A Uniqueness Result for Self-expanders with Small Entropy
aa r X i v : . [ m a t h . DG ] S e p A UNIQUENESS RESULT FOR SELF-EXPANDERS WITH SMALL ENTROPY
JUNFU YAO
Abstract.
In this short note, we prove a uniqueness result for small entropy self-expanders as-ymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integerdegree argument proved by J. Bernstein and L. Wang. Introduction
A properly embedded n -dimensional submanifold Σ in R n +1 is called a self-expander if it satisfies(1.1) H Σ = x ⊥ H Σ is the mean curvature vector of Σ, and x ⊥ is the normal component of the position vector.Self-expanders are self similar solutions of mean curvature flow , that is, the family of hypersurfaces { Σ t } t> = (cid:8) √ t Σ (cid:9) t> satisfying (cid:16) ∂ x ∂t (cid:17) ⊥ = H Σ t Self-expanders are important as they model the behavior of a mean curvature flow coming out ofa conical singularity [1], and also model the long time behaviors of the flows starting from entiregraphs [10].For a hypersurface Σ in R n +1 , Colding-Minicozzi [8] introduced the entropy on Σ(1.2) λ [Σ] = sup y ∈ R n +1 ,ρ> F [ ρ Σ + y ]where F [Σ] is the Gaussian surface area of Σ F [Σ] = (4 π ) − n Z Σ e − | x | d H n ( x )Obviously, this quantity is invariant under dilations and translations. And Huisken’s monotonicityformula [12] shows that this quantity is non-increasing along the mean curvature flow.Next, we talk about the space ACH k,αn introduced by J. Bernstein and L. Wang [2]. A hyper-surface Σ ∈ ACH k,αn , if it is a C k,α properly embedded codimension-one submanifold and C k,α ∗ -asymptotic to a C k,α regular cone C = C (Σ). We refer to [2, Section 2] for technical details.As pointed out in [3] and [5], there may exist more than one self-expanders asymptotic to somespecific cones. While the main theorem here states that for a small entropy cone, there’s only one(stable) self-expander asymptotic to it. More precisely, Theorem 1.1.
There exists a constant δ = δ ( n ) , so that for a given C k,α -regular cone C with λ [ C ] < δ , there is a unique stable self-expander Σ ∈ ACH k,αn with C (Σ) = C .Remark . As pointed out in [7, Theorem 8.21], the outermost flows of any hypercone are self-expanders. Hence, it follows from our result that for a low entropy cone, the inner and outer flowscoincide, so the cone doesn’t fatten. Some regularity results
In [6], the authors defined a space
RMC n , consisting of all regular minimal cones in R n +1 andlet RMC ∗ n be the non-flat elements in RMC n . For any Λ >
1, let
RMC n (Λ) = {C ∈ RMC n : λ [ C ] < Λ } and RMC ∗ n (Λ) = RMC ∗ n ∩ RMC n (Λ)Since all regular minimal cones in R consist of the unions of rays and great circles are the onlygeodesics in S , RMC ∗ (Λ) = RMC ∗ (Λ) = ∅ for all Λ >
1. Now fix the dimension n ≥ >
1. Consider the following hypothesis:(2.1) For all 3 ≤ l ≤ n, RMC ∗ l (Λ) = ∅ Lemma 2.1.
There is a constant
Λ = Λ n > , so that the hypothesis (2 . holds.Proof. We first show that any regular minimal cone C has generalized mean curvature 0 near theorigin. Let X be a smooth vector field compactly supported in R n +1 . Define a smooth cut-offfunction η , so that η = 0 in B (0) and η = 1 outside B (0). Let η r = η ( · r ), then Z C η r (cid:0) div C X (cid:1) d H n = Z C div C (cid:0) η r X (cid:1) d H n − Z C X · ∇ C η r d H n = − Z C X · ∇ C η r d H n (2.2)The construction of η r gives Spt ∇ η r ⊂ B r (0) \ B r (0) and |∇ C η r | ≤ |∇ η r | ≤ Cr where C = C ( n ) is a constant. Let L ( C ) = C ∩ S n be the regular codimension-1 submanifold in S n ,then H n − ( L ( C )) < ∞ . This gives a upper bound for the last term in (2 . | Z C X · ∇ C η r d H n | ≤k X k ∞ | Z C∩ B r (0) \ B r (0) Cr | d H n ≤k X k ∞ · Cr H n − ( L ( C )) Z r r s n − ds ≤ C H n − L ( C ) k X k ∞ r n − So it goes to 0 as r →
0. Letting r → . Z C div C Xd H n = 0for any vector field compactly supported in R n +1 , which means C has generalized mean curvaturenear the origin and it vanishes.Next, we relate the entropy to the density at origin. Observe that1(4 π ) n Z C e − | x | d H n = 1(4 π ) n H n − ( L ( C )) Z ∞ r n − e − r dr = 12 π n H n − ( L ( C ))Γ( n H n − ( L ( C )) nω n = H n ( B ρ (0)) ω n ρ n = Θ( C , ω n = π n Γ( n +1) is the volume of the unit ball in R n . So the density Θ( C ,
0) = F [ C ] ≤ λ [ C ].Thus, by Allard’s regularity theorem [13], if Λ n is sufficiently small, then C is smooth at the origin,and it has to be a hyperplane. (cid:3) UNIQUENESS RESULT FOR SELF-EXPANDERS WITH SMALL ENTROPY 3
Remark . In fact, if we replace the regular cone above by a general stationary integral varifold C with η ,ρ C = C for all ρ >
0, the result still holds. Indeed, we get the smoothness near the originin the same way and the dilation invariance implies it is smooth everywhere.The following is a lemma from [6]. For the sake of completeness, we include a proof here.
Lemma 2.3.
For k ≥ and α ∈ (0 , , let C be a C k,α -regular cone in R n +1 and assume λ [ C ] < Λ n .If V is an E -stationary integral varifold with tangent cone at infinity equal to C , then V = V Σ foran element Σ ∈ ACH k,αn satisfying (1.1).Proof. For every point x ∈ Sptµ V , we will show that there is a tangent plane P x . If that is the case,Θ n ( V, x ) = Θ n ( P x ,
0) = 1. Together with the fact that V has locally bounded mean curvature, theAllard’s regularity theorem applies and V is smooth near x .Suppose a sequence of positive number { λ i } →
0, and η x,λ i V converges to a tangent varifold C x , where η x,λ i ( y ) = y − xλ i . By the nature of convergence and Huisken’s monotonicity formula [12], λ [ C x ] ≤ λ [ V ] ≤ λ [ C ] < Λ n On the other hand, ρ − n k δV k ( B ρ ( x )) ≤ ρ − n Z V ∩ B ρ ( x ) | H V | d H n ≤k H V k ∞ ρ · H n ( V ∩ B ρ ( x )) ρ n It goes to 0 as ρ → + . Then it follows from the knowledge of varifold in [13] that η ,ρ C x = C x for ρ >
0, and that C x is stationary. The previous lemma indicates C x must be a plane. Thus V = V Σ for some smooth self-expander Σ. Following [4], Σ is C k,α ∗ -asymptotic to C . (cid:3) We also need the notion of partial ordering. Roughly speaking, Σ (cid:22) Σ , if the hypersurface Σ is “above” Σ . For the detailed explanation, we refer to [6, Section 4]. The following theorem fromthat paper is a useful tool to construct self-expanders: Theorem 2.4.
For k ≥ and α ∈ (0 , , let C be a C k,α -regular cone in R n +1 and assume either ≤ n ≤ or λ [ C ] < Λ n . For any two Σ and Σ ∈ ACH k,αn , with C (Σ ) = C (Σ ) = C , there exist Σ ± stable self-expanders asymptotic to C with Σ − (cid:22) Σ i (cid:22) Σ + for i = 1 , . The Relationship Between Entropy and Stability
As oberved by S. Guo [11], if the entropy of the cone is sufficiently small, then we have thecurvature bound for the self-expander. More precisely,
Theorem 3.1.
Given κ > , there exists a constant ǫ > depending on n and κ with the followingproperty.If Σ is a self-expander that is asymptotic to a regular cone C with λ [ C ] < ǫ . Then we have k A Σ k L ∞ ≤ κ . Guo pointed out that when κ < √ , all self-expanders are stable. In the following lemma, weuse a Poincar´e type inequality from [3] to get a slightly strengthening by improving the bound to κ ≤ q n +12 . Lemma 3.2.
Let Σ be a self-expander in ACH k,αn satisfying | A Σ | ≤ n +12 . Then Σ is strictly stablein the sense that for all u ∈ C ∞ c (Σ) \{ } , (3.1) h−L Σ u, u i = Z Σ h |∇ Σ u | + ( 12 − | A Σ | ) u i e | x | d H n > JUNFU YAO
Proof.
Following [3, Appendix A], since (∆ Σ + x · ∇ Σ )( | x | + 2 n ) = | x | + 2 n , integrating by partsgives Z Σ (2 n + | x | ) u e | x | d H n = Z Σ h (∆ Σ + 12 x · ∇ Σ )( | x | + 2 n ) i u e | x | d H = − Z Σ ∇ Σ ( | x | + 2 n ) · ∇ Σ ( u ) e | x | d H n = − Z Σ u x ⊤ · ∇ Σ u e | x | d H n ≤ Z Σ ( | x ⊤ | u + 4 |∇ Σ u | ) e | x | d H n . (3.2)So moving | x ⊤ | u to the left hand side, Z Σ (2 n + | x ⊥ | ) u e | x | d H n ≤ Z Σ |∇ Σ u | e | x | d H n Together with | A Σ | ≤ n +12 , h−L Σ u, u i = Z Σ h |∇ Σ u | + ( 12 − | A Σ | ) u i e | x | d H n ≥ Z Σ ( |∇ Σ u | − n u ) e | x | d H n ≥ Z Σ | x ⊥ | u e | x | d H n ≥ u ∈ C ∞ c (Σ) \{ } satisfying h−L Σ u, u i = 0, then the inequlity (3.2) above should bean equality, which means(3.3) u x ⊤ = − ∇ Σ u, in { u > } Fix p ∈ { u > } , r = sup { s > B Σ s ( p ) ⊂ { u > }} and define v = log u in B Σ r ( p ). u beingcompactly supported implies r < ∞ is well-defined and that there is a p ∈ ∂B r ( p ) ∩ { u = 0 } .From (3.3) we know that ∇ Σ ( v + | x | ) = 0, which means v = − | x | + constant . However, thiscontradicts to the fact that v → −∞ as q → p . Hence, h−L Σ u, u i > u . (cid:3) Proof of Theorem 1.1
In this section we use the mountain-pass theorem proved by J. Bernstein and L. Wang to proveTheorem 1.1.
Proof of Theorem 1.1.
The existence result follows from [9, Theorem 6.3] and [4, Proposition 3.3].Hence, we only need to show the uniqueness. Letting κ = q n +12 , Theorem 3.1 ensures the existenceof an ǫ = ǫ ( n ) so that if λ [ C ] < ǫ , then | A Σ | ≤ κ for all self-expanders asymptotic to C . Thenby Lemma 3.2, all self-expanders asymptotic to C are strictly stable. That is, C is a regular value in the sense of [3].Let δ = min { ǫ ( n ) , Λ n − } . As an application of [3, Theorem 1.3], Π − ( C ) is a finite set, whereΠ assigns each element in ACH k,αn to the trace at infinity.Now, let us argue by contradiction. Suppose there were two self-expanders Σ and Σ bothasymptotic to C . Following Theorem 2.4, we can produce two distinct self-expanders Σ ± withΣ − (cid:22) Σ i (cid:22) Σ + for i = 1 ,
2. Applying [5, Theorem 1.1], there is a self-expander Σ = Σ ± withpossibly codimension-7 singular set and Σ − (cid:22) Σ (cid:22) Σ + . Notice that Huisken’s monotonicityformula tells us λ [Σ ] ≤ λ [ C ] < δ ≤ Λ n . Thus, by Lemma 2.3, Σ is actually smooth. Now,replace Σ ± by Σ and Σ − and iterate the preceding argument. So we produce as many self-expandersas we can. And this contradicts the fact that Π − ( C ) is finite. (cid:3) Acknowledgements
The author would like to thank Professor Jacob Bernstein for his patient and generous help.
UNIQUENESS RESULT FOR SELF-EXPANDERS WITH SMALL ENTROPY 5
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