6. Bibliografía
2.1. Envase y embalaje
2.2.5. Usos y aplicaciones del poliestireno expan-
Now that have developed a rigorous surgery procedure, we will show that we can perform surgery on a hypersurface undergoing MCF before a singular time so that the mean curvature stays bounded, unless we otherwise recognise the surface to be a sphere or Sn−1 ×
S. Since we will be performing surgery on an ( ˆα,δ, εˆ 0, L0)-neck we need to be sure the surface develops a neck in regions of high curvature. Huisken and Sinestrari achieve this by introducing a Neck Detection Lemma (Lemma 7.4 of [30]). Before we can discuss this in more detail we introduce some new definitions. Consider a smooth MCF defined on the time time interval [0, t0]. Let g(t) be the metric on M induced by the immersion Xt,
and for a point p∈ M and r >0 let the closed ball of radius r around p (with respect to the metricg(t)) be denotedBg(t)(p, r). Then suppose thatt, θ are such that 0 ≤ t−θ < t ≤ t0. Now we define (the same way Perelman did in [38]) a (backward) parabolic neighbourhood of (p, t) as follows
P(p, t, r, θ) ={(q, s) :q∈Bg(t)(p, r), s∈[t−θ, t]}. (3.5) Now if ti are surgery times, then extend the definition to MCF with surgery, that
is for a family of flows Xi : M
i×[ti−1, ti] → Rn+1. From now on to avoid any
confusion we will denote t+i to distinguish the hypersurface before surgery with the one obtained after the surgery occurs at timeti. Then Huisken and Sinestrari
identify that when considering parabolic neighbourhoods in the case of MCF with surgery, thatP(p, t, r, θ) may not be well defined if surgery occurs between times
t−θ and t, and so they provide the following definition:
Definition 3.13. LetXi be a family of mean curvature flows with surgery, and
let (p, t)∈Mi×[ti−1, ti] for somei >1,θ∈(0, t] andr >0. Then we say that the
ball Bg(t)(p, r) has not been changed by surgeries in the interval [t−θ, t] if there are no points of Bg(t)(p, r) which belong to a region changed by a surgery that occurred at time s∈]t−θ, t]. In this case we define the parabolic neighbourhood as we did in the smooth case and say thatP(p, t, r, θ) does not contain surgeries. Finally to make computation and analysis of necks simpler Huisken and Sines- trari introduce the following notation
ˆ
r(p, t) := n−1
H(p, t), ˆ
3.5. NECK DETECTION 69 For the case of n = 2 we require slightly different parameters due to a different gradient estimate, however the proof of both the neck detection and neck contin- uation follows the same ideas. For this reason we have decided to detail then= 2 case as the n ≥3 case is dealt with extensively in the original paper of Huisken and Sinestrari [30]. Then for reference we will state the two Neck Detection Lem- mas proved by Brendle and Huisken in [8]. The proofs are detailed in section 13 of [8] in full so we refer the reader there for a greater discussion. The key difference between the two lemmas is that version A requires that ˆP(p0, t0, L0+ 4,2θ0) does not contain surgeries, whereas version B allows for a choice of θ much small than
θ0 where ˆP(p0, t0, L0+ 4, θ) is surgery free.
Theorem 3.9 ((Neck Detection Lemma, Version A)). Let δ and δˆbe chosen as above, and let ε¯and Λ be chosen so that the conclusion of Theorem 3.5 holds. Let Mt be a mean curvature flow with surgery satisfying the surgery assumptions,
where ε ≤ ε¯ and L ≥ 1000Λ. Then, given ε0 > 0 and L0 ≥ 100, we can find
η0 > 0 and K0 with the following significance: Suppose that t0 and p0 ∈ Mt0
satisfy:
• H(p0, t0)≥max{K0, H1/Θ}, κ1(p0, t0)/H(p0, t0)≤η0,
• the parabolic neighbourhood Pˆ(p0, t0, L0+ 4,2θ0)does not contain surgeries.
Then (p0, t0) lies at the centre of an ( ˆα,δ, εˆ 0, L0)-neck of size H(p0, t0)−1.
Finally, the constants η0 and K0 may depend on ε0, L0, δ,ˆδ, and the initial data,
but they are independent of the remaining surgery parameters ε, L,and H1. .
Theorem 3.10 ((Neck Detection Lemma, Version B)). Let δ and δˆ be chosen as above, and let ε¯andΛ be chosen so that the conclusion of Theorem 3.5 holds. Let Mt be a mean curvature flow with surgery satisfying the surgery assumptions,
where ε ≤ ε¯and L ≥ 1000Λ. Then, given θ, ε0 > 0 and L0 ≥ 100, we can find
positive numbers η0 and K0 with the following significance: Suppose that t0 and
p0 ∈Mt0 satisfy
• H(p0, t0)≥max{K0, H1/Θ}, κ1(p0, t0)/H(p0, t0)≤η0,
• the parabolic neighbourhood Pˆ(p0, t0, L0+ 4, θ) does not contain surgeries.
Let us dilate the surface {x∈Mt0 :distg(t0)(p0, x)≤L0H(p0, t0)
−1} by the factor
70 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY
C3-norm. Here, Γ is a closed, convex curve satisfying L(Γ) ≤3π and sup Γ|κ− 1| ≤ 1/100 . The constant K0 may depend on θ, ε0, L0, δ,δˆ, and the initial data,
but they are independent of the remaining surgery parameters ε, L, and H1. Now that we have a way of detecting the formation of necks, we need a method for continuing the flow to a point whereby either the neck ‘closes up’ to a convex cap or we can force the curvature below a threshold value so as to make surgery feasible. First though we recount the following useful propositions which are also from [8].
Proposition 3.3 ((Replacement for Lemma 7.12 in [30])). Let Mt be a mean
curvature flow with surgery satisfying the surgery assumptions. Suppose that (p1, t1) is a point in space-time such thatH(p1, t1)≥H1 and the parabolic neigh- bourhood ˆP(p1, t1,L˜+ 4,2θ0) contains at least one point belonging to a surgery region. Then there exists a point q1 ∈ Mt1 and an open set V ⊂Mt1 such that
distg(t1)(p1, q1)≤ ( ˜L+ 4)H(p1, t1)
−1,{x∈ M
t1 : distg(t1)(q1, x)≤500H
−1
1 } ⊂ V , and V is diffeomorphic to a disk. Moreover, the mean curvature is at most 40H1 at each point in V
Proposition 3.4 (Replacement for Lemma 7.19 in [30]). Let Σ be an embedded surface in R3 which is α-noncollapsed, and let y
1 < y2 be two real numbers. We assume that the surface Σ is contained in the cylinder {(x1, x2, x3) ∈ R3 :
x2
1 +x22 ≤ 100, y1 ≤ x3 ≤ y2}. Moreover, we assume that ∂Σ = Γ1 ∪Γ2, where Γ1 ⊂ {x∈ R3 :x3 =y1} and Γ2 ⊂ {x∈R3 :x3 =y2}. Then we have H(x)≥ Θ4 for all pointsx∈Σ satisfyingx3 ∈[y1+ 1, y2] andhν(x), e3i ≥0. Here,ν denotes the outward-pointing unit normal to Σ and Θ = 400/α.
Proposition 3.5. Suppose thatMt is a mean curvature flow with surgeries sat-
isfying the surgery assumptions , whereε ≤ε¯andL≥1000Λ. Moreover, suppose that (p0, t0) satisfiesH(p0, t0 ≥1000H1 and κ1(p0, t0)≤η0H(p0, t0),whereη0 and
H1 are defined as above. Then p0 lies at the centre of an ( ˆα,δ, εˆ 0, L0)-neck. Before we finally present the proof of the neck continuation theorem from [8] we finalise our choice of surgery parameters. In order for us to employ the neck detection lemmas in the proof of the neck continuation theorem we choose
ε0 <ε¯, L0 >1000Λ and we suppose that the mean curvature on an ( ˆα,ˆδ, ε0, L0)- neck varies by factor of 1 +L−01. Then the only surgery parameters left to set are the curvature parameters H1, H2, H3 as well as a parameter η1 that will be required for the neck continuation theorem. We choose η1 < η0, where η0 is
3.5. NECK DETECTION 71 chosen in the Neck Detection Lemma version A. Finally given K1 > K0, where
K0 is from the Neck Detection Lemma Version B we setH1 >1000ΘK1 and then define
H2 := 1000γ0H1 H3 := 10H2
where γ0 is such that theorem 3.12 holds. For a full discussion of these surgery parameters we refer the reader to page 9 and 10 of [8]. While these ensure lower bounds for our curvature parameters, we can take Hi to be arbitrarily large and
hence for large enough curvature parameters we will be able to implement surgery for MCF.
Theorem 3.11 (Neck continuation theorem). Suppose Mt is a hypersurface un-
dergoing mean curvature with a finite number of surgeries and satisfies all our surgery assumptions, where ε ≤ ε¯. Suppose also that (p0, t0) satisfies H1 ≤
H(p0, t0)/1000andκ1(p0, t0)≤η1H(p0, t0), whereη1 andH1 are defined as above.
Then there exists a finite collection of pointsp1, ....pl with the following properties:
• For each i = 0,1, ..., l the point pi lies at the centre of an ( ˆα,δ, εˆ 0, L0)−
neck N(i)⊂M
t0, and H(pi, t0)≥H1.
• For each i= 0,1, ..., l the point pi+1 lies on the neck N(i), and we have
distg(t0)(pi+1, ∂N
(i)\N(i−1))∈[(L
0−100)H(p0, t0)−1,(L0−50)H(p0, t0)−1]. • Finally, at least one of the following holds: either the union N =∪l
i=1Ni
covers the entire surface; or H(pl, t0) ∈ [H1,2H1]; or there exists a closed
curve in N ∩ {x ∈ Mt0 : H(x, t0) ≤ 40H1} which is homotopically non-
trivial in N and bounds a disk D in {x ∈Mt0 :H(x, t0) ≤40H1}; or the
outer boundary ∂N(k)\N(k−1) bounds a convex cap.
We present the proof as given in [8] as this is they key theorem that will allow us to prove the existence of Mean Curvature Flow with surgery. We remark also that this proof follows the same arguments as presented in [30], however the proof of Lemma 7.19 no longer holds for the case when n = 2. The reason is that the gradient estimate in [30] works on all scales, whereas the gradient estimate in Proposition 3.2 becomes weaker when the curvature is much smaller than H1.
We will use Proposition 3.4 to overcome this problem.
Proof. We first construct points that satisfy the first 2 properties, then we must check the third property case by case. To see that this is possible we use Propo- sition 3.5 to conclude that the point p0 lies at the centre of an ( ˆα,δ, εˆ 0, L0)-neck
72 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY
N(0) ⊂M
t0. Then the construction of the points p1, p2, ... is by induction. Sup-
pose that we have constructed points p1, ..., pk and necks N(1), ..., N(k) such that
they satisfy the following properties:
• For eachi= 0,1, ..., k, the pointpi lies at the centre of an ( ˆα,δ, εˆ 0, L0)-neck
N(i) ⊂M
t0, and we have H(pi, t0)≥H1.
• For each i= 1, ..., k−1, the pointpi+1 lies on the neck N(i), and we have distg(t0)(pi+1, ∂N
(i)\N(i−1))∈[(L
0−100)H(pi, t0)−1,(L0−50)H(pi, t0)−1]. IfH(pk, t0)∈[H1,2H1] then one of the cases in the third property holds and hence we are done. Thus, for rest of the proof, we have the following assumption
H(pk, t0)≥2H1. We can now break the proof into multiple cases:
Case 1: Suppose that the there exists a point p∈N(k) such that distg(t0)(p, ∂N
(k)\N(k−1))∈[(L
0−100)H(pk, t0)−1,(L0−50)H(pk, t0)−1], and the parabolic neighbourhood ˆP(p, t0, L0+4,2θ0) contains a point modified by surgery. In this case, Proposition 3.3 implies that there exists a point q ∈ Mt0 and an
open set V ⊂ {x∈Mt0 :H(x, t0)≤40H1} such that
distg(t0)(p, q)≤(L0 + 4)H(p, t0) −1 , {x∈Mt0 : distg(t0)(q, x)≤500H −1 1 } ⊂V,
and V is diffeomorphic to a disk. By our choice of ε0 and L0, we can choose ε0 sufficiently small such that the mean curvature on the neck N(k) varies at most by a factor 1 +L−01. Hence, H(pk, t0)≤(1 +L−01)H(p, t0). Since the set
{x ∈ Mt0 : distg(t0)(p, x) ≤ (L0 − 100)H(pk, t0) −1} is contained in N(k), we conclude that distg(t0)(q, N (k))≤(L 0+ 4)H(p, t0)−1−(L0−100)H(pk, t0)−1 ≤(L0+ 4)(1 +L−01)H(pk, t0) −1− (L0−100)H(pk, t0)−1 ≤200H(pk, t0)−1 ≤100H1−1.
Consequently, there exists a closed curve which is contained in N(k)∩V and is homotopically non-trivial in N(k). Since V is diffeomorphic to a disk, this curve
3.5. NECK DETECTION 73 bounds a disk in V , and we are done.
Case 2: We now assume that the parabolic neighbourhood ˆP(p, t0, L0+4,2θ0) is free of surgeries for all points p ∈ N(k) satisfying dist
g(t0)(p, ∂N
(k)\N(k−1)) ∈ [(L0−100)H(p)k, t0)−1,(L0−50)H(pk, t0)−1]. There are two possibilities now:
Subcase 2.1: Suppose that there exists a point p ∈ N(k) with the property that
distg(t0)(p, ∂N
(k)\N(k−1))∈[(L
0−100)H(pk, t0)−1,(L0−50)H(pk, t0)−1 and
κ1(p, t0) ≤ η0H(p, t0). By Version A of the Neck Detection Lemma, the point
p lies at the centre of an ( ˆα,δ, εˆ 0, L0)-neck N . Moreover, since p ∈ N(k) and
H(pk, t0) ≥ 2H1, we have H(p, t0) ≥ H1. Hence, we can put p(k+1) := p and
N(k+1) :=N and continue the process.
Subcase 2.2: Suppose that κ1(p, t0)> η0H(p, t0) for all points p∈N(k) satis- fying distg(t0)(p, ∂N (k)\N(k−1))∈[(L 0−100)H(pk, t0)−1,(L0−50)H(pk, t0)−1. LetN = Pk i=0N
(i), and letAbe the set of all pointsx∈N satisfying dist
g(t0)(p, ∂N
(k)\N(k−1))≥ (L0−50)H(pk, t0)−1andκ1(x, t0)≤η1H(x, t0). The assumptions of Theorem 3.11 imply that the initial point p0 belongs to A, soA is non-empty. Let us consider a point p∗ which has maximal intrinsic distance from p0 among all points in A.
Subcase 2.2.1: Suppose that the parabolic neighbourhood ˆP(p∗, t0,104,2θ0) contains a point modified by surgery. In this case, Proposition 3.3 that there exists a point q ∈ Mt0 and an open set V ⊂ {x ∈ Mt0 : H(x, t0)≤ 40H1} such
that distg(t0)(p ∗, q)≤104H(p∗, t 0)−1, {x∈Mt0 : distg(t0)(q, x)≤500H −1 1 } ⊂V,
and V is diffeomorphic to a disk. Since H(p∗, t0)≥H1/2, this implies {x∈Mt0 : distg(t0)(p ∗ , x)≤100H(p∗, t0)−1} ⊂ {x∈Mt0 : distg(t0)(q, x)≤204H(p ∗ , t0)−1} ⊂ {x∈Mt0 : distg(t0)(q, x)≤500H −1 1 } ⊂V.
Consequently, there exists a closed curve in N ∩V which is homotopically non-trivial in N . This curve bounds a disk which is contained inV . Hence, we
74 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY can again terminate the process.
Subcase 2.2.2: Suppose, finally, that the parabolic neighbourhood ˆ
P(p∗, t0,104,2θ0) is free of surgeries. In this case, Version A of the Neck De- tection Lemma implies that the point p∗ lies at the centre of an ( ˆα,δ, εˆ 1,100)- neck N∗. Clearly, κ1 ≤ ε1H H at each point on N∗. Consequently, the set
N∗ is disjoint from the set {p ∈ N(k) : dist
g(t0)(p, ∂N
(k)\N(k−1)) ∈ [(L 0 − 100)H(pk, t0)−1,(L0 −50)H(pk, t0)−1]}. Furthermore, since p∗ has maximal dis- tance from p0 among all points in A, we conclude that the part of N that lies between the neck N∗ and the set {p ∈ N(k) : dist
g(t0)(p, ∂N
(k)\N(k−1)) ∈ [(L0−100)H(pk, t0)−1,(L0−50)H(pk, t0)−1]} is strictly convex. Let ω be a unit vector in R3 which is parallel to the axis of the neck N∗. Then we have to show
that: hν, ωi ≥ −ε1 for all points p∈N(k) satisfying distg(t0)(p, ∂N
(k)\N(k−1))∈[(L
0−100)H(pk, t0)−1,(L0−50)H(pk, t0)−1]. Moreover, we have κ1(p, t0)> η0H(p, t0) for all pointsp∈N(k) satisfying distg(t0)(p, ∂N
(k)\N(k−1))∈[(L
0−100)H(pk, t0)−1,(L0−50)H(pk,0)−1]. Putting these facts together (and using the fact that η0 ≥ 10ε1), we conclude that hν, ωi ≥4ε1 for all points p ∈ N(k) satisfying distg(t0)(p, ∂N
(k)\N(k−1)) ∈ [(L 0 − 100)H(pk, t0)−1,(L0 −75)H(pk, t0)−1]. We claim that the boundary curve
∂N(k)\N(k−1) bounds a convex cap. Let us now choose a curve Γ
0 such that Γ0 ⊂ {p ∈ N(k) : distg(t0)(p, ∂N
(k)\N(k−1)) ∈ [(L
0 − 100)H(pk, t0)−1,(L0 − 75)H(pk, t0)−1]} and Γ0 is contained in a plane orthogonal to ω. For each point on Γ0, we solve the ODE
˙
γ = ω
T(γ)
|ωT(γ)|2,
whereωT(γ) denotes the projection ofω to the tangent plane toMt0 at the point
γ. This gives a family of curves Γy ⊂ Mt0, each of which is contained in a plane
orthogonal to ω. The curves Γy are well-defined fory∈[0, ymax). Moreover, there exists a point p ∈ Γ0 such that ν(γ(y, p)) → ω as y → ymax. Then we aim to show the following inequalities
ε1 <hν, ωi<1, κ1 >0, H > 2H1
Θ (?)
hold for all y ∈ [0, ymax). Indeed, the inequalities in (?) are clearly satisfied for y = 0. If one of the inequalities in (?) fails for some y > 0, we consider the smallest value of yfor which that happens. The first inequality in (?) cannot fail
3.6. PROOF OF MEAN CURVATURE FLOW WITH SURGERY 75