• The region Ωt enclosed byMt shrinks as t increases.
• For each t, the surface Mt is outward minimising within the region Ω0. • For each t the inscribed radius and the outer radius of Mt are at least Hα.
• Each surgery procedure involves performing a Λ-surgery on an ( ˆα,δ, ε, Lˆ )− neck of size r ∈ [2H1 1, 2 H1], where ˆα > α, ˆδ ≤ 1 10, L ≥ 1000Λ and H1 ≥ (1000 supM0|A|)2 infM0H .
Where by outward minimising we mean the following:
Definition 3.11 (Outward-Minimising). LetU ⊂Rn+1 be an open set. We say that the set E ⊂Rn+1 is outward-minimising in U if
|∂E∩K| ≤ |∂F ∩K|
for any F ⊃E such thatF\E ⊂⊂U and any compact set K ⊃(F\E).
It is clear from the definition that the outward minimising property holds under surgery as after removing either a piece diffeomorphic to Sn−1 ×
S or a sphere at time T we have
|∂MT+ ∩K| ≤ |∂MT ∩K| ≤ |∂F ∩K|
We note that the outward minimising property is required so that the gradient estimates of Haslhofer and Kleiner [23] are satisfied.
Remark. From our surgery construction we see that the resulting hypersurface is always embedded and the resulting cap is at least C5 with uniform bounds that are independent of the surgery parameters ˆδ,α, ε,ˆ Λ, L.
3.4
Pseudolocality Theorem and Surgery Param-
eters
When performing surgery it will be prove important to obtain gradient bounds shortly after surgery. By proving a Pseudocalocality Theorem for Mean Curvature Flow we can estimate gradient terms after a surgery time and this will allow us to find the surgery parameters.
66 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY For brevity we omit the proofs for the following results concerning estimates on the the derivatives of the curvature as they are all detailed from section 4-12 of [8].
First we recall the following definition:
Definition 3.12. Consider the ball B ⊂ R3 and a one-parameter family of smooth surfacesMt⊂B such that∂Mt⊂∂B. Moreover, suppose that eachMt
bounds a domain Ωt ⊂B. Then we say that the surfacesMtform aregular mean
curvature flow if the family of hypersurfaces form a smooth solution to 1.6 except at finitely many times where one or more connected components of Ωt may be
removed.
Theorem 3.6 (Pseudolocality Theorem). There exists positive constants β0 and
C such that the following holds. Suppose that Mt, t ∈ [0, T], is a regular mean
curvature flow in B4(0) in the sense of Definition 3.12. Moreover, we assume
that the initial surface M0 can be written as an entire graph. If||u||C4 ≤β0, then
|A|+|∇A|+|∇2A| ≤C
for every t∈[0, β0]∩[0, T] and every x∈Mt∩B1(0).
We note also from [14] that even the initial graphM0 need not necessarily be smooth.
An important step in proving the existence of a surgery algorithm, is to find gradient estimates on the second fundamental form. In [30] it shown that the second fundamental form satisfies the inequality
Theorem 3.7. Let Mt in C(R, α) be a solution of mean curvature flow with
surgery with normalised initial datum. Then there exists a constant γ2 ∼ n and
a constant γ3 ∼ n, α such that for suitable surgery parameters the flow satisfies
the uniform estimate
|∇A|2 ≤γ
2|A|4+γR−4
for all t ≥(1/4)R2.
Then using the interior regularity such as in [7] we are able to define bounds on all the derivatives of the second fundamental form. However the proof of these bounds requires the assumption that the constantκη := 12 n+23 − n−11
,is strictly positive forn ≥3. Hence we require a different estimate on the curvature bounds for the case when n = 2. This is achieved in Haslhofer-Kleiner [23].
3.4. PSEUDOLOCALITY THEOREM AND SURGERY PARAMETERS 67
Theorem 3.8 (Haslhofer-Kleiner). Given any α ∈ (0 1
1000], there exists a con-
stants C ∼ α , with the following property. Suppose that Mt, t ∈ [−1,0], is
a regular mean curvature flow in the ball B4(0). Moreover, suppose that each
surface Mt is outward-minimising within the ball B4(0). We further assume that
the inscribed radius and the outer radius are at least α/H at each point on Mt.
Finally, we assume that M0 passes through the origin, and H(0,0) ≤ 1. Then the surface M0 satisfies |∇A| ≤C and |∇2A| ≤C at the origin.
Then if the flow is α−noncollapsed and has been flowed for sufficient enough time such that the requirements in Theorem 3.8 are met, Theorem 3.8 implies that |∇A| ≤CH2. Then we can employ the Pseudolocality Theorem to control the gradient term, |∇A|, shortly after a surgery procedure has been performed. Finally these two results can be combined to obtain the following estimate on |∇A| which holds under surgery for all points in space-time.
Proposition 3.2 (Proposition 2.9 Huisken-Brendle). There exists a constant
C] with the following significance. Suppose that Mt is a mean curvature flow
with surgery satisfying the surgery assumptions. Then |∇A| ≤ C](H +H1)2 and |∇2A| ≤ C
](H +H1)3 for all times t ≥ (1000 supM0|A|)
−2 and all points
x ∈ Mt. The constant C] may depend on the initial noncollapsing constant α,
but is independent of the surgery parameters
Surgery Parameters With the above results we can now fix some of the surgery parameters. We will be considering the estimates for the case n = 2, for a discussion of the surgery parameters for n ≥ 3 see page 208-209 of [30]. We have fixed the constant C] in the above derivative estimate. We now de-
fine the parameters Θ = 400/α, θ0 = 10−6min{αC]1Θ3} and ˆα =
α
1−θ0/8. Thus
by choosing parameters as such, if at point (p0, t0) where H(p0, t0 ≥ HΘ1 we fol- low this point back in time along the flow to a point (p1, t1) then we will have
H(p1, t1) ∈ [12H(p0, t0),2H(p0, t0)] as long as t ∈ (t0 − 2θ0H(p0, t0)−2, t0]. We then choose δ < δ0 where δ0 is the constant from Theorem 3.5. Meanwhile we choose ˆδsufficiently small such that Proposition 2.11 of [8] holds. Then we choose length parameter Λ as well as the parameter ¯ε such that the conclusion of The- orem 3.5 also holds. Then if we perform a Λ-surgery on a ( ˆα,δ, εˆ 0, L0)-neck, the resulting surface will be 1+1δ -noncollapsed (provided ε ≤ ε¯) and L ≥ 1000Λ. Now as discussed in the noncollapsing section, when we choose our parameters as above, Brendle and Huisken prove in proposition 2.13 of [8] that Brendle’s
68 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY cylindrical estimate holds under surgery. The final parameters will be discussed in the following section.