Incorporación de postizos

One of the key advancements of Perelman in his programme of Ricci Flow with surgery for 3 manifolds was the development of a his noncollapsing result under surgery in [38]. Recall from that paper that we say a manifold is κ-noncollapsed at a point p if for any r, such that the norm of the curvature tensor, ||Rijkl||, is

bounded above by r−2 _{at all points of the metric ball centred at x of radius r,} then we have VolB(x, r) ≥ κrn_{. To deal with the case of MCF first recall the}

following definitions.

Definition 3.6. Given an embedded oriented surface M → _R3_{and a point p ∈} M, the inscribed radius at p is defined as the radius of the largest open ball in R3 which is disjoint from M and touches M at p from the inside.

Similarly, the outer radius atp is defined as the radius of the largest open ball in R3 which is disjoint from M and touches M at p from the outside.

The definition of κ-noncollapsed is closely related to the injectivity radius as if ||Rijkl|| ≤ r−2 on B(p, r) and assuming B(p, r) is κ-noncollapsed, then the

injecitivity radius of the manifold at the point p is bounded below by a positive constantCthat depends only onrandκ. This leads us to the following definition for hypersurfaces:

Definition 3.7. A hypersurface Mn _{with everywhere positive mean curvature}

that bounds an open region Ω in_Rn+1 _{isα-noncollapsed if for everyp∈}M _there is an open ball B of radius _Hα_(p) contained in Ω withx∈∂B.

In the same spirit of Perelman, Sheng and Wang prove that a compact mean- convex hypersurface undergoing MCF remains α-noncollapsed for some α >0 in [39].

We note that the proof by Sheng and Wang is different from that of Perelman’s and instead uses the curvature pinching result in Theorem 2.10 and proved by Huisken and Sinestrari in [29],[28]. Sheng and Wang also remark in their paper that the singularity profile of mean convex MCF is analogous to those of Perelman and so it is conceivable that a similar surgery procedure could be devised for MCF in the case of n = 2.

3.2. NONCOLLAPSING RESULT 55

Remark. To match the literature of Brendle and Huisken in [8] we will from now on refer to a hypersurface being α-noncollapsed when dealing with MCF and κ- noncollapsed when talking about Perelman’s noncollapsing result for Ricci Flow. Ben Andrews was then able to prove in [3] that we can take α to be the noncollapsing constant of the initial compact mean-convex hypersurface and then using an elegant argument involving the maximum principle show that this is preserved under smooth MCF. In [38], Perelman constructs an integral he terms the reduced volume which has many similarities to the Huisken’s monotonicity formula 1.12 which he uses to prove κ-noncollapsing with surgeries. However we wish to avoid a detailed discussion of Perelman’s surgery programme for Ricci Flow and instead focus on why noncollapsing is required to prove the existence of Mean Curvature Flow with surgery. In [30], Huisken and Sinestrari develop cylindrical estimates for n ≥ 3 which allow them to identify the formation of necks, however in n= 2 there is no good cylindrical estimate.

Hence we need to put an additional constraint on our hypersurface as Mean Curvature Flow with surgery will clearly not be possible for immersed mean- convex hypersurfaces. Then to ensure our neck-like regions remain embedded we need to restrict the class by imposing an embedded which is precisely what the noncollapsing property attempts to do as it provides a qualitative expression of ‘how embedded’ a surface is. Therefore to extend the surgery argument of Huisken and Sinestrari to the case of n = 2, just as Pereleman extended the surgery procedure for 3-manifolds in the case of Ricci Flow, we must be able to show that the the evolving hypersurface remains α-noncollapsed under surgery for some α >0.

To do this, Brendle and Huisken work with the following estimate, proved by Brendle in [7] that is proved using similar techniques as in the proof of Theorem 2.10:

µ≤(1 +δ)H+C(δ)

where µ is the reciprocal of the inscribed radius and δ > 0. By showing that this estimate still holds true when performing surgeries for a suitable choice of surgery parameters Huisken and Brendle are able to guarantee the neck remains

α-noncollapsed. This however is by no means immediate as we have the problem that when performing surgery the ratio µ/H may deteriorate, but Huisken and Brendle are able to overcome this by choosing parameters such that the ratioµ/H

improves immediately prior to surgery and such that the noncollapsing constant prior to surgery can absorb the error terms.

56 CHAPTER 3. MEAN CURVATURE FLOW WITH SURGERY Now that we a have a notion of noncollapsing for hypersurfaces undergoing MCF we introduce the notion of an α-Andrews flow, which will come into dis- cussion later.

Definition 3.8. Let α > 0 . A smooth α-Andrews flow {Kt ⊆ U}t∈I in an

open set U ⊆ _Rn _{over a time interval I ⊆}

R is a smooth family of mean convex domains moving by MCF, such that for every p∈ ∂Kt the two closed balls ¯Bint and ¯Bext that are tangent at pand have radius α/H(p) satisfy ¯Bint∩U ⊆Kt and

¯

Bext∩U ⊆U\Int(Kt), respectively.

We conclude this discussion with an example of a noncollapsed hypersurface.

Example 3.1 (The Bowl soliton). In [22], Haslhofer proves that the only non- collapsed translating soliton of MCF that is uniformly 2-convex must be the rotationally symmetric bowl soliton, which in dimension 1 is the Grim Reaper curve and in higher dimensions is the parabaloid obtained by rotation of the Grim Reaper curve