Ancient mean curvature flows and their spacetime tracks

arXiv:1901.05481v2 [math.DG] 26 Feb 2019 ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS FRANCESCO CHINI AND NIELS MARTIN MØLLER Abstract. We study properly immersed ancient solutions of the codimension one mean curvature flow in n-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic Omori-Yau maximum principle for ancient mean curvature flows. 1. Introduction Ancient mean curvature flows show up naturally in the study of singularities, as tangent flows from blow-up analysis (for a basic discussion, see e.g. Chapter 4 in [Ma11]). Especially in recent years they have gained much attention and some partial classifications are now available. Convex ancient solutions arise in the case of mean convex mean curvature flows [Wh00], [HS99a], [HS99b] and have also been investigated by [Wa11]. In the case of the curve shortening flow, Daskalopoulos, Hamilton and Sesum provided a complete classification [DHS10] of closed convex embedded ancient curve shortening flows. Note that there also exist nonconvex examples [AY18]. Date: February 27, 2019. Key words and phrases. Mean curvature flow, ancient solutions, minimal surfaces, bi-halfspace theorem, wedge theorem, Omori-Yau principle, maximum principles, nonlinear partial differential equations, parabolic equations. Francesco Chini was partially supported by the Villum Foundation’s QMATH Centre. Both authors were partially supported by Niels Martin Møller’s Sapere Aude grant from The Independent Research Fund Denmark (Danish Ministry). 1 2 FRANCESCO CHINI AND NIELS MARTIN MØLLER In [HS15], Huisken and Sinestrari again studied ancient mean curvature flow under several natural curvature assumptions, namely convexity and k-convexity, and provided some characterizations of the shrinking sphere, assuming convexity. Haslhofer and Hershkovits [HH16] proved the existence of an ancient oval in dimensions n > 1, as conjectured by Angenent in [An13], building on an idea of White [Wh03]. Recently Angenent, Daskalopoulos and Sesum [ADS18] proved the uniqueness of this ancient mean curvature flow under some assumptions, based on their previous work [ADS15] (using the barriers from [KM14]). More precisely they prove that an ancient mean curvature flow which is compact, smooth, noncollapsed, not self-similar and uniformly 2-convex must be the solution constructed in [Wh03]-[An13]-[HH16]. In [BC17] Brendle and Choi classified convex, noncompact, noncollapsed ancient flows in R3 , proving that they agree (up to isometry and up to scaling) with the self-translating bowl soliton. Recently, in [BC18], they extended their result to higher dimensions, under the extra assumption of uniform 2-convexity. In [CHH18] Choi, Haslhofer, Hershkovits classified all 2-dimensional ancient mean curvature flows with low entropy in R3 . In this paper we generalize some of the results from [CM18], which classified the projected convex hulls of all proper self-translaters. This establishes the following string of generalizations: The below timedependent Theorem 5 for ancient flows implies the time-independent self-translating hypersurfaces case [CM18], which again implies the minimal hypersurface case [HM90], which finally implies the Euclidean case of conically bounded minimal surfaces [Om67]. Note that in Theorem 3 and Theorem 5 below, we do not have any curvature or non-collapsing nor entropy assumptions. Here we only need to assume the flows to be properly immersed. We expect these results to be useful in the future investigation of ancient solutions, both for problems of classification and construction of examples, and hence to the investigation of the set of possible singularities in the mean curvature flow. Finally, also in regularity questions for mean curvature flow with boundary, bi-halfspace theorems (a.k.a. wedge theorems) are useful: In January 2019, Brian White posted a paper [Wh19] with a result on boundary regularity (announced some time ago, e.g. in [Wh09], see also [St96]), proved there using a new wedge theorem for self-shrinking Brakke flows. It would be interesting to understand its relation to our smooth results in [CM18] and in Theorem 3 and Theorem 5 below. ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS 3 2. Preliminaries Definition 1. Let M n be a smooth, connected n-dimensional manifold without boundary and let I ⊆ R be a (time) interval. A mean curvature flow is a smooth map F : M × I → Rn+1 such that Ft : M → Rn+1 is an immersion for every t ∈ I, where Ft (x) := F (x, t), and F satisfies the following equation → − ∂F (1) = H. ∂t The mean curvature flow is said to be an ancient, immortal or eternal solution, if respectively after a time translation I = (−∞, 0), (0, ∞) or R. In what follows, we will denote by Mt the manifold M endowed with the pullback metric induced by Ft , i.e. Mt := (M, Ft∗ h·, ·iRn+1 ) . The Levi-Civita connection and the Laplacian on Mt will be denoted by ∇Mt and ∆Mt respectively. Moreover, we will always consider proper mean curvature flows, meaning that for every t ∈ I the map Ft : M → Rn+1 is a proper immersion. We remind the reader that properly immersed hypersurfaces are geodesically complete w.r.t. the induced Riemannian metric (by the Heine-Borel property and Hopf-Rinow). As always, most of our results fail without the properness assumption, see e.g. the examples in [Na96] of minimal surfaces non-properly immersed into ambient balls. 3. Main results Lemma 2 (Omori-Yau Maximum Principle for Ancient MCFs). Let F : M × (−∞, 0) → Rn+1 be a proper ancient mean curvature flow. Let f : M × (−∞, 0) → R be a bounded and twice differentiable function. Then there is a sequence of points (xi , ti ) ∈ M × (−∞, 0) such that (i) limi→∞ f (xi , ti ) = supM ×(−∞,0) f , (ii) limi→∞ |∇Mti f (xi , ti )|
= 0, ∂ (iii) lim inf i→∞ ∂t − ∆Mti f (xi , ti ) ≥ 0. Theorem 3 (Wedge Theorem for Ancient Mean Curvature Flows). Let H1 and H2 be two halfspaces of Rn+1 such that the hyperplanes P1 := ∂H1 and P2 := ∂H2 are not parallel. Then H1 ∩ H2 does not contain any proper ancient mean curvature flow. More precisely, there does not exist any proper ancient mean curvature flow F : M × (−∞, 0) → Rn+1 such that Ft (M) ⊆ H1 ∩ H2 for all times t ∈ (−∞, 0). 4 FRANCESCO CHINI AND NIELS MARTIN MØLLER Remark 4. Theorem 3 holds in particular for proper eternal mean curvature flows, i.e. I = R. There are two particularly important subclasses of eternal mean curvature flows: self-translating solitons and minimal hypersurfaces. Therefore it generalizes Theorem 1 contained in the paper [CM18] by the present authors (see also the corollaries and discussion there), as well as classical theorems by Omori [Om67] (in the Euclidean case) and Hoffman-Meeks [HM90] (in the case without boundary). A third type of ancient solutions are the self-shrinking solitons (where there are many examples, see e.g. [KKM11]-[Mø11]), which even obey a halfspace theorem [CE16] (proved by CavalcanteEspinar using the barriers from [KM14]). As in [CM18], it is interesting to ask which of the cases can actually occur in Theorem 5. For each case we have respectively: Flat planes, reaper cylinders (plus “Angenent ovals” [An92] and “ancient pancakes” [BLT17]), which give slabs. No examples (to our knowledge) of halfspaces. Spheres, cylinders and the bowl soliton for all of Rn+1 . Note of course that by [CE16], self-shrinkers cannot provide examples in the halfspace case (see also the discussion in [CM18]). Note also that Theorem 3 is the “bi-halfspace” result we can expect for ancient mean curvature flows. In fact a “halfspace theorem” version would be false. There are several counterexamples: for instance planes and grim reaper cylinders. Also, a “halfspace” statement would be false, even for those ancient mean curvature flows all of whose timeslices are compact: A counterexample to this is given by the so-called ancient pancake [BLT17] (or for n = 1, Angenent’s ovals [An92]) which is contained for all its evolution in a slab between two parallel hyperplanes, and thus no general halfspace theorem could hold for all ancient solutions. We remark that the statement of Theorem 3 is false for general immortal mean curvature flows, i.e. I = (0, ∞). In fact there are selfexpanding mean curvature flows such that they are contained for their entire evolution in the intersection of two halfspaces with nonparallel boundaries (see f.ex. [SS07]). Note that f.ex. Lemma 13 as stated, and hence the proof of Theorem 3, would have failed if we had instead taken I = (0, ∞). Theorem 5 (Classification of Sets of Reach of Ancient Flows). Consider a proper ancient mean curvature flow. Let R := [ t∈(−∞,0) Ft (M) ⊆ Rn+1 ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS 5 denote its set of reach. Then the convex hull Conv(R) is either a hyperplane, a slab, a halfspace or all of Rn+1 . In the next corollary, we keep track of the time coordinate to get a spacetime track version, which is of course equivalent to Theorem 5. Corollary 6 (Spacetime Tracks of Ancient Flows). Consider for a proper ancient mean curvature flow its spacetime track ST [ ST := {t} × Ft (M) ⊆ R × Rn+1 . t∈(−∞,0) Then Conv(π2 (ST )) is either a hyperplane, a slab, a halfspace or all of Rn+1 , where π2 denotes the projection to the Rn+1 -factor. In the next corollary, the set R ∪ {p∞ } can also be thought of simply as the closure of the set of reach. Corollary 7 (Sets of Reach of Compact Convex Ancient Flows). Consider any compact convex ancient mean curvature flow in Rn+1 , which at time 0 becomes extinct at a point p∞ ∈ Rn+1 . Then R ∪ {p∞ } (the set of points reached, with the singular point added in) is either a slab, a halfspace or all of Rn+1 . Remark 8. Note that Corollary 7 is in agreement with Corollary 6.3 1 in [Wa11] where blow-downs (−t)− 2 Ft (M) as t → −∞ for convex ancient solutions were classified: any Sk × Rn−k , with k = 1, . . . , n or multiplicity two hyperplanes. It is not clear to us whether the halfspace case of Corollary 7 could occur. For instance in the convex case it does not happen in the curve shortening flow (i.e. the case n = 1) because of the classification for closed curves in [DHS10], which shows that the only possible sets of reach are strips and R2 . Moreover Wang (Corollary 6.1 [Wa11]) showed that the set of reach of a (not necessarily compact) convex ancient mean curvature flow arising as a limit flow of a mean convex flow, is the entire Rn+1 . Note that the set of reach there is taken over the whole maximal time interval, which might be (−∞, ∞). 4. Proofs The proof of Lemma 2 is based on the Omori-Yau maximum principle (tracing its roots back to [Om67]–[CY75]) proven by Ma in [Ma17]. The main difference is that here we are interested in ancient mean curvature flows and thus our time interval is not finite, which complicates slightly (but essentially) the proof. On the other hand, because of the applications we have in mind, we focus on the case where the ambient 6 FRANCESCO CHINI AND NIELS MARTIN MØLLER manifold is Euclidean space Rn+1 and the codimension is 1, and the argument we give here is self-contained. Proof of Lemma 2. Let (x̄i , t̄i ) be a sequence in M × (−∞, 0) such that (2) lim f (x̄i , t̄i ) = i→∞ sup f. M ×(−∞,0) Consider the function r : Rn+1 → R defined as r(y) := kyk. This defines a function ̺ on M × (−∞, 0) by ̺(x, t) := r(F (x, t)). Now let (εi )i∈N be the sequence of positive numbers (well-defined even if ̺(x̄i , t̄i ) = 0)   1 1 1 , < ∞. (3) 0 < εi := min i i ̺(x̄i , t̄i )2 Note that limi→∞ εi = 0 and that for every i ∈ N 1 εi ̺(x̄i , t̄i )2 ≤ . i Let us now for i = 1, 2, . . . define fi : M × (−∞, 0) → R by (4) (5) fi (x, t) := f (x, t) − εi (̺(x, t))2 . Note that each fi is bounded from above by supM ×(−∞,0) f < ∞. Claim: Fix a time t ∈ (−∞, 0) and fix i ∈ N. Then there exists a point xit ∈ M where the function fi (·, t) attains its supremum over M. Furthermore, this is locally uniform in the sense that considering τ near t, all the points xiτ ∈ M can be chosen from a fixed compact subset K ⊆ M (with K = Kt possibly dependent on t and on the proximity of τ to t). If M is compact, then the claim is trivial. If M is not compact, it follows from the crucial properness assumption. In fact, let R > 0 be large enough so that Ft (M) ∩ BR 6= ∅, where BR is the ambient open ball of radius R > 0 in Rn+1 centered at 0. Since f is bounded on M × (−∞, 0), we can choose S > R > 0 so that (6) sup M ×(−∞,0) f − εi S 2 < inf M ×(−∞,0) f − εi R 2 Equation (5) now shows that for points p ∈ M \ Ft−1 (BS ) (which is nonempty, because from the properness of Ft and noncompactness of M follows that Ft (M) cannot be contained in any finite radius ambient ball) holds: (7) fi (p, t) ≤ f (p, t) − εi S 2 ≤ sup M ×(−∞,0) f − εi S 2 . ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS 7 Therefore taking the supremum over p ∈ M \ Ft−1 (BS ) yields (8) sup M \Ft−1 (BS ) fi (·, t) ≤ sup M ×(−∞,0) f − εi S 2 < inf M ×(−∞,0) f − εi R 2 , using (6). Thus, finally, using f − εi R2 ≤ fi on Ft−1 (BR ): (9) sup fi (·, t) < inf fi (·, t) Ft−1 (BR ) M \Ft−1 (BS ) From continuity of F , we also have that there exists δ > 0 such that, for every time τ ∈ (t − δ, t + δ), we have Fτ (M) ∩ BR 6= ∅ and thus (9) still holds. Properness of the flow implies that each Fτ−1 (B̄S ) ⊆ M is compact, therefore we have for every τ ∈ (t − δ, t + δ): (10) sup fi (·, τ ) = max fi (·, τ ). F −1 (B̄S ) M It only remains to find a uniform compact K ⊆ M as claimed. This is also guaranteed by the properness of the immersion at each time, as follows. Since Ft−1 (B̄S ) is compact, we can choose K to be any larger compact set such that Ft−1 (B̄S ) ⊆ K ◦ ⊆ M, where K ◦ denotes the interior of K. Consider the closed set C := M \ K ◦ . The two sets Ft (C) and BS are disjoint. Together with compactness of B̄S , and the assumption that Ft is proper (and hence a closed map), which ensures closedness of Ft (C), it then implies distRn+1 (Ft (C), B̄S ) > 0. From the triangle inequality follows that distRn+1 (Fτ (C), BS ) is continuous in τ ∈ (t − δ, t + δ). Therefore after possibly taking δ > 0 smaller holds distRn+1 (Fτ (C), BS ) > 0 for all τ ∈ (t − δ, t + δ). But then as claimed Fτ−1 (BS ) ⊆ K and finally, using (10) we finish the proof of the final part of the claim: (11) ∀τ ∈ I : sup fi (·, τ ) = max fi (·, τ ). M K For any time t ∈ (−∞, 0) and i ∈ N, let us use the claim and denote (12) Li (t) := max fi (x, t) = fi (xit , t), x∈M for some xit ∈ Kt . Note that the function (−∞, 0) ∋ t 7→ Li (t) is bounded from above by L := supM ×(−∞,0) f . By Hamilton’s Trick [Ha86] (see f.ex. Lemma 2.1.3. in [Ma11], which is where we use the uniformicity property (11) and the Kt in the claim), each Li is a locally Lipschitz function of t and therefore continuous. The function Li is also differentiable almost everywhere, and at any of its 8 FRANCESCO CHINI AND NIELS MARTIN MØLLER differentiability times t ∈ (−∞, 0) we have as usual ∂fi i dLi (t) = (x , t). dt ∂t t This implies (using Lemma 9 in the Appendix) that there is a time i ti ∈ (−∞, 0) such that dL (ti ) exists and with xi := xiti there holds dt ∂fi dLi (xi , ti ) = (ti ) ≥ −2εi , ∂t dt (13) and also Li (ti ) − sup Li < εi , (14) (−∞,0) or in other words fi (xi , ti ) − (15) sup fi < εi . M ×(−∞,0) We have from (12) that ∆Mti fi (xi , ti ) ≤ 0, therefore with (13)   ∂ Mti fi (xi , ti ) ≥ −2εi . −∆ (16) ∂t From standard computations and (1) we also have (as the only step in the proof where we use the mean curvature flow equation) the following   ∂ Mt (17) −∆ (̺(x, t))2 = −2n, ∂t for every (x, t) ∈ M × (−∞, 0). Combining (16) and (17), we get with (5) (18)  ∂ − ∆Mti ∂t  f (xi , ti ) ≥ −2(n + 1)εi . This shows Part (iii) of the Lemma. Let us now check that also (i) and (ii) hold. From (5) and (15), and since Li (ti ) = fi (xi , ti ) = maxM fi (·, ti ), we have (see (2) for the definition of (x̄i , t̄i )) (19) f (xi , ti ) ≥ fi (xi , ti ) > sup M ×(−∞,0) fi − εi 1 − εi , i where the final inequality made use of (4). This shows Part (i), by taking the limit for i → ∞ in the string of inequalities (19). ≥ fi (x̄i , t̄i ) − εi ≥ f (x̄i , t̄i ) − ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS 9 Let us now show Part (ii). Observe that ∇Mti fi (xi , ti ) = 0, since xi ∈ M is a maximum point for fi (·, ti ). Thus we have  n+1  ⊤ ∇Mti f (xi , ti ) = 2εi ̺(xi , ti )∇Mti ̺(xi , ti ) = 2εi ̺(xi , ti ) ∇R r (F (xi , ti ) . Noting (20)  ∇ Rn+1  r (F (xi , ti ) ⊤ ≤  ∇ Rn+1  r (F (xi , ti ) ≤ 1, it is enough to show that εi ̺(xi , ti ) −−−→ 0 (note that (4) concerns i→∞ (x̄i , t̄i )). But from (19), we have fi (xi , ti ) > f (x̄i , t̄i ) − 1i − εi , so that εi ̺(xi , ti )2 = f (xi , ti ) − fi (xi , ti ) < Therefore √ sup M ×(−∞,0) f − f (x̄i , t̄i ) + 1 + εi . i εi ̺(xi , ti ) −−−→ 0 and therefore by (2), finally i→∞ εi ̺(xi , ti ) −−−→ 0. i→∞  Proof of Theorem 3. The rest of the proof is very similar to that of Theorem 1 in [CM18], in the case of self-translating solitons without boundary, the proof of which was in turn inspired by an idea for 2dimensional minimal surfaces in R3 by Borbély in [Bo11]. In particular we are going to apply Lemma 2 to a function f which is constructed exactly in the same way as in [CM18]. Let H1 , H2 ⊆ Rn+1 be two halfspaces such that P1 := ∂H1 and P2 := ∂H2 are not parallel. Let us assume by contradiction that there exists a proper ancient mean curvature flow F : M × (−∞, 0) → Rn+1 such that Ft (M) ⊆ H1 ∩ H2 for every t ∈ (−∞, 0). We can assume without loss of generality that 0 ∈ P1 ∩ P2 . Let w1 , w2 ∈ Sn such that Hi = {x ∈ Rn+1 : hx, wi i ≥ 0}. For R > 0, let LR ⊆ Rn+1 be the (n − 1)-dimensional affine subspace obtained by translating P1 ∩ P2 in the direction of w1 + w2 and such that the boundary of the solid cylinder DR := {x ∈ Rn+1 : dist(x, LR ) ≤ R} is tangent to P1 and P2 . Let dR : Rn+1 → R denote the distance function from LR , i.e. dR (x) := dist(x, LR ). Observe that (H1 ∩ H2 ) \ DR consists of two connected components. Let VR be the one where dR is bounded. Let us choose R > 0 large enough such that there exists t ∈ (−∞, 0) such that Ft (M) ∩ VR 6= ∅. 10 FRANCESCO CHINI AND NIELS MARTIN MØLLER Let us now define a function f : M × (−∞, 0) → R as follows ( dR (F (x, t)) if F (x, t) ∈ VR (21) f (x, t) := R otherwise. Observe that by construction f is continuous and bounded. Since we have chosen R > 0 in such a way that F (x, t) ∈ VR for some (x, t) ∈ M × (−∞, 0), we have that (22) 0 < R < sup f = F −1 (VR ) sup M ×(−∞,0) f < ∞. We want to apply the Omori-Yau maximum principle in the form of Lemma 2 to f . Note that f is smooth on the interior of F −1 (VR ) and, because of (22), this is actually enough in order to apply Lemma 2. By standard computations, see e.g. [CM18], one can check that on −1 F (VR ) we have   1 − k∇Mt f k2 ∂ Mt . −∆ f =− (23) ∂t dR Let (xi , ti ) ∈ M × (−∞, 0) be an Omori-Yau sequence given by Lemma 2. From (23), Part (ii) of Lemma 2 and (22), we have that the function f eventually becomes strictly subcaloric at points in the sequence:   1 ∂ Mti f (xi , ti ) = − lim −∆ < 0. i→∞ ∂t supM ×(−∞,0) f On the other hand, this is in contradiction with Part (iii) of Lemma 2, which concludes the proof.  Proof of Theorem 5. This proof proceeds quite like in the case of minimal surfaces [HM90] and self-translaters [CM18]: \ Conv(R) = {H ⊆ Rn+1 : H is a halfspace s.t. R ⊆ H}, the intersection of all halfspaces containing the set of reach. If any such two halfspaces H1 and H2 were not parallel, we would conclude that for all times t ∈ (−∞, 0) the flow is contained in a non-halfspace wedge, Ft (M) ⊆ H1 ∩ H2 , violating Theorem 3. Hence the conclusion follows.  Proof of Corollary 7. Let us first remind the reader that by a convex hypersurface Σn ⊆ Rn+1 we mean one where all principal curvatures κi > 0, i = 1, . . . , n, and that by a theorem of Sacksteder [Sa60], this ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS11 implies that Σ = ∂Ω, for some strictly convex domain in Rn+1 . Knowing this we immediately rule out the “flat plane minus one point” as a possible set of reach. Let now F : M × (−∞, 0) → Rn+1 be a mean curvature flow as in the statement. Following Huisken [Hu84], the flow will become extinct at a “round point” p∞ ∈ Rn+1 at time 0. Let Ωt be the bounded convex body such that Σt = ∂Ωt . We have that the flow sweeps out the interior of each Ωt . These facts easily imply that adding the singular point to R we get a convex set: Namely, suppose that p1 , p2 ∈ R are given. Then there exist t1 , t2 ∈ (−∞, 0) so that pi ∈ Fti (M), and with t0 := min(t1 , t2 ) we have, by the monotonicity of the domains Ωt , that p1 , p2 ∈ Ωt0 . Also, considering the line segment between p1 and p2 it is, by convexity of Ωt0 , contained in Ωt0 . Hence by the property that the flow sweeps the interior of Ωt0 , the line segment is contained in R ∪ {p∞ }. The case where one pi = p∞ follows similarly (or by continuity). Thus, having shown that Conv(R) = R ∪ {p∞ } under these extra assumptions, we apply Theorem 3 to finish the proof of Corollary 7.  12 FRANCESCO CHINI AND NIELS MARTIN MØLLER 5. Appendix In this section, we state and prove the following elementary lemma, needed in the proofs in the paper’s main sections: Lemma 9. Let L : (−∞, 0) → R be a locally Lipschitz function bounded from above. Then for every ε > 0 there exists some t0 ∈ (−∞, 0) such that L is differentiable at t0 and satisfies the following (i) L′ (t0 ) ≥ −ε, (ii) L(t0 ) > sup(−∞,0) L − ε. Proof of the Lemma. Recall that Lipschitz continuity implies absolute continuity. Let us fix ε > 0. Let us first assume that there exists t0 ∈ (−∞, 0) such that (24) L(t0 ) = sup L. (−∞,0) If L is differentiable at t1 , then we are done. Let us assume it is not. Let δ > 0 be such that |L(t) − L(t1 )| < ε for any |t − t1 | < δ. Then Z t1 (25) L′ (t) dt = L(t1 ) − L(t1 − δ) ≥ 0. t1 −δ Therefore there exists t0 ∈ (t1 − δ, t1 ) such that L is differentiable at t0 and such that L′ (t0 ) ≥ 0. Moreover |L(t0 ) − L(t1 )| < ε. Let us now assume that the supremum is not attained. The case where sup(−∞,0) L = limt→0− L(t) can be studied similarly to the above. Therefore let us study the case where sup(−∞,0) L = limt→−∞ L(t). We can assume that there is an interval I := (−∞, τ ) ⊆ (−∞, 0) such that L|I ≥ sup L − ε and such that there are no local maxima and no local minima in I. Namely, otherwise we could proceed as we did above. Note however that for the case of local minima we have to consider intervals of the kind (t1 , t1 + δ) instead. Having no local extrema implies together with continuity that the function L is monotone on I. Since sup(−∞,0) L = limt→−∞ L(t), it must be monotonically decreasing and thus satisfy L′ ≤ 0 at all points of differentiability in I, so Lebesgue-almost everywhere. Moreover Z Z τ ′ (26) L = L′ (t) dt = L(τ ) − sup L ≥ −ε. I ′ −∞ Therefore L |I is summable. There also exists a differentiability point t0 such that L′ (t0 ) ≥ −ε, otherwise we would get a contradiction with summability from L′ (t0 ) < −ε a.e in I.  ANCIENT MEAN CURVATURE FLOWS AND THEIR SPACETIME TRACKS13 References [An92] S. Angenent, Shrinking doughnuts, Progr. 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E-mail address: [email protected]