On the rigidity of mean convex self-shrinkers
aa r X i v : . [ m a t h . DG ] M a r ON THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS
QIANG GUANG AND JONATHAN J. ZHU
Abstract.
Self-shrinkers model singularities of the mean curvature flow; they are definedas the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers S k × R n − k are rigid in a strong sense -that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large,but compact, set must be a round cylinder. Using this result, Colding and Minicozzi wereable to establish uniqueness of blowups at cylindrical singularities, and provide a detaileddescription of the singular set of generic mean curvature flows.In this paper, we show that the bounded curvature assumption is unnecessary for therigidity of the cylinder if either n ≤
6, or if the mean curvature is bounded below by apositive constant. These results follow from curvature estimates that we prove for strictlymean convex self-shrinkers. We also obtain a rigidity theorem in all dimensions for graphicalself-shrinkers, and curvature estimates for translators of the mean curvature flow. Introduction
Mean curvature flow (“MCF”) is an evolution equation where a one-parameter family ofhypersurfaces M t ⊂ R n +1 flows by mean curvature, that is, it satisfies(0.1) ( ∂ t x ) ⊥ = − H n , where x is the position vector, H is the mean curvature and n is the outward unit normal.MCF is the negative gradient flow of the area functional.We call a hypersurface Σ n ⊂ R n +1 a self-shrinker , or more simply a shrinker , if it satisfies(0.2) H = 12 h x, n i . It is easy to see that a self-shrinker is the t = − x = 0. The simplest examples in R n +1 are generalized cylinders S k × R n − k . Here, and henceforth, S k denotes the round sphere ofradius √ k . When n = 1, the only smooth complete embedded self-shrinkers are straightlines through the origin, and the circle of radius √ n ≥
2; in these higher dimensions, there are many more self-shrinkers (see for instance[Ang92], [Cho94] and [KKM15]).By the combined work of Huisken [Hui90], Ilmanen [Ilm97] and White [Whi94], singular-ities of MCF are modeled by self-shrinkers. As such, one of the most important questionsin the study of MCF is to classify the possible singularities. The first major result is due to Huisken ([Hui90], [Hui93]), who showed that the only smooth complete embedded self-shrinkers in R n +1 with H ≥
0, polynomial volume growth and | A | bounded are generalizedcylinders S k × R n − k . Later, Colding-Minicozzi [CM12a] were able to remove the assumptionof bounded curvature | A | . Consequently, they showed that the only generic shrinkers are thegeneralized cylinders S k × R n − k , in the sense that all others can be perturbed away.In this paper we consider self-shrinkers that are either mean convex or graphical on com-pact sets. Our first main result is the following local curvature estimate for uniformly meanconvex shrinkers: Theorem 0.1.
Given n and δ >
0, there exists C = C ( n, δ ) so that for any smooth properlyembedded self-shrinker Σ n ⊂ R n +1 which satisfies( ⋆ ) H ≥ δ on B R ∩ Σ for
R > | A | ( x ) ≤ CRR − | x | H ( x ) , for all x ∈ B R − ∩ Σ . The proof of Theorem 0.1 is inspired by the interior curvature estimates of Ecker-Huisken[EH91], which give local curvature estimates for MCF with bounded gradient. Similar argu-ments may also be found in [AG92] and [CNS88].One central problem in the study of the singularities is the uniqueness of blowups, namely,whether different sequences of dilations might give different blowups. For compact singu-larities of MCF, this uniqueness problem is better understood; see for instance [Sch14] and[Ses08]. The first uniqueness theorem for blowups at noncompact singularities was obtainedby Colding-Ilmanen-Minicozzi [CIM15], who proved that if one blowup at a singularity ofMCF is a multiplicity-one cylinder, then every subsequential limit is also a cylinder, andColding-Minicozzi [CM15], who showed that the axis of the cylinder is also independent ofthe sequence of rescalings. Using this uniqueness in a fundamental way, Colding-Minicozzi[CM14] were able to give a quite complete description of the singular set for MCF havingonly generic singularities.The key to proving the uniqueness at cylindrical singularities is the rigidity theorem of[CIM15, Theorem 0.1], which says that any self-shrinker that is mean convex with bounded | A | on a large compact set must in fact be a cylinder. As an application of Theorem 0.1,we show that their rigidity theorem holds even without the assumption on | A | , so long as n ≤
6. Specifically, we prove the following:
Theorem 0.2.
Given n ≤ λ , there exists R = R ( n, λ ) so that if Σ n ⊂ R n +1 is aself-shrinker with entropy λ (Σ) ≤ λ which satisfies( † ) H ≥ B R ∩ Σ,then Σ is a generalized cylinder S k × R n − k for some 0 ≤ k ≤ n .It is important to emphasize that we do not assume any bound for the curvature | A | inthe above theorem. In this way Theorem 0.2 is analogous to Colding-Minicozzi’s removal ofthe curvature assumption in Huisken’s classification of mean convex shrinkers, and gives a N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 3 quantitative and stronger version of their result, for n ≤
6. This restriction on dimensioncomes from the curvature estimate for shrinkers with positive mean curvature. Namely, anyshrinker in R n +1 , n ≤
6, with
H > | A | ( x ) ≤ C (1 + | x | ) for some C dependingonly on the volume growth and n ; see Lemma 3.5 or Section 3 in [GZ15].If we assume a lower bound of the mean curvature, we obtain the following rigidity theoremthat holds in all dimensions. Theorem 0.3.
Given n , λ and δ >
0, there exists R = R ( n, λ , δ ) so that if Σ n ⊂ R n +1 isa self-shrinker with entropy λ (Σ) ≤ λ which satisfies( ‡ ) H ≥ δ on B R ∩ Σ,then Σ is a generalized cylinder S k × R n − k for some 1 ≤ k ≤ n .Finally, we give a rigidity theorem for graphical shrinkers in all dimensions. In [GZ15], theauthors showed that for n ≤
6, any shrinker in R n +1 which is graphical (in the sense that fora constant unit vector V , the normal part h V, n i >
0) inside a large, but compact, set mustbe a hyperplane. Again the restriction on dimension came from the curvature estimate forgraphical shrinkers (see Theorem 0.4 in [GZ15]). Here, if we assume that h V, n i has a positivelower bound, then in all dimensions we obtain the following theorem as a direct consequenceof the curvature estimate of Ecker-Huisken [EH91, Theorem 3.1] (see also Theorem 4.1) andsome ingredients from [GZ15]. Theorem 0.4.
Given n , λ and δ >
0, there exists R = R ( n, λ , δ ) so that if Σ n ⊂ R n +1 isa self-shrinker with entropy λ (Σ) ≤ λ satisfying • w = h V, n i ≥ δ on B R ∩ Σ for some constant unit vector V ,then Σ is a hyperplane.Let us now briefly outline the structure of this paper. In Section 1, we review some keydefinitions and notation. In Section 2, we prove our main curvature estimate Theorem 0.1.We discuss the proofs of Theorems 0.2 and 0.3 in Section 3, by adapting the iteration andimprovement scheme of Colding-Ilmanen-Minicozzi [CIM15]. We are also able to give shorterproofs using a compactness argument (see Remark 3.7), but we believe that the improvementmethod provides a more effective argument. In particular, the cylindrical estimates of Lemma3.8 may be of independent interest. Finally, in Section 4 we provide the proof of Theorem0.4 as well as some curvature estimates and a Bernstein-type theorem for translators of themean curvature flow. Acknowledgements.
The authors would like to thank Professor William Minicozzi for hisever helpful advice and encouragement. The second author is supported in part by theNational Science Foundation under grant DMS-1308244.1.
Notation and Background
Notation.
Let Σ n ⊂ R n +1 be a smooth hypersurface, ∆ its Laplace operator, A itssecond fundamental form and H =div Σ n its mean curvature. We denote by B R ( x ) the (closed) QIANG GUANG AND JONATHAN J. ZHU ball in R n +1 of radius R centered at x . For convenience we will introduce the shorter notation B R = B R (0).We begin by recalling the following classification of smooth, embedded mean convex self-shrinkers from [CM12a]. Theorem 1.1. ([CM12a]) S k × R n − k are the only smooth complete embedded self-shrinkerswithout boundary, with polynomial volume growth, and H ≥ R n +1 .We will also consider the operators L and L from [CM12a] defined by(1.1) L = ∆ − h x, ∇·i , (1.2) L = ∆ − h x, ∇·i + | A | + 12 . The next lemma records three useful identities from [CM12a].
Lemma 1.2. ([CM12a]) If Σ n ⊂ R n +1 is a smooth self-shrinker, then for any constant vector V ∈ R n +1 we have(1.3) LH = H, (1.4) L h V, n i = 12 h V, n i and(1.5) L| A | = | A | − | A | + 2 |∇ A | . Colding and Minicozzi [CM12a] introduced the entropy λ of a hypersurface Σ, defined as(1.6) λ (Σ) = sup x ,t F x ,t (Σ) = sup x ,t (4 πt ) − n Z Σ e − | x − x | t dµ, where the supremum is taking over all t > x ∈ R n +1 . It was proven in [CM12a]that for a self-shrinker, the entropy is achieved by the F -functional F , , so no supremum isneeded. Note that Cheng and Zhou [CZ13] (see also [DX13]) proved that for self-shrinkers,finite entropy, polynomial volume growth and properness are all equivalent.2. Curvature estimates for strictly mean convex shrinkers
This section is devoted to proving Theorem 0.1. The proof requires some modifications ofEcker-Huisken’s interior estimates for mean curvature flow [EH91] (see also [Eck04]), in whichthe authors derive curvature estimates using the maximum principle under the assumptionthat the flow is locally graphical. For our estimates, the mean convexity will replace thelocal graphical assumption - in particular, the key ingredient is the identity LH = H thatholds on all shrinkers. N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 5
First, in order to apply the maximum principle, we describe the choice of cutoff functionsand detail the relevant computations:Fix n and δ >
0. Let Σ n ⊂ R n +1 be a self-shrinker which satisfies( ⋆ ) H ≥ δ on B R ∩ Σ for
R > v = 1 /H and v = 1 /δ , then we have v ≤ v on B R ∩ Σ. Lemma 1.2 gives that LH = H .Hence, v satisfies the equation(2.1) ∆ v = 2 |∇ H | H − ∆ HH = 12 h x, ∇ v i + 2 |∇ v | v + (cid:16) | A | − (cid:17) v. We now fix the function(2.2) h ( y ) = y − ky , where k = (2 v ) − . Simple computations give that(2.3) h ′ ( y ) = 1(1 − ky ) and h ′′ ( y ) = 2 k (1 − ky ) . For convenience, in what follows we will abuse notation slightly and write h = h ( v ), h ′ = h ′ ( v ) and so on.Let f = | A | h . Then we have(2.4) ∆ f = h ∆ | A | + | A | ∆ h + 2 h∇| A | , ∇ h i . Note that(2.5) ∇ h = h ′ ∇ v = 2 h ′ v ∇ v, and ∆ h = h ′ ∆ v + h ′′ |∇ v | . Combining this with Lemma 1.2 gives that∆ f = h (cid:16) |∇ A | + (1 − | A | ) | A | + 12 h x, ∇| A | i (cid:17) + | A | (cid:16) h ′′ |∇ v | + h ′ ∆ v (cid:17) + 2 h∇| A | , ∇ h i . (2.6)Now we estimate the right hand side of the equation (2.6). First, we have(2.7) 2 h∇| A | , ∇ h i = h∇ h, ∇ f i h − | A | |∇ h | h + 4 h ′ | A | v h∇| A | , ∇ v i . Using the absorbing inequality gives that(2.8) 4 h ′ | A | v h∇| A | , ∇ v i ≤ h ′ ) | A | v |∇ v | h + 2 h |∇| A || . This implies(2.9) 2 h∇| A | , ∇ h i ≥ h∇ h, ∇ f i h − h ′ ) | A | v |∇ v | h − h |∇| A || . We also have that(2.10) h ′′ |∇ v | + h ′ ∆ v = 4 h ′′ v |∇ v | + h ′ h v h x, ∇ v i + 4 |∇ v | + (2 | A | − v + 2 |∇ v | i QIANG GUANG AND JONATHAN J. ZHU and(2.11) 12 h x, ∇ f i = h h x, ∇| A | i + | A | h x, ∇ h i = h h x, ∇| A | i + | A | h ′ v h x, ∇ v i . Therefore, we obtain that∆ f ≥ h∇ h, ∇ f i h + 12 h x, ∇ f i + (1 − | A | ) | A | h + 2 h ′ v | A | − h ′ v | A | + h h ′′ v + 6 (cid:16) h ′ − ( h ′ ) v h (cid:17)i | A | |∇ v | . (2.12)Now by the choice of h (compare (2.3)), we have(2.13) h − h ′ v = − kh , and(2.14) 4 h ′′ v + 6 (cid:16) h ′ − ( h ′ ) v h (cid:17) = 2 k (1 − kv ) h. Inserting these inequalities into (2.12) implies that(2.15) ∆ f ≥ h∇ h, ∇ f i h + 12 h x, ∇ f i − f + 2 kf + 2 k |∇ v | (1 − kv ) f. Here we used that h ′ v | A | ≤ f . We will set(2.16) a = ∇ hh and d = 2 k |∇ v | (1 − kv ) . Lemma 2.1.
Let x ∈ R n +1 and ρ >
0, and set φ ( x ) = ( µ ( x )) , where ( µ ( x )) + =max( µ ( x ) ,
0) and µ ( x ) = ρ − | x − x | . If Σ n is a shrinker, then on B ρ ( x ) ∩ Σ we have(2.17) ∆ φ = 24 µ | ( x − x ) T | − nµ + 6 µ H h x − x , n i . In particular, we have the estimate(2.18) | ∆ φ ( x ) | ≤ µρ + 6 nµ + 3 µ ρ | x | ≤ (24 + 6 n ) ρ + 3 ρ | x | . Proof.
Since ∇ φ = − ρ − | x − x | ) ∇| x − x | = − µ ( x − x ) T , we have∆ φ = − µ ( x − x ) T )= − h µ h∇ µ, ( x − x ) T i + µ (cid:16) n − h x − x , n i H (cid:17)i = 24 µ | ( x − x ) T | − nµ + 6 µ H h x − x , n i . (2.19)The second claim follows easily from the shrinker equation and the fact that µ ≤ ρ . (cid:3) We are now ready to prove our main curvature estimate.
N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 7
Proof of Theorem 0.1.
Now fix a point x ∈ B R − ∩ Σ and set ρ = R − | x | . Let φ be the function defined in Lemma 2.1. We will work on B ρ ( x ) ∩ Σ. Using (2.15) gives that∆( φf ) = φ ∆ f + f ∆ φ + 2 h∇ φ, ∇ f i≥ φ h h a + x , ∇ f i − f + 2 kf + df i + f ∆ φ + 2 h∇ φ, ∇ f i . (2.20)Note that(2.21) h a, ∇ ( φf ) i = φ h a, ∇ f i + f h a, ∇ φ i and(2.22) h∇ φ, ∇ ( f φ ) i = f |∇ φ | + φ h∇ φ, ∇ f i . This implies ∆( φf ) ≥ h a + x , ∇ ( φf ) i − h a + x , ∇ φ i f + φ h ( d − f + 2 kf i + f ∆ φ + 2 φ h∇ φ, ∇ ( f φ ) i − |∇ φ | φ f. (2.23)Now we set F ( x ) = φ ( x ) f ( x ) and consider its maximum on B ρ ( x ) ∩ Σ. Since F vanishes on ∂B ρ ( x ) ∩ Σ, F achieves its maximum at some point y ∈ B ρ ( x ) ∩ Σ. At the point y , wehave(2.24) ∇ F ( y ) = 0 and ∆ F ( y ) ≤ . In the following, we will work at the point y . By (2.23) and f ( y ) >
0, we have(2.25) h a + y , ∇ φ i + 2 |∇ φ | φ ≥ φ ( d −
1) + 2 kφf + ∆ φ. Note that(2.26) | a | = 4 (cid:16) h ′ h (cid:17) v |∇ v | = 2 kv d ≤ | y | k d. This yields that(2.27) h a, ∇ φ i ≤ ( d + 1) φ + | a | d + 1) |∇ φ | φ ≤ ( d + 1) φ + | y | k |∇ φ | φ . Combining (2.27) with (2.25) gives that(2.28) 2 kφf ≤ − ∆ φ + 2 φ + (cid:16) | y | k (cid:17) |∇ φ | φ + | y | |∇ φ | . By the definition of φ , we have(2.29) φ ≤ ρ , |∇ φ | ≤ ρ and |∇ φ | φ ≤ ρ . Combining this with Lemma 2.1, | y | ≤ R and (2.28) yields that(2.30) F ( y ) = φ ( y ) f ( y ) ≤ C ( ρ + Rρ + R ρ ) , QIANG GUANG AND JONATHAN J. ZHU where C is a constant depending on n and δ .Since F achieves its maximum at y , we have F ( x ) ≤ F ( y ). This implies(2.31) ρ | A | ( x ) H ( x ) − k = F ( x ) ≤ F ( y ) ≤ C ( ρ + Rρ + R ρ ) . In particular, we have(2.32) | A | ( x ) ≤ C (cid:16) Rρ (cid:17) H ( x ) . Since x is an arbitrary point in B R − ∩ Σ, this completes the proof of Theorem 0.1.3.
Rigidity theorems for mean convex shrinkers
In this section we prove Theorems 0.2 and 0.3 by adapting the iteration and improvementscheme used to prove [CIM15, Theorem 0.1]. For convenience of the reader, we briefly outlinethis scheme here; recall that the two key ingredients are the so-called iterative step [CIM15,Proposition 2.1] and the improvement step [CIM15, Proposition 2.2] (compare Proposition3.2 below). In the iterative step, it is shown that if a self-shrinker is almost cylindrical(quantified by H and | A | ) on a large scale, then it is still close to a cylinder on a larger scale,albeit with some loss in the estimates. It is important here that the scale extends by a fixedmultiplicative factor. Proposition 3.1. (Iteration; [CIM15, Proposition 2.1]) Given λ < n , there existpositive constants R , δ , C and θ so that if Σ n ⊂ R n +1 is a shrinker with λ (Σ) ≤ λ , R ≥ R , and • B R ∩ Σ is smooth with H ≥ / | A | ≤ B (1+ θ ) R ∩ Σ is smooth with H ≥ δ and | A | ≤ C .On the other hand, in the improvement step, it is shown that if a shrinker is close to acylinder on some scale, then the estimates can be improved so long as we decrease the scaleby a fixed amount. We will show that the initial closeness in the improvement step onlyneeds to be quantified by H — using our curvature estimate Theorem 0.1 and a compactnessresult of shrinkers, we can show that the bounded curvature assumption in the improvementstep (Proposition 2.2 of [CIM15]) can be removed, which in turn implies Theorem 0.2. Ourimprovement step is stated as follows: Proposition 3.2. (Improvement) Given n and λ , let δ ∈ (0 , /
4) be given by Proposition3.1. Then there exists R = R ( n, λ ) so that if Σ n ⊂ R n +1 is a shrinker with λ (Σ) ≤ λ and • H ≥ δ on B R ∩ Σ,then H ≥ / | A | ≤ B R − ∩ Σ.The main argument in the improvement step is to control the derivatives of the tensor τ = A/H . These estimates are shown to decay exponentially as R α e − R/ for some α , allowingone to extend good cylindrical estimates from a fixed scale 5 √ n to almost the whole ball N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 9 of radius R . For us, instead of assuming | A | ≤ C for some constant C as in [CIM15], ourcurvature estimates give that | A | ≤ CR for shrinkers with positive mean curvature H in B R .In the proof of Proposition 3.2, we show that this is still enough to control the derivativesof τ , possibly with a worse exponent α of R . But the exponential factor still decays muchfaster than any polynomial factor, so the polynomial factor can be eventually absorbed intothe exponential factor as long as we choose R sufficiently large. The remaining details of ourproof will be deferred to Section 3.2.To complete the iteration and improvement scheme, we first apply Proposition 3.2, thenapply Proposition 3.1 and repeat the process. The multiplicative factor extends the scaleby more than the fixed decrease if R is large enough, so we get strict mean convexity onall of Σ, which must therefore be a cylinder by the classification of mean convex shrinkers(Theorem 1.1). Thus we have: Proposition 3.3.
Given n and λ <
2, let δ ∈ (0 , /
4) be given by Proposition 3.1. Thenthere exists R = R ( n, λ ) so that if Σ n ⊂ R n +1 is a shrinker with entropy λ (Σ) ≤ λ whichsatisfies • H ≥ δ on B R ∩ Σ,then Σ is a generalized cylinder S k × R n − k for some 1 ≤ k ≤ n .We also need the following compactness theorem for self-shrinkers which plays an impor-tant role in our argument: Lemma 3.4 (Compactness) . Let Σ i ⊂ R n +1 be a sequence of shrinkers with λ (Σ i ) ≤ λ and(3.1) | A | ( x ) ≤ C (1 + | x | ) on B i ∩ Σ i . Then there exists a subsequence Σ ′ i that converges smoothly and with multiplicity one to acomplete embedded shrinker Σ with | A | ( x ) ≤ C (1 + | x | ) and lim i →∞ λ (Σ ′ i ) = λ (Σ) . (3.2) Proof.
The key is that the a priori bound on | A | is uniform on compact subsets. Thus, asin Lemma 2.7 in [CIM15], for any R we may obtain smooth convergence in B R by coveringwith a finite number of balls. Passing to a diagonal argument gives the overall smoothconvergence to a smooth, complete, embedded shrinker Σ with λ (Σ) ≤ λ . Again arguing asin [CIM15], if multiplicity is greater than one then the limit Σ must be L -stable. But thereare no such shrinkers with polynomial volume growth (see Theorem 0.5 in [CM12b]), so themultiplicity must be one. (cid:3) Now we are ready to prove Theorem 0.3.
Proof of Theorem 0.3.
Since we assumed H ≥ δ on B R ∩ Σ, the curvature estimate Theorem0.1 gives in particular that | A | ≤ CH ≤ C | x | on B R/ ∩ Σ. Applying the compactnessLemma 3.4 we get that Σ is smoothly close to S k × R n − k in B R/ . Thus for R sufficiently large we may assume λ (Σ) ≤ λ <
2, and H ≥ δ on B R/ ∩ Σ. The result then follows fromProposition 3.3. (cid:3)
Proof of Theorem 0.2.
For the proof of Theorem 0.2, we will need the followingcurvature estimate from Section 3 in [GZ15] (see in particular Theorem 0.4 and Remark 3.6therein). The key fact was that LH = H on any shrinker Σ, which implies an almost-stabilityinequality for Σ if the eigenfunction H is positive. Lemma 3.5.
Given n ≤ α >
0, there exists C = C ( n, α ) so that if Σ n ⊂ R n +1 is ashrinker with λ (Σ) ≤ α and H > B R ∩ Σ for some
R >
2, then on B R − ∩ Σ we have(3.3) | A | ≤ C (1 + | x | ) . Remark 3.6.
Note that for properly embedded self-shrinkers with finite genus in R , Song[Son14] (see also [Wan15]) gave the linear growth of the second fundamental form.Now we give the proof of Theorem 0.2 via Proposition 3.3. Proof of Theorem 0.2 using Proposition 3.3.
First, the Harnack inequality gives that either H ≡ H >
0. If H ≡ B R , then Σ is a hyperplane in B R . Thus by the rigidityof the hyperplane (for example, Theorem 0.1 in [GZ15] or Theorem 0.4, or even directly byBrakke’s theorem [Bra78]), Σ must be a hyperplane R n if R is sufficiently large.Next, we assume H > B R . Lemma 3.5 then gives a curvature estimate on B R − ∩ Σ. Bythe compactness of Lemma 3.4, we can assume that Σ is smoothly close to S k × R n − k in B R for some k ≥
0, where R can be taken as large as we wish. If k = 0, then again the rigidityof the hyperplane means that Σ must be a hyperplane, although this is a contradiction sincein this case we assume H > B R ∩ Σ. So k ≥
1, and consequently H is approximately p k/ B R ∩ Σ, then Theorem 0.2 follows directly from Proposition 3.3. (cid:3)
Remark 3.7.
In the above proofs of Theorems 0.2 and 0.3, the smooth closeness (obtainedvia compactness) also implies a bound for | A | on a large ball, so at that point we could alsoappeal directly to Theorem 0.1 in [CIM15]. The compactness Lemma 3.4 can also give ashorter proof of our main rigidity theorem for graphical shrinkers in [GZ15], but in both caseswe feel that the more effective proofs given may provide a more complete understanding.3.2. Proof of the improvement step.
In this subsection, we prove Proposition 3.2 bysketching the necessary modifications of the proof of Proposition 2.2 in [CIM15].As discussed earlier, the central argument is the very tight estimate on the tensor τ = A/H ,that decays exponentially in R . Thus, our main modification is the following lemma, whichremoves the curvature bound of Corollary 4.12 in [CIM15] by accepting a slightly largerpower of R , although we still have the exponential decay. Lemma 3.8.
Given n , λ and δ >
0, there exists a constant C τ > λ (Σ) ≤ λ , R ≥
2, and • B R +1 ∩ Σ is smooth with H ≥ δ > N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 11 then sup B R − ∩ Σ |∇ τ | + R − (cid:12)(cid:12) ∇ τ (cid:12)(cid:12) ≤ C τ R n +4 e − R/ . (3.4) Proof.
First, Theorem 0.1 gives there exists a constant C = C ( n, δ ) such that | A | ≤ CRH in B R . Hence, Proposition 4.8 in [CIM15] with s = 1 / Z B R − / ∩ Σ |∇ τ | e −| x | / ≤ C R n +4 e − ( R − / / . (3.5)Since e −| x | / ≥ e − R − R +14 on B R − , it follows that Z B R − ∩ Σ |∇ τ | ≤ C R n +4 e − R . (3.6)This gives the desired integral decay on ∇ τ . We will combine this with elliptic theory toget the pointwise bounds. The key is that τ satisfies the elliptic equation L H τ = 0 (seeProposition 4.5 in [CIM15]), that is,(3.7) ∆ τ − h x, ∇ τ i + h∇ log H , ∇ τ i = 0 . Note that we have(3.8) |∇ log H | = 2 |∇ H | H ≤ | A || x | H ≤ CR | x | , where we used that |∇ H | ≤ | A || x | and | A | ≤ CRH .Therefore, the two first order terms in the equation (3.7) come from x T in L and ∇ log H ;both grow at most quadratically. Now we can apply elliptic theory on balls of radius 1 /R to get for any p ∈ B R − ∩ Σ that (cid:0) |∇ τ | + R − |∇ τ | (cid:1) ( p ) ≤ C R n Z B R ( p ) ∩ Σ |∇ τ | . (3.9)Combining this with the integral bounds (3.6) gives the lemma. (cid:3) Now we sketch the proof of Proposition 3.2.Fix n , λ >
0, and δ >
0. Let
R > R n +1 , λ (Σ) ≤ λ and H ≥ δ on Σ ∩ B R . By Lemma 3.8, the tensor τ = A/H satisfies(3.10) |∇ τ | + (cid:12)(cid:12) ∇ τ (cid:12)(cid:12) ≤ ε τ on B R − ∩ Σ , where ε τ := C R n +8 e − R/ and the constant C depends only on n , δ and λ . As in [CIM15], the key point is that ǫ τ can still be made small for large R , due to the decaying exponential factor.Now fix small ε >
0, to be chosen as needed, but depending only on n . Combining thecompactness of Lemma 3.4 with the classification of mean convex shrinkers [CM12a], there exists a constant R = R ( n, λ , δ , ε ) so that if R ≥ R , then B √ n ∩ Σ is C ε -close to acylinder S k × R n − k for some 1 ≤ k ≤ n . The remainder of Proposition 3.2 follows from theproof of Proposition 2.2 in [CIM15].4. Bernstein type theorems
Rigidity of the hyperplane self-shrinker.
In this subsection, we will prove Theorem0.4 which gives a Bernstein type theorem for self-shrinkers in all dimensions. The key is thatthe positive lower bound of w = h V, n i enables us to obtain a curvature estimate in alldimensions. This is the content of the next theorem. Theorem 4.1.
Given n and δ >
0, there exists C = C ( n, δ ) so that for any smooth properlyembedded self-shrinker Σ n ⊂ R n +1 which satisfies • w = h V, n i ≥ δ on B R ∩ Σ for some constant unit vector V and R > | A | ≤ C, on B R/ ∩ Σ . Theorem 4.1 is essentially a corollary of Theorem 3.1 in [EH91], and the proof is similar toTheorem 0.1 — the essential component being that Lw = w (see Lemma 1.2). CombiningTheorem 4.1 and some ingredients from [GZ15], we can now prove Theorem 0.4. Proof of Theorem 0.4.
Given n , λ and δ , Theorem 4.1 gives a curvature bound C . Since Σis graphical and satisfies a curvature bound, Theorem 2.2 in [GZ15] allows us to make | A | assmall as we want by choosing R sufficiently large. In particular, we can choose R such that | A | ≤ / B R/ ∩ Σ. Now Theorem 0.4 follows directly from the compactness of Lemma3.4, Brakke’s Theorem [Bra78] (see also [Whi05]) and the fact that any complete shrinkerwith | A | < / (cid:3) Curvature estimates and a Bernstein type theorem for translators.
In thissubsection, we will sketch that the methods used in Section 2 can also be applied to prove acurvature estimate for translators with positive lower bound for the normal part of a constantvector field. As an immediate corollary, we obtain the Bernstein type theorem for translatorswhich was proved by Bao and Shi [BS14] by using different methods.Recall that a smooth hypersurface Σ n ⊂ R n +1 is called a translating soliton , or translator for short, if it satisfies the equation(4.2) H = −h y, n i , where y ∈ R n +1 is a constant vector. For simplicity, we may assume y = e n +1 , so thattranslators satisfy the equation(4.3) H = −h e n +1 , n i . The curvature estimate for translators is the following:
Theorem 4.2.
Given n and δ >
0, there exists C = C ( n, δ ) so that for any smooth properlyembedded translator Σ n ⊂ R n +1 which satisfies N THE RIGIDITY OF MEAN CONVEX SELF-SHRINKERS 13 • w = h V, n i ≥ δ on B R ( x ) ∩ Σ for some constant unit vector V and x ∈ R n +1 ,we have(4.4) | A | ( x ) ≤ C (cid:16) R + 1 R (cid:17) w ( x ) , for all x ∈ B R/ ( x ) ∩ Σ . Proof.
Since the proof is very similar to Theorem 0.1, we will only sketch the argument.The stability operator L for translators is defined by L = ∆ + h e n +1 , ∇·i + | A | . We havethe following identities (see for instance [IR14])(4.5) L h V, n i = 0 and L | A | = 2 |∇ A | − | A | . Set v = 1 /w , v = 1 /δ and f = | A | h . Similar computations and estimates as in the proofof Theorem 0.1 give that∆ f ≥ h∇ h, ∇ f i h − h e n +1 , ∇ f i − h | A | + 2 h ′ v | A | + h h ′′ v + 6 (cid:16) h ′ − ( h ′ ) v h (cid:17)i | A | |∇ v | . (4.6)Choosing h ( y ) = y − ky , where k = (2 v ) − . We then obtain that(4.7) ∆ f ≥ h∇ h, ∇ f i h − h e n +1 , ∇ f i + 2 kf + 2 k |∇ v | (1 − kv ) f. Let φ ( x ) = (( R − | x − x | ) + ) . We set F ( x ) = φ ( x ) f ( x ) and consider its maximum on B R ( x ) ∩ Σ. Assume F achieves its maximum at some point y ∈ B R ( x ) ∩ Σ.Using ∇ F ( y ) = 0, ∆ F ( y ) ≤ φ , we have(4.8) F ( y ) = φ ( y ) f ( y ) ≤ C ( R + R ) , where C is a constant depending on n and δ .Since F achieves its maximum at y , we have F ( x ) ≤ F ( y ) for all x ∈ B R/ ( x ) ∩ Σ. Thisimplies for any x ∈ B R/ ( x ) ∩ Σ(4.9) (cid:16) R (cid:17) | A | ( x ) w ( x ) − k ≤ F ( x ) ≤ F ( y ) ≤ C ( R + R ) . Now the theorem follows directly. (cid:3)
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
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