The level set flow of a hypersurface in R 4 of low entropy does not disconnect
aa r X i v : . [ m a t h . DG ] J a n THE LEVEL SET FLOW OF A HYPERSURFACE IN R OF LOW ENTROPYDOES NOT DISCONNECT
JACOB BERNSTEIN AND SHENGWEN WANGA
BSTRACT . We show that if Σ ⊂ R is a closed, connected hypersurface with entropy λ (Σ) ≤ λ ( S × R ) , then the level set flow of Σ never disconnects. We also obtain a sharpversion of the forward clearing out lemma for non-fattening flows in R of low entropy.
1. I
NTRODUCTION
A family of hypersurfaces Σ t ⊂ R n +1 evolves by mean curvature flow (MCF) if itsatisfies(1.1) (cid:18) ∂∂t x Σ t (cid:19) ⊥ = H Σ t here a hypersurface is a smooth submanifold of codimension one and x Σ t is the positionvector, H Σ t is the mean curvature vector and ⊥ is the projection onto the normal of Σ t .A fundamental property of MCF is that the flow of a closed hypersurface must develop asingularity in finite time. If one considers the level set flow (see Chen-Giga-Goto [5] andEvans-Spruck [6–9]), then one obtains a canonical set theoretic weak mean curvature flowthat persists through singularities and, for closed initial data, vanishes in finite time. Bydefinition, as long as the flow is smooth, then the topology does not change, however thisneed not be the case for the level set flow after the first singularity.When n = 1 , it follows from Gage-Hamilton [10] and Grayson [11] that the flowdisappears when it becomes singular. In particular, the flow remains connected until itdisappears. In contrast, when n > , non-degenerate neck-pinch examples show that thereare flows that become singular without disappearing. In these examples, the level set flowdisconnects after the neck-pinch singularity. In [2], the first author and L. Wang showedthat, when n = 2 and the entropy of the initial surface is small enough (see (2.1) below),then the flow also disappears at its first singularity. This result makes use of a classificationof singularity models in R of low entropy from [2] and whether such a classification existsin higher dimension is unknown. In the present note we show that when n = 3 and theinitial hypersurface is closed, connected and of low entropy, then even if the flow formsa singularity before it disappears, its level set flow remains connected until its extinctiontime. Theorem 1.1.
Let Σ ⊂ R be a closed, connected hypersurface and let { Γ t } t ∈ [0 ,T ] be thelevel set flow with initial condition Γ = Σ and extinction time T . If λ (Σ) ≤ λ ( S × R ) ,then, for all t ∈ [0 , T ] , Γ t is connected. Moreover, if W [ t ] = R \ Γ t , then W [ t ] has atmost two connected components for all t ∈ [0 , T ] . Mathematics Subject Classification.
A technical feature of the level set flow is that it may “fatten”, i.e., develop non-emptyinterior. If this occurs in Theorem 1.1, then there will be a T ∈ [0 , T ) so that W [ t ] hastwo components for t ∈ [0 , T ) and one component for t ∈ [ T , T ] .In [17], the second author showed that, for mean curvature flows of low entropy, ifthe flow reaches the point x at time t , then, the flow remains near x after t until itdisappears. This is a forward in time analog of the standard, unconditional, clearing outlemma – e.g., [8, Theorem 3.1] – that says that if the flow reaches x at time t , thenthe flow must be near x at earlier times. Theorem 1.1 allows us to sharpen the resultfrom [17] and prove the forward clearing out lemma in R with the optimal upper boundon the entropy. Corollary 1.2.
There exist uniform constants
C > and η > , so that if { M t } t ∈ [0 ,T ] is aa non-fattening level set flow in R that starts from a smooth closed hypersurface M ⊂ R with λ ( M ) < λ ( S × R ) , x ∈ M t and M t + R = ∅ , then for all ρ ∈ (0 , R C ) , H ( B ρ ( x ) ∩ M t + C ρ ) ≥ ηρ . Here H denotes the 3-dimensional Hausdorff measures.Remark . The entropy assumption can be seen to be sharp by considering the translatingbowl soliton in R and, in the closed setting, by considering a sequence of unit spheres atincreasing distance from one another and joined by a thin tube.2. N OTATION AND B ACKGOUND
Let B R ( x ) be the open ball in R n +1 centered at x and, for a set K ⊂ R n +1 , let T r ( K ) = [ x ∈ K B r ( x ) be the r -tubular neighborhood of K . For any ρ > , x ∈ R n +1 and subset Ω ⊂ R n +1 , set Ω + x = { x + x ∈ R n +1 : x ∈ Ω } and ρ Ω = { ρx : x ∈ Ω } . Following [4], the entropy of a closed hypersurface, Σ , is defined by(2.1) λ (Σ) = sup ( y ,ρ ) ∈ R n +1 × R F ( ρ Σ + y ) where F is the Gaussian area of Σ given by(2.2) F (Σ) = (4 π ) − n Z Σ e − | x | d H n . The entropy and Gaussian area readily extend to the less regular objects studied in geomet-ric measure theory. Clearly, λ ( R n ) = 1 . If S n is the unit n -sphere in R n +1 , then Λ k = λ ( S k ) = λ ( S k × R n − k ) = F ( √ k S k ) and so, by a computation of Stone [16],(2.3) > Λ > > Λ > . . . > Λ n > . . . → √ . Let us now briefly recall some background results in the theory of (weak) mean curva-ture flow – our primary sources are [6–9] and [13]. We begin with the level set flow, whosemathematical theory was developed by Chen-Giga-Goto [5] and Evans-Spruck [6–9].Let Γ be a non-empty compact subset of R n +1 . Select a Lipschitz function u so that Γ = { x : u ( x ) = 0 } and so that u ( x ) = − C when | x | ≥ R for some constants HE LEVEL SET FLOW OF A LOW ENTROPY HYPERSURFACE DOES NOT DISCONNECT 3
C, R > . For such a u , { u ≥ a > − C } is compact. In [6], Evans-Spruck establishedthe existence and uniqueness of viscosity solutions to the initial value problem:(2.4) ( u t = Σ n +1 i,j =1 ( δ ij − u x i u x j | Du | − ) u x i x j on R n +1 × (0 , ∞ ) u = u on R n +1 × { } . Setting Γ t = { x : u ( x, t ) = 0 } , define { Γ t } t ≥ to be the level set flow of Γ = Γ . Asshown in [6], the Γ t depend only on Γ and are independent of the choice of u . The level setflow has a uniqueness property and satisfies an avoidance principle. As such, for any closedinitial set, the level set flow vanishes after a finite amount of time. Furthermore, as long asthe initial set is a closed hypersurface, the level set flow agrees with the classical solutionto (1.1) as long as the latter exists. A technical feature the level set flow is that some timeslices may develop non-trivial interior – a phenomena called “fattening”. Importantly,initial sets are generically non-fattening – see for instance [13, Theorem 11.3]In addition to the level set flow, we will also need to consider the measure theoreticversion of MCF introduced by Brakke. An n -dimensional Brakke flow (or Brakke motion ), K , in R n +1 is a family of Radon measures K = { µ t } t ∈ I , that satisfies (1.1) in the senseof being a negative gradient flow, see [13] for the precise definition. The Brakke flow is integral if for almost every t ∈ I , µ t ∈ IM n ( R n +1 ) , that is, µ t is an integer n -rectifiableRadon measure. The Hausdorff n -measure, H n restricted to any classical solution of (1.1)is an integral Brakke flow.Denote the parabolic rescaling and translation of a Brakke flow K = { µ t } by D ρ K = n µ ρ, ρ − t o and K − ( x , t ) = n µ ,x t + t o where µ ρ,x ( A ) = ρ n µ ( ρ − A + x ) . It follows from the Brakke’s compactness theorem [13, 7.1] and the Huisken monotonicityformula [12, 14] that given an integral Brakke flow K = { µ t } t ∈ I with uniformly boundarea ratios, for any t > inf I and x ∈ R n +1 and any sequence ρ i → ∞ there existsa subsequence ρ i j → ∞ so that D ρ ij ( K − ( x , t )) converges (in the sense of Brakkeflows – see [13]) to a Brakke flow T = { ν t } t ∈ R . We call such a flow a tangent flow to K at ( x , t ) and denote the set of all possible limits (for different sequences of scalings)by Tan ( x ,t ) K . By Huisken’s monotonicity formula, T ∈
Tan ( x ,t ) K is backwardlyself-similar. If ν − = H n ¬ Υ for a smooth hypersurface Υ , then Υ satisfies the equation(2.5) H Υ + x ⊥ Any hypersurface, Υ , that satisfies (2.5) is called a self-shrinker and is asymptoticallyconical if lim ρ → ρ Υ = C in C ∞ loc ( R n +1 \ { } ) for some regular cone C . For instance,any hyperplane through the origin is an asymptotically conical self-shrinker.A feature of Brakke flows is that they may suddenly vanish. In order to handle technicalissues that arise from this possibility we will need Ilmanen’s enhanced motions [13, 8.1][18]. Following the formulation in [18], a pair ( τ, K ) is an enhanced motion , if τ ∈ I locn +1 ( R n +1 × R ) is a locally ( n + 1) -dimensional integral current in space-time and K = { µ t } t ∈ R is a Brakke flow that together satisfy(1) ∂τ = 0 and ∂ ( τ t ≥ s ) = τ s and τ t ∈ I n ( R n +1 ) for each time slice t (2) ∂τ t = 0 for all t (3) t τ t is continuous in the flat topology(4) µ τ t ≤ µ t for all t JACOB BERNSTEIN AND SHENGWEN WANG (5) V µ t = V τ t + 2 W t for some integral varifold W t for a.e. t . In other words, they arecompatible for a.e. t as defined in [18].Here τ is the called the undercurrent and K is the overflow . Likewise ( τ, K ) is an enhancedmotion with initial condition τ ∈ I n ( R n +1 × { t } ) if the above holds for all t ≥ t and ∂τ = τ . An enhanced motion ( τ, K ) is a matching motion if µ τ t = µ t for a.e. t for whichthis makes sense.Associated to each E ⊂ R n +1 × R of locally finite perimeter, there is a unique ( n +2) -dimensional integral current [ E ] ∈ I locn +2 ( R n +1 × R ) . Similarly, given an orientedcodimension- k submanifold Σ ⊂ R n +1 × R there is a unique [Σ] ∈ I lock ( R n +1 × R ) . If ∂ ∗ E is the reduced boundary of E , then [ ∂ ∗ E ] = ∂ [ E ] ∈ I locn +1 ( R n +1 × R ) . As such,there is an integer ( n + 1) -rectifiable Radon measure H n ¬ ∂ ∗ E – see [13] for details. Weextend the notion of canonical boundary motion from [1] – see also [3, 13]. These flowsare special cases of flows introduced by Ilmanen in [13] that synthesis the level set flowand Brakke flow in a natural way and are key to our approach. Definition 2.1. A canonical boundary motion is a triple ( E , E, K ) consisting of an openbounded set E ⊂ R n +1 × { } with ∂E a smooth closed hypersurface, an open boundedset E ⊂ R n +1 × [0 , ∞ ) of finite perimeter and a Brakke flow K = { µ t } t ≥ so:(1) E = { ( x, t ) : u ( x, t ) > } , where u solves equation (2.4) with u chosen so E = { x : u ( x ) > } and ∂E = { x : u ( x ) = 0 } ;(2) The level set flow of ∂E is non-fattening;(3) For t ≥ , each E t = { x : ( x, t ) ∈ E } is of finite perimeter and µ t = H n ¬ ∂ ∗ E t .If, in addition,(4) { u = 0 } = ∂ ∗ E in R n +1 × (0 , ∞ ) ,where u is from Item (1), then ( E , E, K ) is a strong canonical boundary motion . Remark . Observe, { u > } = E ⊂ ¯ E ⊂ { u ≥ } for a canonical boundary motionand ¯ E = { u ≥ } for a strong canonical boundary motion. If Γ t = { x ∈ R n +1 | u ( x, t ) =0 } , then { Γ t } t ≥ is the level set flow of Γ = Σ and is non-fattening. Clearly, ∂E t ⊂ Γ t ,but equality need not hold – even for strong canonical boundary motions.By [13, 11.4], for a E with the property that the level set flow of ∂E is non-fattening,there are E and K so ( E , E, K ) is a canonical boundary motion. In general, the non-fattening condition is not enough to ensure the existence of a strong canonical boundarymotion, however, in [13, 12.11], Ilmanen shows such existence for “generic” E .Finally, we introduce the following notation for a level set flow { Γ t } t ≥ in R n +1 , n ≥ , W [ t ] = R n +1 \ Γ t W [ s, r ] = { ( x, t ) | x ∈ ( R n +1 \ Γ t ) , s ≤ t ≤ r } = [ t ∈ [ s,r ] W [ t ] n ( t ) = { connected components of W [ t ] } ∈ N ∪ {∞} . As Γ t is compact and n ≥ , there is exactly one unbounded component of W [ t ] , denotedby W − [ t ] . Let W + [ t ] = W [ t ] \ W − [ t ] be the bounded components and set W ± [ s, r ] = [ t ∈ [ s,r ] W ± [ t ] . HE LEVEL SET FLOW OF A LOW ENTROPY HYPERSURFACE DOES NOT DISCONNECT 5
3. P
ROOF OF T HEOREM
FOR STRONG CANONICAL BOUNDARY MOTIONS
In this section we show Theorem 1.1 for flows that are strong canonical boundary mo-tions. We begin with several preliminary results. The first is an elementary topologicalresult – we include a proof for the sake of completeness.
Lemma 3.1.
Let Γ ⊂ R n +1 be a compact set. If R n +1 \ Γ has exactly two components, W ± , and Γ = ∂W ± , then Γ is connected.Proof. Suppose that Γ is not connected. Let K be one component of Γ and K ′ = Γ \ K = ∅ . Observe that both K and K ′ are compact and so there is a r > so that T r ( K ) ∩ T r ( K ′ ) = ∅ and, hence, T r (Γ) is not connected. Let ˆ W ± = W ± ∪ T r (Γ) . Clearly, ˆ W ± are open sets with ˆ W + ∩ ˆ W − = T r (Γ) . For each x ∈ Γ , W ± ∩ B r ( x ) = ∅ as Γ = ∂W ± .As the union of intersecting connected sets is connected, W ± ∪ B r ( x ) is connected. Itreadily follows that both ˆ W − and ˆ W + are connected. Finally, by the Mayer-Vietoris longexact sequence for reduced homology, as R n +1 = ˆ W + ∪ ˆ W − is simply connected andboth ˆ W ± are connected, T r (Γ) = ˆ W + ∩ ˆ W − must be connected. This contradicts ourchoice of r and proves the lemma. (cid:3) Another elementary fact is that the level set flow remains connected up to and includingits first disconnection time.
Lemma 3.2.
Let { Γ t } t ∈ [0 ,T ] be a level set flow of compact sets in R n +1 . If Γ t is connectedfor t ∈ [0 , t ) , then Γ t is connected.Proof. By the definition and basic properties of level set flow lim t → t − Γ t = Γ t in Haus-dorff distance. On the one hand, by the avoidance principle, Γ t ⊂ T √ n ( t − t ) (Γ t ) . On the other, as the space-time track of the level set flow, R n +1 × [0 , T ] \ W [0 , T ] , is closedand Γ t is compact, for every ǫ > , there is a δ > so that if < t − t < δ , then Γ t ⊂ T ǫ (Γ t ) . Hence, if Γ t is disconnected, then for t < t close enough to t , Γ t isdisconnected, proving the claim. (cid:3) The next result summarizes and extends of [3] and provides a description of the regu-larity properties of strong canonical boundary motions flows in R of low entropy. Proposition 3.3.
Let (cid:16) E , E, K = { µ t } t ≥ (cid:17) be a strong canonical boundary motion in R . Suppose the flow has extinction time T and Σ = ∂E satisfies λ (Σ ) < Λ .(1) For each t ∈ [0 , T ) , there are a finite, possibly empty, set of points x , . . . , x m ( t ) ∈ R so that µ t = H ¬ Σ t where Σ t is a hypersurface in R \ { x , . . . , x m } .(2) For an open dense subset I ⊂ [0 , T ] , if t ∈ I , then µ t = H ¬ Σ t where Σ t is aclosed hypersurface.(3) Let ( x , t ) ∈ R × (0 , T ] be a point at which K has positive Gaussian den-sity, if { ν t } t ∈ R = T ∈
Tan ( x ,t ) K , then ν − = H ¬ Υ where Υ is a smoothself-shrinker and either Υ is closed or it is asymptotically conical. Moreover,whichever holds depends only on ( x , t ) and not on the choice of tangent flow.(4) For each ( x , t ) ∈ R × (0 , T ] for which Tan ( x ,t ) K contains an asymptoticallyconical shrinker, there is an R = R ( x , t , ∂E ) > so that for all R ∈ (0 , R ]Σ t ( x , R ) = spt( µ t ) ∩ B ∗ R ( x ) = Σ t ∩ B ∗ R ( x ) = ∂E t ∩ B ∗ R ( x ) = ∂ ∗ E t ∩ B ∗ R ( x ) , JACOB BERNSTEIN AND SHENGWEN WANG is a connected hypersurface that divides B ∗ R ( x ) into two components, one con-tained in E t and one disjoint from it. Here B ∗ R ( x ) = B R ( x ) \ { x } .Proof. Note first that as ( E , E, K ) is a strong canonical boundary motion, ( E, K ) is acanonical boundary motion in the sense of [3] – see Theorem 2.3 and the discussion at thebeginning of Section 4 of [3]. As such, Items (1) and (2) are both immediate consequencesof [3, Theorem 4.3] – see [3, Corollary 4.4] and the proof of [3, Theorem 4.5] for details.Item (3) follows from [3, Proposition 4.1 and Lemma 4.2].It remains to show Item (4). First, set ǫ = Λ − λ ( ∂E ) > . Next observe that if ( x , t ) is a singular point of K , then, by hypothesis it is a non-compact singularity and soby [3, Theorem 4.2(2)], there is a α = α ( ǫ ) > and a ρ = ρ ( x , t ) > so that for all ( ρ, t ) ∈ (0 , ρ ) × ( t − ρ , t + ρ ) , A t ( x , t , ρ ) = Σ t ∩ (cid:16) B αρ ( x ) \ ¯ B αρ ( x ) (cid:17) = spt( µ t ) ∩ (cid:16) B αρ ( x ) \ ¯ B αρ ( x ) (cid:17) is a connected non-empty hypersurface that is proper in B αρ ( x ) \ ¯ B αρ ( x ) . The sameis true if ( x , t ) is not a singular point as then Tan ( x ,t ) K consists of a static hyperplane.For ρ ∈ (0 , ρ ) , let A ( x , t , ρ ) = [ t ∈ ( t − ρ ,t + ρ ) A t ( x , t , ρ ) × { t } this is a connected non-empty hypersurface that is proper in the hollow space-time cylinder C ( x , t , ρ ) = (cid:16) B αρ ( x ) \ ¯ B αρ ( x ) (cid:17) × ( t − ρ , t + ρ ) . Clearly, A t ( x t , t , ρ ) = A ( x , t , ρ ) ∩ { x = t } and this intersection is transverse.By Item (3) of the definition of canonical boundary motion, spt( µ t ) = ∂ ∗ E t , and so A ( x , t , ρ ) = ∂ ∗ E ∩ C ( x , t , ρ ) . As A ( x , t , ρ ) is smooth, every point is in the reduced boundary and so A ( x , t , ρ ) = ∂ ∗ E ∩ C ( x , t , ρ ) . Hence, by Item (4) of the definition of a strong canonical boundary motion, A ( x , t , ρ ) = ∂ ∗ E ∩ C ( x , t , ρ ) = ∂ ∗ E ∩ C ( x , t , ρ ) = ∂E ∩ C ( x , t , ρ ) . Together with the fact that that A ( x , t , ρ ) meets { x = t } transversally, this means A t ( x , t , ρ ) = ∂ ∗ E t ∩ (cid:16) B αρ ( x ) \ ¯ B αρ ( x ) (cid:17) = ∂E t ∩ (cid:16) B αρ ( x ) \ ¯ B αρ ( x ) (cid:17) . Set R = 2 αρ and, for any R ∈ (0 , R ) , let Σ t ( x , R ) = ∞ [ i =0 A t ( x , t , − R ) . By the above, Σ t ( x , R ) is a connected non-empty hypersurface proper in B ∗ R ( x ) and,moreover, Σ t ( x , R ) = ∂ ∗ E t ∩ B ∗ R ( x ) = ∂E t ∩ B ∗ R ( x ) = spt( µ t ) ∩ B ∗ R ( x ) = Σ t ∩ B ∗ R ( x ) . Finally, as Σ t ( x , R ) is connected, non-empty and proper in B ∗ R ( x ) , B ∗ R ( x ) \ Σ t ( x , R ) has two components. On the one hand, Σ t ( x , R ) ⊂ ∂E t implies at least one of these is asubset of E t . On the other, Σ t ( x , R ) ⊂ ∂ ∗ E t means the other is disjoint from E t . (cid:3) HE LEVEL SET FLOW OF A LOW ENTROPY HYPERSURFACE DOES NOT DISCONNECT 7
Next we use the above regularity properties to strengthen the relationships betweenthe level set flow and its interior for strong canonical boundary motions of low entropy –compare with Remark 2.2.
Proposition 3.4.
Let ( E , E, K = { µ t } t ≥ ) be a strong canonical boundary motion in R with λ ( ∂E ) < Λ and let { Γ t } t ∈ [0 ,T ] be the level set flow with Γ = Σ . If there is a t ∈ (0 , T ] , so for all ( x, t ) ∈ R × (0 , t ] , Tan ( x,t ) K is either trivial or consists of onlyasymptotically conical tangent flows, then for all s ∈ [0 , t ] , Γ s = spt( µ s ) = ∂E s = ∂ ( R \ ¯ E s ) . If, in addition, Γ s is connected, then E s = W + [ s ] and Γ s = ∂W ± [ s ] . Proof.
As the level set flow is the biggest flow, spt( µ t ) ⊂ Γ t – see [13, 10.7]. Pick a s ∈ (0 , t ] and a x ∈ Γ s . Let T ∈
Tan ( x ,s ) K be a tangent flow to K at the point ( x , s ) . By Item (4) of the definition of strong canonical boundary motion, ( x , s ) ∈ ∂ ∗ E . Hence, there is a sequence ( x i , s i ) ∈ ∂ ∗ E with s i > and lim i →∞ ( x i , s i ) =( x , s ) . As ( x i , s i ) ∈ ∂ ∗ E , the Gaussian density of K at ( x i , s i ) is at least and so, by theupper semicontinuity property of Gaussian density, the Gaussian density of K at ( x , s ) ispositive and so T is non-trivial. Hence, by Item (3) of Proposition 3.3 and the hypothesis, T = { ν t } t ∈ R is asymptotically conical.Thus, Item (4) of Proposition 3.3 implies that there is a R > so for all R ∈ (0 , R ) , spt( µ s ) ∩ B ∗ R ( x ) is non-trivial. As spt( µ s ) is closed, this means that x ∈ spt( µ s ) andhence, spt( µ t ) = Γ t for all t ∈ (0 , t ] proving the first equality. To see the second equality,first note that, by definition, ∂E s ⊂ Γ s . Now suppose that x ∈ Γ s . By what we havealready shown we know that x ∈ spt( µ s ) and Item (4) of Proposition 3.3 holds at ( x , s ) .Hence, there is a R > so for all R ∈ (0 , R ) , spt( µ s ) ∩ B ∗ R ( x ) = ∂E s ∩ B ∗ R ( x ) and this intersection is non-empty. As the topological boundary of a set is closed, x ∈ ∂E s and so Γ s = ∂E s , proving the second equality. As the other component given by Item (4)of Proposition 3.3 is disjoint from ¯ E s , the same argument proves the third equality.To complete the proof, first observe that, by definition, E s ⊂ W + [ s ] and ∂E s ⊂ ∂W + [ s ] ⊂ Γ s . As ∂E s = Γ s this immediately implies Γ s = ∂W + [ s ] . Similarly, bydefinition ∂W − [ s ] ⊂ Γ s , and, for any x ∈ ∂W − [ s ] . Hence, Item (4) of Proposition3.3 implies that there is an R > so that B ∗ R ( x ) ∩ Γ s divides B ∗ R ( x ) into exactly twocomponents, U ± ( x ) , with ∂U ± ( x ) ∩ B ∗ R ( x ) = Γ s ∩ B ∗ R ( x ) and so that, up to relabel-ing, U + ( x ) ⊂ E s and U − ( x ) ∩ E s = ∅ . As x ∈ ∂W − [ s ] and W − [ s ] ∩ E s = ∅ , U − ( x ) ⊂ W − [ s ] and so ∂W − [ s ] ∩ B ∗ R ( x ) = Γ s ∩ B ∗ R ( x ) . Hence, as x ∈ ∂W − [ s ] ⊂ Γ s , B R ( x ) ∩ Γ s ⊂ ∂W − [ s ] and so ∂W − [ s ] is an open non-empty subset of Γ s . As Γ s isassumed to be connected, this means Γ s = ∂W − [ s ] . Finally, let Ω = W + [ s ] \ E s . As ∂E s = Γ s = ∂W + [ s ] , ∂ Ω ⊂ Γ s . For each x ∈ Γ s , Item (4) of Proposition 3.3, impliesthat, for R sufficiently small, B R ( x ) \ Γ s consists of two components one disjoint from E s and one contained in E s . As B R ( x ) ∩ W − [ s ] = ∅ the component disjoint from E s iscontained in W − [ s ] and so is disjoint from Ω . Likewise, the component contained in E s is disjoint from Ω by construction. Hence, Ω ∩ B R ( x ) = ∅ and so x ∂ Ω . As x wasarbitrary, this means ∂ Ω = ∅ which implies Ω = ∅ . That is, E s = W + [ s ] . (cid:3) We use the preceding results and ideas from [19] to show that strong canonical boundarymotions remain connected until they disappear. That is, we show Theorem 1.1 for strongcanonical boundary motions.
JACOB BERNSTEIN AND SHENGWEN WANG
Proposition 3.5.
Let ( E , E, K = { µ t } t ≥ ) be a strong canonical boundary motion in R with E connected and λ [ ∂E ] < Λ . If { Γ t } t ∈ [0 ,T ] is the level set flow with Γ = ∂E and extinction time T , then Γ t is connected and n ( t ) = 2 for all t ∈ [0 , T ) .Proof. As E is connected and bounded and ∂E = Σ is compact, W + [0] = E . As Σ is a hypersurface, there is a δ > so that Γ t is a smooth flow for t ∈ [0 , δ ] and so Γ t isconnected, n ( t ) = 2 and W [ t ] = E t for t ∈ [0 , δ ] . Let t dis = sup { t ∈ (0 , T ) | n ( s ) = 2 and Γ s is connected for all ≤ s < t } be the first possible disconnection time. Clearly, t dis > δ and if t dis = T , then we aredone. In what follows we suppose t dis < T and derive a contradiction.First observe that, by construction, t dis must be a singular time, but not the extinctiontime of the flow. As such, for any ( x, t ) ∈ R × (0 , t dis ] , for which K has positive Gaussiandensity all tangent flows to K at ( x, t ) are asymptotically conical. Indeed, by Proposition3.3, if a tangent flow at ( x, t ) was closed, then, as Γ t was connected for t < t < t dis , for t < t and t close enough to t , spt( µ t ) would also be a closed connected hypersurface.This would imply that the whole flow becomes extinct at t , contradicting the fact that t dis < T is not the extinction time.By Lemma 3.2 and the definition of t dis , Γ t is connected for all t ∈ [0 , t dis ] . Hence, byProposition 3.4, for all t ∈ [0 , t dis ] , Γ t = spt( µ t ) = ∂W ± [ t ] . We conclude that n ( t dis ) =2 . Indeed, if n ( t dis ) ≥ , then as W − [ t ] is connected, there is a component, Ω , of W + [ t dis ] so Ω ′ = W + [ t dis ] \ Ω is non-empty. As E t dis = W + [ t dis ] = Ω ∪ Ω ′ , Ω ∩ Ω ′ = ∅ and Ω , Ω ′ are both open, Γ t dis = ∂E t dis = ∂ Ω ∪ ∂ Ω ′ . Hence, as Γ t dis is connected, there is an x ∈ ∂ Ω ∩ ∂ Ω ′ . By Item (4) of Proposition 3.3, there is an R > so that B ∗ R ( x ) ∩ E t dis hasexactly one non-empty component, namely, B ∗ R ( x ) ∩ Ω = B ∗ R ( x ) ∩ Ω ′ . This contradicts Ω ∩ Ω ′ = ∅ and implies n ( t ids ) = 2 .Next observe that there is a δ > so that there are no compact singularities in thetime interval [ t dis , t dis + δ ] . Indeed, by Item (3) of Proposition 3.3, singularities arecompact if and only if they are collapsed. Furthermore, by [1, Proposition 4.10] the limitof collapsed singularities is also a collapsed singularity. Hence, if there is no such δ , then,by Proposition 3.3, the flow would have a compact singularity at t = t dis and this hasalready been ruled out.For each t ∈ [0 , T ] , let C [ t ] be the set of components of W [ t ] . By [19, Theorem 5.2], forany ≤ t < s ≤ T , there is a well-defined map π s,t : C [ s ] → C [ t ] given by π s,t (Ω s ) = Ω t if and only if there is a time-like continuous path in W [ t, s ] , connecting a point in Ω s × { s } to a point in Ω t × { t } . Using Proposition 3.4 and the fact that there are no compactsingularities in [ t dis , t dis + δ ] , it is clear that for all t dis ≤ t < s ≤ t dis + δ the map π s,t is surjective. Hence, n ( t ) is a non-decreasing function on [ t dis , t dis + δ ] . By Item(2) of Proposition 3.3 and Proposition 3.4, there is a s ∈ ( t dis , t dis + δ ) so that Γ s is asmooth closed hypersurface and so n ( s ) < ∞ . Hence, setting k = inf { n ( t ) | t ∈ ( t dis , s ) } ,the definition of t dis and the monotonicity of n ( t ) implies that ≤ k < ∞ and there is a δ ∈ (0 , δ ) so that n ( t ) = k for t ∈ ( t dis , t dis + δ ) . For any t dis < t < s < t dis + δ ,the fact that π s,t is surjective and n ( s ) = n ( t ) = k is finite implies that π s,t is a bijection.We claim that k > . Indeed, for any t ∈ ( t dis , t dis + δ ) , if n ( t ) = 2 , then W + [ t ] = E t .This is because there always exactly one unbounded component, W − [ t ] , whereas E t isalways a bounded component of W + [ t ] . By Proposition 3.4, as there are no compactsingularities in [0 , t dis + δ ] , Γ t = ∂W ± [ t ] and so Lemma 3.1 implies Γ t is connected.Hence, if k = 2 , then not only is n ( t ) = 2 in ( t dis , t dis + δ ) , but Γ t is connected. Thiscontradicts the definition of t dis and so we must have k > . HE LEVEL SET FLOW OF A LOW ENTROPY HYPERSURFACE DOES NOT DISCONNECT 9
Now choose any t ′ ∈ [ t dis , t dis + δ ) . As n ( t ′ ) = k > n ( t dis ) , the pigeonholeprinciple implies that there must be two points x , x from different components, Ω ′ , Ω ′ of W [ t ′ ] so that ( x , t ′ ) , ( x , t ′ ) are each connected via time-like paths in W [ t dis , t ′ ] to thesame component of W [ t dis ] × { t dis } . Label the two paths, p ( s ) , p ( s ) , so that p (1) =( x , t ′ ) , p (1) = ( x , t ′ ) . As p (0) , p (0) are in the same component of W [ t dis ] × { t dis } ,there is a path p in W [ t dis ] so that ( p (0) , t dis ) = p (0) , ( p (1) , t dis ) = p (0) . By theavoidance principle, there is a universal constant C > so that if B r ( y ) ∩ Γ t dis = ∅ ,then ( y, t ) ⊂ W [ t ] for any t ∈ [ t dis , t dis + Cr ] . As p ([0 , is compact, we can choose < r < dist( p [0 , , Γ t dis ) . Hence, the avoidance principle gives p ([0 , × [ t dis , t dis + Cr ] ⊂ W [ t dis , t dis + Cr ] As such, if δ = min n t ′ − t dis , Cr , δ o , then for any t ∈ ( t dis , t dis + δ ) , ( x , t ′ ) , ( x , t ′ ) can also be connected via time-like paths in W [ t, t ′ ] to the same components of W [ t ] . Thatis, π t ′ ,t (Ω ′ ) = π t ′ ,t (Ω ′ ) which contradicts the previously established fact that π t ′ ,t is abijection for such t, t ′ and so proves the proposition. (cid:3)
4. P
ROOF OF T HEOREM
Theorem 4.1.
Let Σ be a smooth closed connected hypersurface in R with λ [Σ] ≤ Λ . If { Γ t } t ∈ [0 ,T ] is the level set flow with Γ = Σ and extinction time T , then, for all t ∈ [0 , T ] , Γ t is connected and n ( t ) ≤ . Moreover, if E + = W + [0 , T ] and E − = W − [0 , T ] ∪ (cid:0) R × ( T, ∞ ) (cid:1) , then E ± are both sets of locally finite perimeter in R × [0 , ∞ ) and there are Brakke flows K ± so that ( τ ± = ± (cid:0) ∂ [ E ± ] + [ W ± [0] × { } ] (cid:1) , K ± ) are both matching motions with initial condition [Σ × { } ] . Finally, ∂ ∗ E ± = ∂E ± in R × (0 , ∞ ) .Proof. First observe that we may assume λ (Σ) < Λ . Indeed, suppose that λ (Σ) = Λ andconsider, { Σ t } t ∈ [0 ,δ ] , the classical solution to (1.1) with Σ = Σ . As Σ is closed, λ (Σ) = F [ ρ − (Σ − x )] for some ρ > and x ∈ R n +1 . Hence, by the Huisken monotonicityformula, either λ [Σ δ ] < Λ or Σ = ρ Υ + x where Υ is a closed self-shrinker. In the lattercase, the theorem is immediate (as the flow will remain smooth until disappearing), whilein the former, one can prove the result for Σ δ and then use the fact that the flow was smoothto conclude it also for Σ .As Σ is a closed connected hypersurface in R , standard topological results, e.g., [15],imply that there is a connected bounded domain E ⊂ R with ∂E = Σ . Let n be theunit normal to Σ that points into E . As Σ is smooth, there is an ǫ > so for | s | < ǫ Σ s = { p + s n ( p ) | p ∈ Σ } is a foliation of T ǫ (Σ) by hypersurfaces . By shrinking ǫ , if needed, we can also ensurethat λ (Σ s ) < Λ for | s | < ǫ . Pick a Lipschitz function u : R → R with the property that(1) { u = s } = Σ s for | s | < ǫ ,(2) { u ≤ − ǫ } is the unbounded component of R \ T ǫ (Σ) ; and (3) { u ≥ ǫ } is the bounded component of R \ T ǫ (Σ) .Let u be the solution to 2.4 with initial data u . As such, if Γ st = { x | u ( t, x ) = s } , then for | s | < ǫ , { Γ st } t ≥ is the level set flow with Γ s = Σ s . For each i ≥ , pick s ± i ∈ ( − ǫ, ǫ ) so that s − i < s − i − < < s i +1 < s i and lim i →±∞ s i = 0 . Let E i = { u > s i } and E i = { u > s i } . By [13, 12.11], one can choose the s i so that for i = 0 , there are Brakkeflows K i so that (cid:0) E i , E i , K i (cid:1) are all strong canonical boundary motion.By Proposition 3.5, each Γ it = Γ s i t = { u = s i } is connected and for t ∈ [0 , T i ) , where T i is the extinction time of the flow, divides R into two components W ± i [ t ] which satisfy Γ it = ∂W ± i [ t ] and W + i [ t ] = E it = { x | u ( t, x ) > s i } . Consider the open sets U + [ t ] = ∞ [ i =1 W + i [ t ] = { x | u ( x, t ) > } and U − [ t ] = ∞ [ i =1 W −− i [ t ] = { x | u ( x, t ) < } . As each W ± [ t ] is connected and U ± [ t ] is their nested union, it follows that both the U ± [ t ] are also connected. Moreover, as Γ t = { x | u ( x, t ) = 0 } = R \ (cid:0) U + [ t ] ∪ U − [ t ] (cid:1) ,W ± [ t ] = U ± [ t ] . For i ≥ let, G i [ t ] = R \ (cid:0) W + i [ t ] ∪ W −− i [ t ] (cid:1) = { x | s − i ≤ u ( x, t ) ≤ s i } and observe that each G i [ t ] is a compact set, G i +1 [ t ] ⊂ G i [ t ] and T ∞ i =1 G i [ t ] = Γ t .For t ∈ [0 , T ] , each G i [ t ] is connected. Indeed, T − i , the extinction time of (cid:8) Γ − it (cid:9) t ≥ must satisfy T − i > T and so, when t ≤ T , Γ − it and W ±− i [ t ] are both non-empty andconnected. In particular, there is exactly one component, G − i [ t ] , of G i [ t ] that contains Γ − it = ∂W ±− i [ t ] . Let G + i [ t ] = G i [ t ] \ G − i [ t ] , so G + i [ t ] is closed and disjoint from G i [ t ] .Observe that W −− i [ t ] ∪ G − i [ t ] is a closed non-empty subset of W − i [ t ] = W − i [ t ] ∩ Γ it = { u ≤ s i } that is disjoint from G + i [ t ] . As G + i [ t ] is also a closed subset of W − i [ t ] , W − i [ t ] = W −− i [ t ] ∪ G − i [ t ] ∪ G + i [ t ] and the closure of a connected set is connected, G + i [ t ] = ∅ , andso G i [ t ] is connected. As the nested intersection of compact connected sets is connected,it follows that Γ t is connected and so we’ve proved the first part of the theorem.To prove the second part of the theorem we begin observe that for i ≥ , E i = W [0 , T ] is a set of finite perimeter while F − i = { u < s − i } = R × [0 , ∞ ) \ ¯ E − i , is a set of locally finite perimeter. Moreover, there are matching motions (cid:0) τ i = ∂ [ E i ] + W + i [0]] , K i (cid:1) and (cid:0) τ − i = − (cid:0) ∂ [ F − i ] + [ W −− i [0]] (cid:1) , K − i (cid:1) with initial conditions [Σ s ± i × { } ] . As λ (Σ s ± i ) < Λ < , [17, Theorem 3.4] impliesthat, up to passing to a subsequence, the two sequences of matching motions convergeto matching motions ( τ + , K + ) and ( τ − , K − ) both with initial condition [Σ × { } ] . Itfurther follows from standard compactness results for sets of locally finite perimeter, that E i converges as a set of finite perimeter to E + = W + [0 , T ] = [ t ∈ [0 ,T ] U + [ t ] = { u > } HE LEVEL SET FLOW OF A LOW ENTROPY HYPERSURFACE DOES NOT DISCONNECT 11 which is also a set of finite perimeter, while F − i converges as a set of locally finite perime-ter to F − . One readily verifies that F − = W − [0 , T ] ∪ (cid:0) R × ( T, ∞ ) (cid:1) = [ t ∈ [0 ,T ] U − [ t ] ∪ (cid:0) R × ( T, ∞ ) (cid:1) = { u < } . Set E − = F − and observe that τ ± = ± ( ∂ [ E ± ] + [ W ± [0]]) follows from the continuityof the boundary operation.It remains only to verify the claim about the reduced boundary. To that end observe thatin R × (0 , ∞ ) ∂ ∗ E + ⊂ ∂E + . We now suppose that ( x, t ) ∈ ∂E + and t > . By definition, for any r > , B r ( x, t ) ∩ E + = ∅ . In particular, for i sufficiently large B r ( x, t ) ∩ W + i [0 , T ] = ∅ . As x ∈ Γ t ,we have x W + i [0 , T ] and so there is some point ( y r , t r ) ∈ B r ( x, t ) ∩ ∂W + i [0 , T ] . As (cid:0) E i , E i , K i (cid:1) is a strong canonical boundary motion, it has only one compact singularity (atthe terminal time T i < T ) and we can assume t r < T i . Hence, by Proposition 3.4 that y r ∈ spt( µ it r ) and so ( y r , t r ) has positive Gaussian density for K i . As K i converges to K + , theupper semicontinuity of Gaussian density implies that ( y, t ) is a point of positive Gaussiandensity for K + . As ( τ + , K + ) is a matching motion starting from Σ and τ + is the reducedboundary of a set of finite perimeter, ( y, t ) ∈ ∂ ∗ E + . That is, ∂ ∗ E = ∂E + in R × (0 , ∞ ) .Arguing in exactly the same way shows that ∂ ∗ E − = ∂E − in R × (0 , ∞ ) (cid:3) Corollary 4.2.
Let Σ be a smooth closed connected hypersurface in R with λ [Σ] ≤ Λ .If { Γ t } t ∈ [0 ,T ] , the level set flow of Σ with extinction time T , is non-fattening, then there isa unique strong canonical boundary motion ( E , E, K ) , with ∂E = Σ .
5. F
ORWARD CLEARING OUT
In this section apply Theorem 1.1 to prove Corollary 1.2.
Proof of Corollary 1.2.
If the Corollary is not true, then there exist C i → , η i > , R i > , < ρ i < R i C i satisfying η i C i → and a sequence of non-fattening level set flows { M i,t } t ≥ with M i, , closed hypersurfaces with λ ( M i, ) < Λ , M i,t = ∅ for t ∈ ( t , t + R i ) and so that the flows reach the space-time point ( x , t ) , but satisfy H ( B ρ i ( x ) ∩ M t + C i ρ i ) < η i ρ i . By Corollary 4.2, the M i,t agree with the slices of a strong canonical boundary motion ( E i, , E i , K i = { µ i,t } ) . In particular, by Proposition 3.4, µ i,t = H ¬ M i,t and so µ i,t ( B ρ i ( x )) < η i ρ i .Rescale the flows to get a new flow ˜ K i = D Ciρi ( K i − ( x , t )) and let { ˜ M i,t } be thecorresponding rescaling of the level set flow { M t } . By Brakke’s compactness theorem [13,7.1], up to passing to a subsequence, ˜ K i converges to a limit flow ˜ K = { ˜ µ t } , and moreover,by [17, Theorem 3.5], ( T i , K i ) converge to a matching motion ( ˜ T , ˜ K ) .We also have by rescaling ˜ µ i, (cid:16) B Ci (0) (cid:17) < η i ( C i ) → That is, ˜ µ ( R ) = 0 and so the limit flow ˜ K must be extinct before t = 1 . As ( ˜ T , ˜ K ) is a matching motion, this means that ˜ K must develop a collapsed singularity at some t e ≤ . By the classification of singularities given in Proposition 3.3, this singularityhas compact support. Hence, by Brakke’s regularity theorem, for large enough i , the flow { ˜ M i,t } must develop a compact singularity at some time ˜ t i < , and hence { M i,t } mustdevelop compact singularity at some time t i < t + 2 C i ρ i < t + R i < t + R i . Since M i,t + R i = ∅ and there is a compact singularity before the extinction time, there must bedisconnection before time t + R i , contradicting Theorem 1.1. (cid:3) R EFERENCES[1] Jacob Bernstein and Lu Wang. A sharp lower bound for the entropy of closed hypersurfaces up to dimensionsix.
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EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
ALTIMORE , MD 21218, USA
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
ALTIMORE , MD 21218, USA
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