we assume that is, after scaling to make , close in the norm to a piece of a Bowl soliton which includes the tip (where the tip lies well inside in the interior of that piece of a Bowl soliton, at a definite distance from the boundary of that piece of the Bowl soliton) and that every point in is symmetric, where . Then is symmetric.
Proof.
Without loss of generality assume and and . For the sake of the proof, keep in mind that the statement in the Proposition is of local nature and it only matters what is happening on a large parabolic cylinder , while the behavior of our solution outside of this neighborhood does not matter.
The assumptions in the Proposition imply that if we take sufficiently small, using that the Hessian of the mean curvature around the maximum mean curvature point in a Bowl soliton is strictly negative definite (note that by our assumption we may assume the maximum of in is attained at a unique interior point . Moreover, the Hessian of the mean curvature at is negative definite. Hence, varies smoothly in . We now conclude that if , then
(2.1) 
The proof of (2.1) is the same as the proof of Lemma 5.2 in [6].
We claim that there exists a uniform constant with the property that every point with lies on an neck and satisfies . Indeed, knowing the behavior of the Bowl soliton, it is a straightforward computation to check previous claims are true on the Bowl soliton, with a constant for example, . By our assumption, is close to the Bowl soliton and hence the claims are true for our solution as well.
If , the Proposition follows immediately from Proposition 2.5. Thus, we may assume that . Then we have the following claim.
Claim 2.7.
Suppose that is an ancient solution of mean curvature flow. Given any positive integer , there exist a large constant and a small constant with the following property: if the parabolic neighborhood is close in the norm to a piece of the Bowl soliton which includes the tip, and every point in is symmetric, then every point with and symmetric. is
Proof.
Assume is the maximal curvature of the tip of the Bowl soliton in the statement of the Claim. Note that may depend on , but is independent of . Define
The proof is by induction on and is similar to the proof of Proposition 5.3 in [6]. For we have . By above discussion we have that lies on an neck and . This implies
Assume the claim holds for . We want to show it holds for as well, that is, if the parabolic neighborhood is close in the norm to a piece of the Bowl soliton which includes the tip, and every point in is symmetric, then every point with and symmetric. Suppose this is false. Then there exists so that and and is not symmetric. By Proposition 2.5, there exists a point such that either is not symmetric or does not lie at the center of an neck. Note that if we choose so that is
In view of the induction hypothesis, we conclude that . We can now follow the proof of Proposition 5.3 in [6] closely to obtain a contradiction. In that part of the proof one needs to use (2.1), which is proved in Lemma 5.2 in [6]. It is clear from the proof in [6] that if we take a bigger parabolic cylinder around of size , in order to still have (2.1) one needs to require that is close to a Bowl soliton, where needs to be taken very small, depending on . This is clear from the proof of (2.1) that can be found in [6]. ∎
In the following, will denote a large integer, which will be determined later. Moreover, assume that and . Using the Claim, we conclude that for every point
for a uniform constant that is independent of and . Lemma 4.3 in [6] allows us to control how the axis of rotation of varies as we vary the point . More precisely, as in [6], if and are in
Hence, we can find a normalized set of rotation vector fields
at the point . From this we deduce that , for all points
The goal of the remaining part of this section is to show how we can employ Propositions 2.5 and 2.6 to prove Theorem 1.5.
Observe that by the crucial work of Haslhofer and Kleiner in [12] we know that a strictly convex noncollapsed ancient solution to mean curvature flow sweeps out the whole space. Hence, the well known important result of X.J. Wang in [17] shows that the rescaled flow, after a proper rotation of coordinates, converges, as time goes to , uniformly on compact sets, to a round cylinder of radius .
This has as a consequence that is a neck with radius . The complement has two connected components, call them and , both compact. Thus, for every , the maximum of on is attained at least at one point in and similarly for .
For every , we define the tip points and as follows. Let , for be a point such that
Denote by , for .
Throughout the rest of the section we will be using the next observation about possible limits of our solution around arbitrary sequence of points with , when rescaled by .
Lemma 2.8.
Let , be an Ancient Oval satisfying the assumptions in Theorem 1.5. Fix a . Then for every sequence of points and any sequence of times , the rescaled sequence of solutions subconverges to either a Bowl soliton or a shrinking round cylinder.
Proof.
By the global convergence theorem (Theorem 1.12) in [12] we have that after passing to a subsequence, the flow converges, as , to an ancient solution , for , which is convex and uniformly 2convex. Note that on the limiting manifold. By the strong maximum principle applied to we have that everywhere on , where . If is strictly convex, by the classification result in [6] we have that it is a Bowl soliton. If the limit is not strictly convex, by the strong maximum principle it splits off a line and hence it is of the form , where is an dimensional ancient solution. On the other hand the uniform 2convexity assumption on our solution implies the inequality , for a uniform constant . Thus, Lemma 3.14 in [12] implies that the limiting flow is a family of round shrinking cylinders . ∎
We will next show that points which are away from the tip points in both regions , are cylindrical.
Lemma 2.9.
Let , , be an Ancient Oval satisfying the assumptions of Theorem 1.5 and fix . Then, for every there exist and , so that for all and the following holds
(2.2) 