Ancient and Eternal Solutions to Mean Curvature Flow

Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

Alexander Mramor and Alec Payne Department of Mathematics, University of California Irvine, Irvine, CA 92617 Courant Institute, New York University, New York City, NY 10012 mramora@uci.edu,ajp697@nyu.edu
Abstract.

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in for . These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in for . These eternal solutions are not solitons, are -invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As , the profile curves converge to a grim reaper for and become flat for . Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded -invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.

1. Introduction

Ancient solutions to mean curvature flow, i.e. solutions existing on the time interval , , play an important role in the singularity analysis of the flow as the natural blowup limits after rescaling about a singularity, making their study central in defining and understanding weak notions of the flow. A special type of ancient solution is the eternal solution, which is an ancient solution that exists for all time, i.e. it exists on the time interval . Eternal solutions are a subset of ancient solutions but are much more rigid and less is known about them. There are many known ancient solutions to mean curvature flow in Euclidean space, but only a small number of them are eternal, particularly if one excludes the translating solitons. Eternal solutions arise naturally as the blowup limits of Type II singularities, whereas non-eternal ancient solutions arise as the blowup limits of Type I singularities (see [32] for a description of Type I and Type II singularities). From a more analytic perspective, mean curvature flow is the natural analogue of the heat equation in the setting of submanifold geometry, and ancient solutions are the natural analogues of global solutions to elliptic equations, distinguishing ancient solutions in this sense.

It is useful then to have a wide variety of examples of ancient solutions to the flow to help understand the phenomena that could be realized by solutions to mean curvature flow. Ancient solutions can be split up among those which are solitons and those which are not solitons. By “soliton”, we mean a solution to mean curvature flow which evolves by a combination of rigid motions and homotheties. And by “non-soliton,” we mean a solution which is not a soliton. There are far more soliton ancient solutions known than non-soliton ancient solutions. Ancient solutions may also be described as either convex or nonconvex. An ancient solution is called convex if every timeslice of the flow is a convex surface. Convexity is an important characteristic of an ancient solution as the ancient solutions which arise as blowup limits of mean convex mean curvature flow are convex [23, 46].

Some examples of convex ancient solitons include the standard shrinking spheres and cylinders, the Abresch-Langer curves [1], and some of the rotating and shrinking solitons to curve shortening flow [21]. Examples of nonconvex ancient solitons include the Angenent torus [5], desingularizations of the sphere and the Angenent torus [30] and the sphere and the plane [35, 36, 37], the high genus min-max constructions of Ketover [31], and many of the rotating and shrinking solitons to curve shortening flow [21]. Among the convex eternal solitons are the grim reaper, the bowl soliton [4], the strictly convex translating solitons lying in slabs [8, 26], and the non-rotationally symmetric entire translators of Wang [44]. Finally, some examples of nonconvex eternal solitons include any non-flat minimal surface, the winglike translators or translating catenoids [14], the periodic Scherk-like translators [27], translators associated to minimal surfaces like the Costa-Hoffman-Meeks surface [15], the Yin-Yang spiral in one dimension [2], the purely rotating solitons in dimensions two and higher that are analogous to Yin-Yang spirals [29], the nonconvex translating tridents [34, 39], and the multitude of exotic immersed self shrinkers [16, 17]. Despite the richness of the ancient solitons, relatively few examples of non-soliton ancient solutions are known. We will focus on non-soliton ancient solutions throughout this paper.

Non-soliton Ancient Solutions in Euclidean Space
(Strictly) Convex Nonconvex
Eternal
  • Assuming strict convexity and that the curvature attains its maximum at a point in spacetime, non-soliton convex eternal solutions do not exist [22]. In [46], White conjectured that any nonflat convex eternal solution is a translating soliton.

  • The ancient sine curve in [33], also known as the hairclip, as well as the closely related truncated versions of the ancient sine curve [47]

  • The examples of Corollary 1.4 in ,

Non-eternal
  • The Angenent oval [5]

  • The ancient ovals of White [46] and Haslhofer-Hershkovits [24]

  • The “ancient pancakes” in a slab due to Bourni-Langford-Tinaglia [7] and Wang [44]

  • Glued grim reapers forming immersed “ancient trombones” in  [6]

  • The embedded examples of Corollary 1.2 in ,

Table 1.

In Table 1, we give a survey of the known non-soliton ancient solutions according to whether they are eternal or non-eternal and whether they are convex or nonconvex. We restrict ourselves to the codimension one case in Euclidean . Note that any ancient solution can be extended to higher dimensions by simply multiplying by isometric factors of , although we leave out these possibilities. In fact, any weakly convex ancient solution can be taken to be a strictly convex ancient solution multiplied by  [29].

Our first theorem, inspired in part by the interesting work of Choi-Mantoulidis [13], concerns the existence of ancient solutions flowing out of certain unstable minimal surfaces in for . These provide examples for the bottom right box in Table 1. In the following theorem, we will need a technical assumption, which is satisfied by a large class of minimal surfaces. We say that a surface satisfies the uniform tubular neighborhood assumption if there exists a tubular neighborhood of uniform width such that the boundary of this tubular neighborhood is smooth and embedded. That is, there exists some such that is smooth and embedded, where is a unit normal. In other words, satisfies the uniform tubular neighborhood assumption if it does not asymptotically approach itself.

Theorem 1.1.

For , let be an unstable111In line with [19], we mean than on some bounded domain , for the Jacobi operator. 2-sided properly embedded minimal surface which has uniformly bounded curvature, satisfies the uniform tubular neighborhood assumption (as defined above), and satisfies one of the following options:

  1. it is asymptotically flat, or

  2. it is periodic with compact fundamental domain, or

  3. it is periodic and asymptotically flat in its fundamental domain.

Then, there exist two distinct ancient solutions and to mean curvature flow such that and smoothly and uniformly converge to from opposite sides of as . These ancient solutions are embedded and have a sign on mean curvature yet are nonconvex and are not solitons.

In the proof of Theorem 1.1, the assumptions of asymptotic flatness or periodicity plays a subtle role in the asymptotics of the ancient solution, but note that these assumptions include a wide class of minimal surfaces. Some nontrivial examples of minimal surfaces satisfying the assumptions of this theorem are catenoids and the Costa-Hoffman-Meeks surfaces. Note also that minimal surfaces with infinitely many ends such as the Riemann examples are covered by item (3).

The ancient solutions constructed in Theorem 1.1 seem to be the first known instances of nonconvex embedded ancient solutions in which are not solitons. To the authors’ knowledge the only previously constructed nonconvex non-eternal ancient solutions that are not solitons are the immersed curves in of Angenent and You [6]. In fact, to the authors’ knowledge, all previously known mean convex non-eternal non-soliton ancient solutions to mean curvature flow have been convex. Since the ancient solutions of Theorem 1.1 have a sign on mean curvature, i.e. they are mean convex with the correctly chosen normal field, we have the following corollary.

Corollary 1.2.

There exist mean convex, yet nonconvex, non-soliton ancient solutions to mean curvature flow in , .

Our second theorem concerns the existence of an eternal solution flowing out of the catenoid in for each . This provides an example for the top right box in Table 1. In the following theorem, let be a catenoid in for . Center so that it is rotationally symmetric about an axis passing through the origin. The catenoid splits into two connected components, the “inside” and the “outside”. Let be the unit normal on the neck of the catenoid such that points away from the origin. Then, let the “outside” of the catenoid be the connected component that points into.

Theorem 1.3 (The Reapernoid).

For each , there exists a mean convex222This is with respect to the normal on compatible, as , with the normal to the catenoid (see the discussion above the theorem). -invariant eternal solution to mean curvature flow in with uniformly bounded curvature such that for each , is a subset of the outside of the catenoid (as defined above) and converges smoothly and uniformly to as .

As , becomes infinitely far from its axis of rotation. For , the profile curve of will converge as to a grim reaper of the same width as . For , the pointed limit of the profile curve of is a line and the curvature of approaches zero as .

Figure 1. A sketch of the regimes of the profile curves of the higher-dimensional reapernoid, the eternal solution of Theorem 1.3. For , the eternal solution has a profile curve close to that of the catenoid, and for , it has a profile curve close to that of a grim reaper.

By a theorem of Richard Hamilton [22], a strictly convex eternal solution which achieves its spacetime maximum of curvature must be a translating soliton. The eternal solution constructed in Theorem 1.3, which will be referred to as the reapernoid as a reminder of its asymptotics, seems to be the first known instance of a non-soliton eternal solutions in , , which does not split off a line. To the authors’ knowledge, the only previously known non-soliton eternal solutions to mean curvature flow are curves in or their isometric products with . Thus, we find the following corollary.

Corollary 1.4.

For , there exists an eternal solution to mean curvature flow in that is not a soliton and does not split off a line.

Our final result concerns a partial uniqueness statement for the reapernoid eternal solution of Theorem 1.3. Its proof is not particularly difficult but naturally leads into a host of further questions which we discuss in the concluding remarks:

Theorem 1.5.

Suppose is a connected embedded nonflat eternal solution to mean curvature flow in which

  1. is -invariant,

  2. has a sign on mean curvature, and

  3. has uniformly bounded curvature for all time.

Then is either the catenoid itself or it has the asymptotics of the eternal solution of Theorem 1.3 up to scale. That is, converges to the catenoid from the outside as , and as , the profile curve of converges to a grim reaper for or becomes flat for .

In other words, this shows that the reapernoid, the eternal solution of Theorem 1.3, is asymptotically unique among -invariant eternal flows with uniformly bounded curvature and a sign on mean curvature. Note that the translating bowl soliton is excluded from the above conditions because it is -invariant but not -invariant. Also, in Theorem 1.5, we do not prove that the grim reapers found in the limit as are necessarily of the same width as the catenoid, as in Theorem 1.3.

Acknowledgements: The authors would like to thank Kyeongsu Choi and Christos Mantoulidis for responding to comments and writing an interesting and inspiring paper [13] where they construct ancient flows out of compact minimal surfaces in non-Euclidean ambient spaces, using different methods than those of this paper. The authors would also like to thank Mat Langford and Shengwen Wang for their comments and suggestions. Finally, the authors thank their advisors, Richard Schoen and Bruce Kleiner, respectively, for their support and advice.

2. Preliminaries, Old and New

In this section we collect some standard and nonstandard facts on mean curvature flow and minimal surfaces, which will be used later to streamline the proofs. Let be an -dimensional orientable manifold and let be an embedding of realizing it as a smooth closed -sided hypersurface of Euclidean space, which by abuse of notation we also refer to as . Then the mean curvature flow is given by the image of satisfying

(2.1)

where is a unit normal and is the mean curvature. It turns out that (2.1) is a nonlinear heat-type equation, since for the induced metric on ,

(2.2)

That the left-hand side is the Laplacian motivates the assertion that the mean curvature flow is the natural analogue of the heat equation in submanifold geometry. One can easily see that the mean curvature flow equation (2.1) is degenerate. Despite this, solutions to (2.1) always exist for short time and are unique provided that the initial data has bounded second fundamental form. There are several ways to deduce this by relating (2.1) to a nondegenerate parabolic PDE. Solutions to mean curvature flow satisfy many properties that solutions to heat equations do, such as the maximum principle and smoothing estimates. One important consequence of the maximum principle is the comparison principle (also known as the avoidance principle), which says that two initially disjoint hypersurfaces will remain disjoint over the flow. More generally, for two noncompact hypersurfaces with uniformly bounded geometry, if the flows and are initially distance apart they remain so under the flow (see, for instance, Remark 2.2.8 of [32]). In fact, in the cases of interest to us in this paper, such a comparison principle for two noncompact hypersurfaces can be proven independently. We are interested in applying the comparison principle between noncompact hypersurfaces of uniformly bounded geometry that are either asymptotically flat or periodic (or a combination of both). Indeed, between periodic surfaces with compact fundamental domain, the standard comparison principle generalizes immediately. And for asymptotically flat surfaces, pseudolocality (see Chen-Yin [12]) keeps the ends arbitrarily stationary, meaning that the separated flows must have an interior minimum of distance if they approach each other.

Now we give some preliminary facts more specific to the proofs below:

2.1. Preliminaries for the proof of Theorem 1.1

Singularities along the flow can only occur at points and times where the norm of the second fundamental form blows up. Hence, to rule out singularities, we need curvature estimates. Our method to find curvature estimates in this section is the Brakke-White regularity theorem [10]. The version of this theorem stated below is due to Brian White [45]:

Theorem 2.1 (Brakke, White).

There are numbers and with the following property. If is a smooth mean curvature flow starting from a hypersurface in an open subset of the spacetime and if the Gaussian density ratios are bounded above by for , then each spacetime point of is smooth and satisfies:

(2.3)

where is the infimum of among all spacetime points .

In the above theorem we recall that the Gaussian density ratio is given by

(2.4)

By Huisken’s monotonicity formula [28], this quantity is monotone nondecreasing in . So, to get curvature bounds via the regularity theorem, we only need to sufficiently bound a range of the densities in an open set for some time interval with .

The curvature estimates in the proof of Theorem 1.1 will depend on Proposition 2.2, which is proven with the Brakke regularity theorem. Note that the bounds on below are precisely curvature estimates for :

Proposition 2.2.

Let be a flow for , . Let and be smooth properly embedded hypersurfaces that are disjoint and have uniformly bounded by . Suppose that

  1. , , and all satisfy one of assumptions (1)-(3) of Theorem 1.1, i.e. they are all either asymptotically flat, periodic with compact fundamental domain, or are periodic and asymptotically flat in their domain,

  2. lies between hypersurfaces and for ,

  3. is a graph of a function over with , and

  4. the distance between and is uniformly bounded by ,

Then there is and depending on and but not such that if , the flow of will be a graph of a function over with for . Thus, will exist with uniformly bounded curvature as long as it lies between and .

Proof.

We first note by continuity of the flow there is some small (depending on the bound ) so if condition (2) above is satisfied it will remain so for on for some function defined on with .

To deal with later times we will use the Brakke regularity theorem. More precisely, from the bound on for and the bounds that come from choosing small enough, we find that we can obtain bounds on which approach as . Note that the bounds on depend only on , the bound, and . Then, choosing small enough, we find that the area of as a graph over some ball in is an arbitrarily small multiple of the area of that ball in . We may then apply the Brakke regularity theorem over uniformly small balls to find that at time , and, in particular using [18], one may continue the smooth flow. Replacing above with and replacing with the corresponding doubling time , we get for times . Note that the doubling time depends only on and and not on itself. This follows from assumption (1) above since is initially periodic or asymptotically flat and will remain so for as long as it exists. This means that must attain an interior maximum of curvature and thus its doubling time depends only on and . Then, choose small enough to find small enough bounds on for to apply the Brakke regularity theorem over the same uniformly small balls as before. This gives that for . Then, we may iterate the argument using while keeping the same as long as the flow exists between and . ∎

2.2. Preliminaries for the proof of Theorem 1.3

The curvature estimates in the proof of Theorem 1.3 need a different approach than those of Theorem 1.1. The following result of Ecker-Huisken [18] (cf. Corollary 3.2 (ii)) will be used in the proof of Theorem 1.3:

Theorem 2.3 (Ecker-Huisken [18]).

Let be a fixed vector in and let and . Let be a ball in a hyperplane orthogonal to . Suppose that a mean curvature flow may be written as a compact graph over for time . Then, for ,

(2.5)

where and is a unit normal to .

In the proof of Theorem 1.3, we will be working directly with the catenoid, so we will list important facts about the geometry of the catenoid and rotationally symmetric flows in general.

Let be a catenoid in of fixed radius centered around the origin. Arrange the catenoid so that it may be represented as the rotation of a positive graph around the -axis. Arrange and scale the catenoid so that it is symmetric about the -axis and . That is, the minimum point of the graph is located at . The catenoid is the unique nonflat minimal surface, up to scaling, that arises as a surface of rotation in this way. The catenoids of different scales are given by for . For , the catenoid of radius has a finite half-width given by

(2.6)

For , the catenoid has an infinite half-width, so let . Note that others have defined as twice what we have defined it here, but we will have need of this normalization later on.

We define the outward unit normal to to be the unit normal that points away from the axis of rotation. Also, separates into two connected components: the “inside” and the “outside” of the catenoid. The outside of the catenoid is defined to be the connected component that outward unit normal to points into. And the “inside” is the other component.

The mean curvature flow of a surface given by the rotation of a graph is particularly simple; it is equivalent to the flow of the graph satisfying the following equation.

(2.7)

Indeed, if a surface is initially given by the rotation of a graph, then its flow will be given by the rotation of a graph for as long as it exists. Abstractly, this follows from the Sturmian theory of such flows, developed in [3]. For the graph of the profile curve of , the right hand side of (2.7) vanishes, which is consistent with the fact that is a minimal surface. An important observation is also that, as increases, the flow is better and better approximated by the curve shortening flow of the graph of u. Indeed, (2.7) without the second term on the right-hand side is just the curve-shortening flow of a graph.

Finally, for an embedded rotationally-symmetric surface, consider the points such that the unit normal satisfies , where is a unit normal perpendicular to the axis of rotation and pointing away from the axis of rotation. Then, the mean curvature at is the following:

(2.8)

where is the curvature of the profile curve with respect to and is the angle the tangent vector to the curve makes with the positively-oriented -axis. Note that in the case that the curve is a graph with the appropriately chosen normal, every point will satisfy the condition .

3. Proof of Theorem 1.1

Throughout this section unless otherwise stated, will denote a minimal surface as assumed in Theorem 1.1.

To construct our ancient solutions we will proceed as typical: we first construct “old-but-not ancient” solutions to the flow existing on , , recentering the time coordinate to get flows existing on , and take a limit of flows to obtain an ancient solution. To take the limit we need to have good enough estimates, and to show we get something nontrivial, we need to know that the limit flow is nonempty and not another “known” ancient solution to the flow, like the original minimal surface .

First we will prove the following general lemma which will help to construct the old-but-not-ancient flows and immediately give Lemma 3.3.

Lemma 3.1.

Let be a hypersurface in such that exists with uniformly bounded geometry for . For every and , there exists such that if is a hypersurface with , then

(3.1)

for all .

Proof.

Suppose not. Then, for some and , there is a sequence of such that yet there exists such that . Note that since has uniformly bounded geometry independent of , we have that the flow of will a priori exist on some short time interval independent of . Then, the sequence converges to in so by continuity of the flow under perturbations of the initial conditions in , we have that for .

Since has uniformly bounded geometry, we have that each , for large , has uniformly bounded geometry as well. However, since is a compact time interval, we may find a subsequence of converging to such that subconverges to a limit surface such that . However, by the above argument, must be converging to . This is a contradiction. ∎

To define one family of the old-but-not ancient solutions, , we will consider perturbations of by small constant variations normal to , for a given choice of unit normal (here we use the two-sided hypothesis). We can similarly define another family by switching the orientation of the normal throughout to obtain the second claimed family.

More precisely, let be given by . By the uniform tubular neighborhood assumption and bounded curvature, for all small, will be smooth and embedded. Note that the uniform tubular neighborhood assumption gives that there exists one such that is smooth and embedded, but with our assumptions, this implies that is smooth and embedded for all small . For , will be defined to be for an appropriate choice of . Apply Lemma 3.1 to the minimal surface with some fixed choice of and the time . This gives some such that will exist for time and will be -close to in for this same time. We will further refine this choice of after Lemma 3.4.

Note that is (weakly) mean convex with respect to for all small enough such that is smooth and embedded. Indeed, suppose that it is not. Then, if there is a point such that , for some small time , . This follows because points away from , so if has negative mean curvature at with respect to , the flow will force it to become closer to for some short time. However, by the comparison principle between noncompact hypersurfaces (see Section 2), for all time . This is a contradiction, so is mean convex. By Corollary 4.4 of Ecker-Huisken [18], will remain mean convex for as long as it exists.

In order to take a limit of the approximate solutions , we need curvature bounds. These will come from Lemma 3.2. We must also know that for large enough , corresponding to small enough , will exist for long enough. This is given by Lemma 3.3. In order to extract a limit that is distinct from , we need to know that there exists some such that for all small enough, will flow to be distance away from , at some point, after some amount of time. This will come from Lemma 3.4.

Now, we find the following immediate consequence of Proposition 2.2:

Lemma 3.2.

There is and so that as long as the flow is a subset of the interior of the region between the minimal surface and the smooth embedded surface for , , then on and is the graph of a function over with for .

The above lemma shows that the flows will exist with uniformly bounded curvature as long as the flow is between and . Now that we have set , apply Lemma 3.1 to the minimal surface , setting to be , to find the following:

Lemma 3.3.

Let be the constant obtained from Lemma 3.2. Let be the first time . Then as , .

This lemma tells us that , for small enough, will exist for as long as it is a subset of the region between and , with curvature bounds in this region independent of (from Lemma 3.2). This allows us to extract an ancient limit flow from . The only thing remaining is to ensure that the limit flow will be different from . To do this, it suffices to show that for every sufficiently small , . That is, for all small enough , will eventually flow to intersect , for chosen sufficiently small. This is the heart of the proof of Theorem 1.1 and where the assumption of instability is used.

Lemma 3.4.

Let be as chosen above. Let be the first time that . Then after possibly taking smaller, for all , .

Proof.

Suppose this is not the case for . Then, there is some so that for all time . By Lemma 3.2, will have uniform curvature bounds for all time. This means that will exist for time with uniformly bounded curvature and will remain between and .

As proven above, is mean convex, so is moving monotonically away from . If is minimal, then relabel it to and proceed to the next paragraph. Suppose that is not minimal. By the uniform curvature bounds on , we may pass to a limit along any subsequence of to find that smoothly converges to some smooth limit surface which we denote by . In fact, is a minimal hypersurface.

Indeed, suppose is not minimal. Since is mean convex, we know that must be mean convex as well. Then, there is such that . We may find a smooth curve such that and is the spacetime track of a point converging to . For large enough, . Since the flow moves monotonically by mean convexity, we have by integration that . This contradicts the fact that converges to . Thus, is a smooth complete minimal hypersurface disjoint from yet is between and .

We then reset to be . If the statement is true for then we are done; otherwise, we iterate the argument. Labeling , we must have the conclusion either be true for some choice of , or we obtain a sequence of distinct minimal surfaces approaching from one side.

In the latter case, denote by each of the found above using . By the curvature estimates coming from Lemma 3.2 they are graphical over for large enough since . Of course, since , we have that converges from one side to . This then gives rise to a positive solution to the Jacobi operator on as in [43]. By Theorem 1.1 in [19] we must then have on any bounded domain of , contradicting the instability of . ∎

Hence for small enough, we know for every integer there will be so that . Let be such that and define to be . We have that the flow will exist for with curvature—and hence by Shi’s estimates, the derivatives of curvature—bounded uniformly, independent of . Recentering the time parameter by for each of the , we have that is defined for . By definition of , will intersect . Let . Recenter each so that is taken to the origin. Then, take a subsequential limit in the smooth topology to find an ancient solution .

There is a catch though: it is conceivable that does not flow out of if the diverge to spatial infinity. For example, although this will soon be ruled out, if were a catenoid and the diverged, the limit of recenterings of would be a plane.

If is periodic with compact fundamental domain, the recenterings do not matter so we suppose that is asymptotically flat (the precise rate does not matter). In this case we will show that all the are contained in a compact set. Indeed, suppose not. Then the limit of the surface under recenterings will be flat and we will obtain in the limit an ancient solution flowing out of a plane distance from the origin but which, at , intersects the origin.

Going far enough back in time, there will be a time for which is at most distance from . Considering an appropriate translate of by distance though, we see by the comparison principle then that the flow will never be distance more than from , giving a contradiction.

We get that the all lie in a bounded domain, so after recentering them all to the origin, the corresponding recenterings of result in moved by a finite translation. The ancient solution is not , since it must be bounded away from at the origin at time by distance . It is also certainly not a minimal surface because it flows out of but is distance from at , so is not stationary.

We note that if we took the unit normals with the opposite orientation from , we would obtain a distinct ancient solution. These are distinct because they are approaching from opposite sides (since is -sided). With respect to , this other ancient solution has negative mean curvature.

Finally, we can see that these ancient solutions are not solitons. If the ancient solution approaches the minimal surface from one side as , this means that may not be translating as it must be slowing down to approach . These ancient solutions may not be just rotating, as that would imply they would not be on just one side of . Neither are these ancient solutions homothetically shrinking, as this would imply that they do not approach any surface as . Finally, we have that combinations of these rigid motions are also impossible. Homothetic shrinking in combination with any other rigid motion is ruled out for the same reason. And these solutions cannot be translating and rotating at the same time, as both motions occur at some constant rates, which would imply that cannot be converging to any surface as . This completes the proof of Theorem 1.1.

4. Proof of Theorem 1.3

In this section, we will construct the eternal solution described in Theorem 1.3, which exists in for all . This eternal solution will be constructed using a catenoid, similar to what was done in Section 3.

Recall the notation set in Section 2.2. We let be a catenoid in , , normalized to have radius . The catenoid has width (which is infinite for ) and is given by a graph which is reflection symmetric about , as described in the preliminaries. Moreover, we split up into two components: the inside and the outside of the catenoid . The outside of the catenoid is the component of that the outward unit normal points into. The outward unit normal is the normal to that points away from the axis of rotation.

By Theorem 1.1 there exists an ancient solution to mean curvature flow, such that uniformly converges to as and is a subset of the outside of for all time. This ancient solution is embedded and is not a soliton. Since smoothly converges to as , we may equip with a unit normal that is compatible with the outward unit normal to . This is the outward unit normal to , and by Theorem 1.1, is mean convex with respect to its outward unit normal. Recall that the approximate solutions used to construct are of the form for . Since is -invariant and all have this symmetry as well, we get that is -invariant with respect to the same axes of symmetry of . Similarly, may be represented as the rotation of a graph , since can be, where is symmetric about . We know that will remain a subset of the outside of for all time since all lie outside and the mean convexity of will force the flow to nest and thus avoid for as long as it exists. This means that for as long as exists. And by (2.8) combined with mean convexity, we have that is convex for all time.

The last useful property of is that it will remain asymptotic to for as long as it exists. The approximate solutions are asymptotically flat and so by pseudolocality (see Chen-Yin [12]), they must remain arbitrarily close to outside a large enough ball. This means that for as long as the flow exists, it will remain asymptotic to . This means that , as the limit of these approximate solutions, will remain asymptotic to for as long as it exists.

With this in hand, we will show that the ancient solution is in fact eternal. We will find its asymptotics later.

Proposition 4.1.

For , let be the ancient solution to mean curvature flow as described above. Then exists for all time, , and it is spatially asymptotic to for all time slices.

Proof.

In order to prove that exists for all time , we will show that for each , there is a bound on for .

Let be the profile curve of , and let be the profile curve of , considered as graphs over the same axis with the same axes of symmetry. Since is mean convex and initially satisfies , we have that for as long as it exists.

As mentioned in Section 2.2, note that (2.7) without the last term is merely the curve shortening flow of the graph . Since the second term is negative, a solution to (2.7) is a subsolution to graphical curve-shortening flow. This means that a solution to (2.7) starting at will avoid the curve shortening flow of an appropriately chosen graph such that for some independent of . Indeed, since is a convex curve which converges uniformly to as , we may place a grim reaper of some half-width strictly above such that avoids . The grim reaper will be a graph over and its curve shortening flow will be given by , where is a constant. We have that will remain asymptotic to for as long as it exists and will remain asymptotic to . So, if ever intersects , there will be a point such that the difference in height between and will reach a strict local minimum at . Applying the avoidance principle using that is a subsolution of (2.7), we have that will avoid for as long as it exists.

Suppose that is a distance from at . Then, we have that for , for some constant depending only on , , and . Indeed, since , cannot be too large at since that would imply would intersect , which is a contradiction. Here, the dependence on and is irrelevant and such choices can be fixed from the outset.

We may use the time-dependent bound on over in combination with Theorem 2.3 to bound for around the tip, which we identify with . Identify the - plane that is in with the plane . Consider the hyperplane perpendicular to the normal such that contains . The normal is the outward normal to at the point identified with the tip in the - plane. Consider a ball of radius in centered around . Since is normalized to radius , this means for all time and so will remain graphical over for all time. Here we use the fact that for all time that it exists, so the flow will not collapse onto the axis of rotation.

Now, consider the quantity , where is the unit normal (independent of time) to corresponding to the normal to . By the bound over , there is a bound on depending on in . Using the bound on , we may apply Theorem 2.3 with , the hyperplane , and some . This gives a bound for the part of that is graphical over .

With this in hand, we move on to finding a bound on for the rest of . By the fact that there is a bound on around the tip of , we have that there is a bound depending on for the speed of the tip. That is, .

By symmetry of about , we can just consider one side, so let us consider over . Consider the unit vector in the - plane which we identify with . Consider small but fixed. Let be the hyperplane that is orthogonal to . Let be the projection of onto , where the last coordinates of are zero. Then, we may find a fixed small enough such that for each , will be a graph333Technically, will be a double-sheeted graph over by considering the part of lying over , but this does not affect the application of Theorem 2.3, which will only be applied to the part of corresponding to . over the ball for a time depending on . This is possible because and the tip moves at a speed only depending on , as shown above.

Figure 2. An approximate profile of along with the tilted support plane . Note that in reality, is not rotationally symmetric about the -axis.

Moreover, the quantity , which is just with respect to , is uniformly bounded over such . This is because is at an angle with respect to the unit normal at the tip , so will grow at a linear rate with respect to as approaches its asymptote. Thus, applying Theorem 2.3 to using , the bound on , and an appropriate choice of (given the choice of for the previous application of Theorem 2.3), we get a bound on for over .

Putting all of this together, there is a bound for . Since is bounded for any , this gives that will exist for by applying the short time existence theorem at any finite time to extend the flow. So, is an eternal solution as claimed. ∎

To understand the asymptotics of the flow, we will need to relate the motion of the profile curve to the curve shortening flow, which will require the following lemma.

Lemma 4.2.

Let be the eternal solution of Proposition 4.1. As , becomes infinitely far from its axis of rotation.

Proof.

Equivalently, we will prove that will become infinitely far from the axis. Suppose that this is not the case. Since is the unique minimum of the flow, we have that . This implies that .

If is uniformly bounded for as , then converges to a smooth minimal surface as . However, must be given by the rotation of a graph symmetric about . So, is a catenoid symmetric about , but this must intersect , which is a contradiction.

Now, suppose that is not uniformly bounded as . Let be the tilted plane as above. Let be the projection of onto . Here, we are technically taking the part of with the last coordinates zero. Since the tip of is stationary as , i.e. , we have that is a graph over small balls centered on with controlled derivative for all time . We may now apply Theorem 2.3 to over . We find that for any distance , the set of points on which is distance greater than from the tip has uniformly bounded , depending on , for all .

Since we are supposing as , this leaves the possibility that the curvature is blowing up as near the tip . In other words, we may find times and points such that , where are all within a uniform distance in from the tip point . By reflection symmetry across , these points come in pairs . Note that such points are uniformly bounded away, by mean convexity of the flow, from the axis of rotation and hence we must have . In particular, for sufficiently large.

Because this implies that . Thus, must achieve an interior minimum of at some point on the graph of between and for all large enough. It is easy to see something even stronger in fact: one can actually find , , in lieu of the discrete , such that . Then, since , there is on the graph of between and such that for all sufficiently large times , achieves an interior minimum of at . To see the existence of such for sufficiently large times, if did not exist, we would find a sequence of times where has uniformly bounded curvature, which would converge to a minimal catenoid, as before. This is a contradiction as such a catenoid must intersect yet must also be distinct from .

Using that the tip lies between and on the graph of , we see it has lower bounded by . The , by mean convexity and the strict maximum principle, must have and must be increasing for all time, as long as . This implies that must be bounded away from zero for all time, contradicting the fact that . Thus, must become infinitely far from its axis of rotation. ∎

Since the profile curve moves infinitely far from the origin, it behaves like the curve shortening flow. To understand the asymptotics, we will need to take a pointed limit of the profile curve as , but this requires uniform curvature bounds, which we find in the following lemma.

Lemma 4.3.

Let be the eternal solution of Proposition 4.1. Then, there exists such that for .

Proof.

Suppose not. Then, we may find a sequence of times such that achieves the supremum

(4.1)

and .

We first pick such that realizes the supremum . It is possible to pick all positive by the symmetry of about . Then, define

(4.2)

By (2.7), we have that satisfies the following equation: