Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow
Alexander Majchrowski

 

A Chandra Survey of Milky Way Globular Clusters I: Emissivity and Abundance of Weak X-ray Sources
    Alexander Majchrowski
    
=================================================================================================

equationsection










	
	Recently Brendle-Huisken introduced a fully nonlinear flow G, <cit.>. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, <cit.> into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface M_0 in ℝ^n+1 undergoing this G-flow.
	In order to do this we need to make some adjustments to Brendle and Huiskens gradient estimate from <cit.>, after we have done this we can go on to argue as in <cit.>, in order to classify two-convex surfaces undergoing G-flow.
	

	

§ INTRODUCTION

	
	In 1984 Gerhard Huisken showed that for an n-dimensional (n≥2) compact, uniformly convex surface without boundary smoothly embedded in Euclidean space, ℝ^n+1 undergoing mean curvature flow will converge to a round point in finite time. The normalised equation will converge to a sphere as t→∞, <cit.>.
	
	In 1986 he was able to prove the same result for a surface M^n, n
	≥2 embedded in a Riemannian manifold N^n+1, however the requirements for this to happen were much stricter. We had to impose conditions on the sectional curvature of our surface, on the covariant derivative of the curvature tensor and the injectivity radius on our Riemannian manifold N^n+1, as well as having a stronger pinching condition on ℳ_0.
	
	The next natural step was to study 2-convex hypersurfaces embedded in Euclidean space. This was done by Huisken-Sinestrari in 2009, <cit.> using their surgery algorithm. They were able to show that any such surgery would terminate after a finite number of steps and that ℳ_0 was diffeomorphic to either S^n or some finite connected sum of S^n-1× S^1.
	
	Ideally one would then go on to prove a similar result for mean curvature flow of 2-convex hypersurfaces embedded in a Riemannian Manifold. However in this setting 2-convexity is not preserved by the flow. Inspired by Andrews work on harmonic mean curvature flow, <cit.>, Brendle-Huisken introduced the following flow which has the advantage of preserving 2-convexity in the Riemannian setting.
	

	Fixing n≥ 3  consider a closed, embedded hypersurface M_0 in ℝ^n+1. M_0 is κ-2-convex if λ_1+λ_2≥ 2κ, where λ_1≤⋯≤λ_n denote the principal curvatures. We evolve M_0 with normal velocity
	
    G_κ=(∑_i<j1/λ_i+λ_j-2κ)^-1.

	We call this Brendle-Huisken G-flow, but will refer to it as G-flow in later sections.
	Brendle-Huisken were able to extend the surgery algorithm of Huisken-Sinestrari to this fully nonlinear flow in both the Euclidean setting and Riemannian setting. In order to do this they obtained a convexity estimate, cylindrical estimate and gradient estimate for this flow. 
	
   In the Euclidean setting we take κ=0. In order to argue as in Section 7, <cit.> for this flow, adjustments have to be make for the gradient estimate. As we are not able to integrate these estimates in their current form to obtain results related to our backward parabolic neighbourhoods being surgery free as in <cit.>. This is crucial in our proof of the neck detection lemma. Some small adjustments were also required in our proof of the neck detection lemma for this setting.
	
   After making these changes we can follow Section 7 of <cit.> to obtain the other necessary results for when certain conditions in the neck detection lemma are not met. Firstly the we may not know that the backward parabolic neighbourhood about a point is surgery free, in this case we can obtain the required result as long as the curvature at our point is large enough compared to the curvature of the regions changed by previous surgeries. We must also deal with the case when λ_1/G is not small, however the proof here does not rely on gradient estimates and is instead a general property of hypersurfaces as shown by Huisken-Sinestrari in <cit.>.
   
   Putting this all together we obtain our main theorem.
   
   
   	Let ℳ_t be a smooth Brendle-Huisken G-flow of a closed, compact 2-convex hypersurface. Given our neck parameters, there exists a constant G^* depending on ℳ_0 such that if G_max(t_0)≥ G^*, then the hypersurface at time t_0 either contains an (ϵ,k,L)-hypersurface neck or is convex.
   
   
    We will assume that the reader is comfortable with the notation and definitions regarding necks. If not, please refer to <cit.>, <cit.> or <cit.>.
   
   Then the arguments from Section 8 of <cit.> carry over unchanged and we  can obtain results relating to the existence and classification of surgically modified flows.
   
    There are currently many open questions regarding this  G-flow. I am also currently working on adapting Joseph Lauer and John Head's method to show that this 2-convex G-flow with surgeries converges to the weak solution of a level set flow for G as we take our surgery parameter to infinity, <cit.>.
    
    Ancient solutions, solutions which are defined for all negative times have not been studied either, in the convex or 2-convex case.
    
    Another option to consider would be to start putting details together on a suitable algorithm with estimates for dealing with 3-convex flow with surgeries for the G-flow, which in this case would be
    
    G_κ=∑_i<j<k(1/λ+i+λ_j+λ_k-3κ)^-1.
 
    
    This last problem would be hardest to work with, as no surgery algorithm for the 3-convex case currently exists for mean curvature flow or Ricci flow.
   

	
	 
	Acknowledgements:
	I would like to thank Gerhard Huisken for introducing and advising me to work on this fully nonlinear flow G. 
	
	I would also like to thank Xu-Jia Wang for giving me an opportunity to present a talk on this topic at the Geometric and Nonlinear Partial Differential Equations Conference at Murramarang.
	
	Last but not least, I would like to thank my supervisor Zhou Zhang and the School of Mathematics at The University of Sydney for their help and support.
	

§ EVOLUTION EQUATIONS AND NECESSARY ESTIMATES

	
	In this section we go over some preliminary results obtained from <cit.>, we will need the evolution equation for G, convexity estimate, cylindrical estimate, as well as our new gradient estimate which allows us to control the size of the curvature in the neighbourhood of a given point.
	
	Firstly we give some introductory results regarding  G-flow, stated by Brendle-Huisken in <cit.>.
	
    
    	Given G as above we have the following properties:
    	
    		
  * G_κ≤ C_1H, where C_1=C_1(n)>0.

    		
  * ∂ G_κ/∂ h_ij≤ g_ij, where C_2=C_2(n)>0.

    	
    
    
    
    	
    		
  * Clear.
    		
  * This is equivalent to observing ∂ G/∂λ_i being bounded for each i..
    		
    ∂ G/∂λ_i   =∂/∂λ_i(1/λ_1+λ_i+⋯+1/λ_i-1+λ_i+1/λ_i+1+λ_i+⋯+1/λ_n+λ_i)^-1
       =-1(1/λ_1+λ_i+⋯+1/λ_n+λ_i)^-2×(-1/(λ_1+λ_i)^2+⋯+-1/(λ_1+λ_i)^2)
       ≤(1/λ_1+λ_i+⋯+1/λ_n+λ_i)^-2×(1/λ_1+λ_i+⋯+1/λ_1+λ_i)^2
       =1

    		where in the 2nd last step we have used 2-convexity
    	
    			
    
    Now we are able to obtain the necessary evolution equations for G.		
	
	
		If ℳ_t evolves by G-flow, the associated quantities above satisfy the following equations:
			
				
  * ∂/∂ tg_ij=-2Gh_ij 
				
  * ∂/∂ tg^ij=2Gh^ij 
				
  * ∂/∂ tν=∇ G 
			     
  * ∂/∂ th_ij=D_iD_j G-Gh_ilg^lmh_mj 
				
  * ∂/∂ tG=∂ G/∂ h_ij(D_iD_jG+h_ikh_jkG) 
				
  * ∂/∂ tH=Δ|G|^2+∇|H|^2G. 
				
  * ∂/∂ tdμ≤ -G^2/C.
			
	
	
	
		
			
  * 
    ∂/∂ tg_ij   = ∂/∂ t⟨∂ F/∂ x_i,∂ F/∂ x_j⟩
       =⟨∂/∂ x_i(-Gν), ∂ F/∂ x_j⟩+⟨∂/∂ x_j(-Gν), ∂ F/∂ x_i⟩
       =-G⟨∂/∂ x_iν, ∂ F/∂ x_j⟩­G⟨∂ F/∂ x_i, ∂/∂ x_jν⟩
       =-2Gh_ij

			
  * Obtained by differentiating g_ilg^lj=δ_i^j.
			
  * 
    ∂/∂ tν   =⟨∂/∂ tν,∂ F/∂ x_i⟩∂ F/∂ x_jg^ij
       =-⟨ν,∂/∂ t∂ F/∂ x_i)⟩∂ F/∂ x_jg^ij
       =⟨ν,∂/∂ x_i(Gν)⟩∂ F/∂ x_jg^ij
       =∂/∂ x_iG∂ F/∂ x_jg^ij=∇ G.

			
  * In this proof we will make use of the Gauss-Weingarten equations.
			
    ∂/∂ t h_ij   =-∂/∂ t⟨∂^2 F/∂ x_i∂ x_j,ν⟩
       =⟨∂^2/∂ x_i∂ x_j(Gν),ν⟩-⟨∂^2F/∂ x_i∂ x_j,∂/∂ tν⟩
       =⟨∂^2/∂ x_i∂ x_j (Gν),ν⟩-⟨∂^2F/∂ x_i∂ x_j, ∂/∂ x_lG∂ F/∂ x_m g^lm⟩
       =∂^2/∂ x_i∂ x_j G+ G⟨∂/∂ x_i( ∂/∂ x_jν), ν⟩ -⟨∂^2F/∂ x_i∂ x_j,∂/∂ x_lG∂ F/∂ x_m g^lm⟩
       =∂^2/∂ x_i∂ x_j G+G⟨∂/∂ x_i(h_jlg^lm∂ F/∂ x_m),ν⟩
       -⟨Γ_ij^k∂ F/∂ x_k-h_ijν, ∂/∂ x_lG∂ F/∂ x_m g^lm⟩
       =∂^2/∂ x_i∂ x_jG-Γ_ij^k ∂/∂ x_kG+Gh_jmg^ml⟨γ_il^o∂ F/∂ x_o-h_ilν,ν⟩
       =D_iD_j G - Gh_ilg^lmh_mj.

			
  * Here we just use the fact that ∂ G/∂ t=∂ G/∂ h_i^j∂/∂ th_i^j as well as part (ii) and (iv).
			
  * 
    ∂/∂ t H   = ∂/∂ t(h_ijg^ij)
       =g^ij∂/∂ t h_ij+2Gh^ijh_ij
       =g^ij(D_iD_jG-Gh_ilg^lmh_mj)+2Gg^ikg^jlh_klh_ij
       =Δ G+|h|^2G.

			
  * ∂/∂ tdμ=1/2√( g_ij)tr(-2Gh_ij)=-GHμ. Using part (i) of Proposition <ref> the result follows.
		
	
	
	The convexity estimate (<cit.>, Corollary 7.7) is necessary in order to know that the nearly singular regions of the surface become asymptotically convex as a singular time is approached.
	
	
		Suppose that ℳ_t, t∈[0,T) is a surgically modified G-flow starting from a closed, embedded, 2-convex hypersurface ℳ_0 then for any δ>0
		
    λ_1≥δ G- C

		where C is a positive constant that depends only on δ,n and T.
	
	
	Next we need a cylindrical estimate (<cit.>, Theorem 3.1) which implies that at points where λ_1 is small, we have curvature close to the curvature of a cylinder. 
	
	
		Let ℳ_t be a family of closed, two-convex hypersurfaces moving with speed G, then ∀η>0 there exists a constant C=C(δ,T,n)>0 such that
		
    H≤(n-1)^2(n+2)/4(1+δ)G+C_η,T.

	
	
	The following is the gradient estimate taken from <cit.>, Theorem 7.12.  
	
	
		Far a closed, embedded, two-convex hypersurface M_0=δΩ_0. We can find a constant G_#, depending only on M_0 such that the following holds. Suppose that Ω_t, t∈[0,T), is a one-parameter family of sooth open domains with the property that the hypersurfaces M_t=∂Ω_t form a surgically modified flow starting from M_0 with surgery scale G_*≥ G_#. Then we have
		
    α^2G^-2|∇ h|+α^3G^-3|∇^3 h|≤Λ

		for all points in spacetime satisfying G≥ G_#. Here α=α(T,n) is the constant in Proposition 7.8 (<cit.>), and Λ=Λ(T,n) is the constant appearing in Corollary 7.11 <cit.>.
	
	
	Now we want to modify this gradient estimate using the following lemma, this will allow us to integrate our gradient estimates and obtain necessary results related to the backward parabolic neighbourhood as done in <cit.>.
	
	
		The inequalities α^2G^-2|∇ h|≤ C and α^3G^-3|∇^2 h|≤ C from Theorem <ref> imply the following,
		
			
  * K_1|∇ G|≤|∇ h|
			
  * K_2|∂_t G|≤|∇^2 h|
		
		for some constants K_1,K_2.
	
	
	
		For the n-dimensional case we look at the following. We know that
		
    (λ I -h_ij)   =(λ-λ_1)(λ-λ_2)⋯(λ-λ_n) 
    ⇒[ λ-h_11      …  -h_1n;      ⋮      ⋱      ⋮;  -h_n1      … λ-h_nn ]   =λ^n-λ^n-1(λ_1+⋯+λ^n)+⋯-λ_1…λ_n.

		
		
		Here we will need to introduce some notation. Let Q_k(h_ij) denote a k-degree polynomial in terms of of h_ij's, no lower degree can appear. Using a degree argument and equating terms on either side we will obtain
		
    λ^pQ_n-p(h_ij)=λ^pλ_j^n-p.

		This guarantees that we can rewrite our principal curvature values in terms of the second fundamental form.
		Rewriting G as follows,
		
    G=∏_i<j(λ_i+λ_j)/∑_i<j1/λ_i+λ_j∏_i<j(λ_i+λ_j)

		applying the result of (<ref>) to our rewritten G we see that we can write out the λ^p using our h_ij^p terms, 
		
    G=Q_n(h_ij)/Q_n-1(h_ij)

		
		Moreover from our definition of G, (<ref>) and 2-convexity, we can see that 
		
    λ_1+λ_2/n≤ G≤λ_1+λ_2

		lastly from Proposition 2.4 in <cit.> we know that H≤β G for some constant C. This tells us that
		
    λ_1+⋯+λ_n=H≤β G 
       ⇒λ_i≤β_1G
       ⇒ h_ij≤β_2G  for some constants β_1,β_2.

		
		The last step is clear as we know (h_ij)=O(λ_i)O^T, where O is an orthonormal matrix.
		So,
		
    ∇ G   =Q_1(∇ h_ij)Q_n-1( h_ij)Q_n-1(h_ij)-Q_1(∇ h_ij)Q_n-2(h_ij)Q_n(h_ij)/Q_2n-2( h_ij)
       =Q_1(∇ h_ij)Q_2n-2( h_ij)/Q_2n-2(h_ij)
       ≤β_3G^n-2Q_1(∇ h_ij) /β_4 G^2n-2 by (<ref>) and (<ref>) where β_3,β_4 are constants.
       ⇒ |∇ G|≤|β G^2/α^2|

		were in the last line we have applied Proposition <ref>. This proves (i).
		
		Now we  prove part (ii).  From <ref> (v) we know that
		
    ∂_tG=∂ G/∂ h_ij(∇_i∇_j G-h_ijh_jkG).

		We can control ∂ G/∂ h_ij using part (ii) of Proposition <ref>.
		
		Next by applying (<ref>) we can bound the h_ijh_jkG term by β G^3 for some constant β.	
		
		Lastly ∇_i∇_j G will give terms of the form ∇^2h_ij and ∇ h_ijG. Using our result from part (i) as well Proposition <ref> we see that |∂_t G|≤ |KG^3| for some constant K. This completes the proof of part (ii).
	
	
	By apply Lemma <ref> to Theorem <ref> we can obtain our new gradient estimate stated below.
	
	
		Let M be a fully nonlinear flow with surgeries. Then we can find c^#>0, G^#>0 such that for all p∈ M and t>0,
		
    G(p,t)>G^#>0 ⇒ |∇ G(p,t)|≤ c^#G^2(p,t), |∂_t G(p,t)|≤ c^#G^3(p,t),

		where c^# only depends on the dimension of n,T.
	
	
	These estimates allow us to control the size of the curvature in a neighbourhood of a given point.  
	
	

§ THE NECK DETECTION LEMMA


	Using our new gradient estimate we will be able to obtain a result relating to the size of the curvature in a neighbourhood of a given point.
	
	
		Let F:ℳ→ℝ^n+1 be an n-dimensional immersed surface. Suppose that there are c^#, G^#>0 such that |∇ G(p)|≤ c^# G^2(p) for any p∈ℳ such that G(p)≥ G^#. Let p_0∈ℳ satisfy G(p_0)≥γ G^# for some γ>1. Then
		
    G(q)   ≥G(p_0)/1+c^#d(p_0,q)G(p_0)≥G(p_0)/γ for all q
    s.t.  d(p_0,q)   ≤γ-1/c^#1/G(p_0).

	
	
	
		Consider points q∈ℳ such that G(q)<G(p_0)/γ.  Take q_0 to be a point with this property with minimal distance from p, and set d_0=d(p_0,q_0)H(p_0) and θ_0=min{d_0,γ-1/c^#}. Now pick any point q∈ℳ with d(p_0,q)≤θ_0/H(p_0), and let ξ:[0,d(p_0,q)]→ℳ be a geodesic from p_0 to q.
		
		Then from our definition of θ_0 it follows that G(ξ(s))≥G(p_0)/γ≥ G^# for any s∈[0,d(p_0,q)]. Then we can apply Lemma <ref> to obtain |∇ G(ξ(s))|≤ c^#G^2(ξ(s)) and 
		
    d/dsG(ξ(s))≥ -c^#G^2(ξ(s))
for all s∈[0,d(p_0,q)] since it is a geodesic. Integrating this inequality we obtain
		
    G(xi(s))≥G(p_0)/1+c^#sG(p_0), s∈[0,d(p_0,q)],

		which implies
		
    G(q)≥G(p_0)/1+c^#d(p_0,q)G(p_0)≥G(p_0)/1+c^#θ_0.

		This holds for all q such that d(p_0,q)≤θ_0/G(p_0). Now suppose d_0<γ-1/c^#, then d_0=θ_0 and (<ref>) holds for q=q_0. But that implies  G(q_0)>G(p_0)/γ which is a contradiction. Therefore d_0≥γ-1/c^#, which implies θ_0=γ-1/c^#, which makes (<ref>) our assertion.
		
		In the case where G(q)≥G(p_0)/γ for all q∈ℳ, then we have |∇ G|≤ c^#G^2 everywhere, and our result follows more directly from the same argument.
	
	    
	    Next we introduce a backward parabolic neighbourhood. This will be essential in dealing with necks.
	    
	  
	    	Given t,θ such that 0≤ t-θ<t≤ T_0, we define the backward parabolic neighbourhood of (p,t) by,
	    	
    𝒫(p,t,r,θ)={(q,s)|q∈ℬ_g(t)(p,r),s∈[t-θ,t]}.

	    	where ℬ_g(t)(p,r)⊂ℳ is the closed ball of radius r w.r.t. the metric g(t).
	  
	    
	Before we can now apply Lemma 7.2 as in <cit.>. We need to also define r̂_G=(n-1)(n-2)/2G and 𝒫̂_G=𝒫(p,t,r̂_G(p,t)L,r̂^2(p,tθ), we can observe that if (p,t) lies on a neck then r̂_G(p,t) is approximately equal to the radius of the neck.
	
	
		Let c^# and G^# be the constant from the above Corollary Define d^#=(2(n-1)^2(n-2)^2c^#)^-1. Then the following properties hold.
		
			
  * Let (p,t) satisfy G(p,t)≥ 2G^#. Then, given any r,θ∈(0,d^#] such that 𝒫̂_G(p,t,r,θ) does not contain surgeries, we have
			
    G(p,t)/2≤ G(q,s)≤ G(p,t)

			for all (q,s)∈𝒫̂_G(p,t,r,θ).
			
  * Suppose that, for any surgery performed at time less than t, the regions modified b surgery have mean curvature less than K, for some K≥ G^#. Let (p,t) satisfy G(p,t)≥ 2K. Then the parabolic neighbourhood
			
    𝒫(p,t,1/8c^#K,1/8c^#K^2)
  does not contain surgeries. In particular, the neighbourhood 𝒫̂_G(p,t,d^#,d^#) does not contain surgeries and all points (q,s) contained there satisfy (i).
		
	
	
	
		Firstly we prove part (ii).
		
		Suppose the neighbourhood of in (<ref>) is modified by surgeries. Take a point (q,s) which is modified by surgery, with s the maximal time at which we can find such a point. Then by assumption we have G(q,s+)≤ K. Integrating the estimate on ∂_tG from Theorem <ref>,
		
    ∫_s^t∂ G/G^3≤∫_s^t c^#∂ t
    
    		-1/G^2(q,t)   +1/G^2(q,s)≤ c^#(t-s)
    1/G^2(q,t)   ≥1/G^2(q,s)-2c^#(t-s)≥3/4K^2

		where in the last line we used our assumption on H(q,s) and that t-s≤1/8c^#K^2.
		Then we integrate along a geodesic from q to p at time t and use the estimate on ∇ G 
		
    1/G(p,t)≥1/G(q,t)-c^#d_g(t)(p,q)≥√(3)-1/4/2K>1/2K,

		where in the last line we used our estimate on G^2(q,t) and that d_g(t)≤1/8c^#K. This contradicts our assumption  that G(p,t)≥ 2K. In this argument we have had to assume that G≥ G^# along the integration paths in order to apply the results of Theorem <ref>. If this is not true, we can choose the last point along the path with G≤ G^# and integrate from that point on, obtaining a contradiction in the same way. Now we use the definition of d^# to see that 𝒫̂_G(p,t,d^#,d^#) is contained in the neighbourhood (<ref>),
		
    (n-1)(n-2)/2G(p,t)d^#≤1/8K(n-1)(n-2)c^#≤1/8c^#K
    and    (n-1)^2(n-2)^2/2G(p,t)^2d^#≤1/16K^2c^#≤1/8c^#K^2.
	
		Therefore 𝒫̂_G(p,t,d^#,d^#) does not contain surgeries and part (i) can be applied to this neighbourhood.
		
		To prove part (i), we integrate the same inequalities and use the assumption that 𝒫̂_G is surgery free.
	
	
	
	
		Let ℳ_t, t∈[0,T) be the G-flow with surgeries, starting from an initial manifold ℳ_0. Let ϵ,θ,L>0, and k≥ k_0 be given (where k_0≥2 is the parameter measuring the regularity of the necks where surgeries are performed). Then we can find η_0, G_0 with the following property. Suppose that p_0∈ℳ_0 and t_0∈[0,T) are such that:
		
			
  * G(p_0,t_0)≥ G_0,λ_1(p_0,t_0)/G(p_0,t_0)≤η_0
			
  * The neighbourhood 𝒫̂(p_0,t_0,L,θ) does not contain surgeries.
		
		Then
		
			
  * The neighbourhood 𝒫̂(p_0,t_0,L,θ) is an (ϵ,k_0-1,L,θ)-shrinking curvature neck;
			
  * The neighbourhood 𝒫̂(p_0,t_0,L-1,θ/2) is an (ϵ,k,L-1,θ/2)-shrinking curvature neck.
		
		With constants η_0(α,ϵ,k,L,θ).
	
	
	
		Here we argue by contradiction. Suppose that for some values of ϵ,L,θ the conclusion doesn't hold. No matter how we pick η_0 or G^*. We take a sequence {M_t^j}_j≥1 of solutions to the flow. A sequence of times t_j and points p_j such that ,H_j,G_j,λ_1,j denote the values of H,G and λ_1 at F_j(p_j,t_j)∈ M_t_j^j satisfy,
		
		
			
  * λ_1+λ_2≥α_0 G, this is clear from (<ref>).
			
  * parabolic nbhd not changed by surgery
			
  * G_j→∞, λ_1,j/G_j→0 as j→∞.
			
  * (p_j,t_j) doesn't lie at the centre of an (ϵ,k_0-1,L,θ)-shrinking neck.
		
		G_j→∞ since curvature of the flows uniformly bounded at t=0. We now continue with a parabolic rescaling such that, H(p_j,t_j)=1 and (p_j,t_j) is translated to the origin 0∈ℝ^n+1 and t_j becomes 0, we define it as
		
    F̃(p,τ)=1/H_j[F_j(p,r̂^2τ+t_j)-F(p_j,t_j)]

		
		denoting by ℳ̅_τ^j our rescaled surface.
		
		Our gradient estimates give bounds on |A| and derivatives up to k_0.  This tells us we have a limit surface to the flow M̃_τ^∞.
		
		Passing to the limit in the convexity estimate from Lemma (<ref>) we find that λ̃_1≥ 0 which implies that λ̃_i≥0 for all i=1,...n. On the other hand we know that λ_1+λ_2≥0 so λ̃_j>0 for j≥2 and S̃_i>0 for all i≤ n-1, where S_i is defined in <cit.>. This together with (iii) tells us λ̃_1(0,0)=0.
		
		
		Looking at the quotient Q̃_n=S̃_n/S̃_n-1 we know it is non negative everywhere. Arguing as in <cit.> and using the strong maximum principle, we know if it is positive somewhere in the interior of P̃^∞(0,0,d,d), it is positive everywhere at all later times. But Q̃_n(0,0)=0, showing that λ̃_1≡0 in this set.

		Scaling by H I know that my principal curvatures on a cylinder are λ_1=0 and λ_j=1/n-1 for all j≥2. 
		
    ⇒ H=1 ⇒ G   =((n-1)(n-1)+(n-1)(n-2)/2(n-1)/2)^-1
       =((n-1)^2(n+2)/4)^-1.

		On a cylinder H-((n-1)^2(n+2)/4G=0. I know passing to the limit in my cylindrical estimate yields H-(n-1)^2(n+2)/4G≤0 ⇒ G≥4/(n-1)^2(n+2), we want to see that 4/(n-1)^2(n+2) is the maximum value G can attain.
		
		We want to see, G(0,a_1,...,a_n-1)≤ G(0,1/n-1,...,1/n-1) with equality when the a_i's are equal.
		
		Picking from (0,a_1,...,a_n-1) we know,
		
    G^-1=1/a_1+⋯+1/a_n-1+∑1/a_i+a_j+λ(1-a_1+⋯+a_n-1)

		where λ is the Lagrange multiplier with the constraint that the sum of our a_i's=1.
		Taking partial derivatives we see that for any two a_i,a_k,
		
    ∂ G^-1/∂ a_i   =-1/a_i^2-∑_j≠ i1/(a_i+a_j)^2-λ=0
    ∂ G^-1/∂ a_k   =-1/a_k^2-∑_j≠ k1/(a_k+a_j)^2-λ=0
    ⇒ -1/a_i^2   +1/a_k^2-∑_j≠ i,k1/(a_i+a_j)^2+∑_j≠ i,k1/(a_k+a_j)^2=0

		⇒ a_i=a_k and all a_i's are equal. Therefore we have a maximum when they are equal, on the boundary we will have a minimum. This tells me that G=C(n)H for some constant C=C(n), so on the limit I have that my fully nonlinear G-flow is the same as mean curvature flow. This allows me to argue as in [<cit.>, Lemma 7.4] and [<cit.>, Theorem 5.1] to obtain that ℳ̃_t^∞ on 𝒫̃^∞)(0,0,d,d) is a portion of a shrinking cylinder.
		
		
		Now we can iterate the procedure to that the whole neighbourhoods 𝒫̅^∞_j(0,0,L,θ) of the rescaled flow converge to a cylinder. From the first step we know that, for j large enough, the curvature on 𝒫̅_j^∞(0,0,d,d) is close to the curvature of a unit cylinder. Then, using the gradient estimates we have uniform bounds on G̅_j also on some larger neighbourhoods, i.e. 𝒫̅_j^∞(0,0,2d,2d), we can repeat the previous argument to prove convergence to a cylinder there. After a finite number of iterations we can obtain convergence of the neighbourhoods 𝒫̅_j^∞(0,0,L,θ). The immersions converge in the C^k_0-1-norm and this ensures that for j large enough, the neighbourhoods are (ϵ,k_0-1,L,θ)-shrinking necks. This contradicts assumption (iv) and proves part (i) of the lemma.
			
		
				To prove part (ii) of the lemma we argue in a similar fashion. Again we argue by contradiction and take a sequence of rescaled flows. Consider the smaller parabolic neighbourhoods 𝒫̅(0,0,L-1,θ/2) and apply interior regularity results from <cit.> to find bounds in the C^k+1 norm as well. This yields compactness in the C^k-norm, which yields the desired result.
	
	
	
		Part (i) of the lemma concerns the whole parabolic neighbourhood which is surgery free, but can be arbitrarily close to surgery, the points of the neighbourhood are even allowed to be modified by a surgery at the initial time t_0-θ r_0^2. Therefore the description goes up to k_0-1 derivatives. Part (ii) is concerned with a smaller parabolic neighbourhood, where we can use interior parabolic regularity and as many derivatives as we wish.
	
	
	
		Given ϵ,θ>0, L≥ 10 and k>0 an integer, we can find η_0, H_0>0 such that the following holds. Let p_0,t_0 satisfy (ND1) and (ND2) of Lemma <ref>. Then
		
			
  * The point (p_0,t_0) lies at the centre of a cylindrical graph of length 2(L-2) and C^k+2-norm less than ϵ;
			
  * The point (p_0,t_0) lies at the centre of a normal (ϵ,k,L-2)-hypersurface neck.
		
	
	
	
		Using Proposition 3.5 and Theorem 3.14 from <cit.> both properties are true is (p_0,t_0) lies at the centre of an (ϵ',k',L-1)-curvature neck for suitable ϵ',k'. Therefore it is clear that the properties hold if (p_0,t_0) lies at the centre of an (ϵ',k',L-1,θ/2)-shrinking curvature neck. Thus is suffices to apply part (ii) of the neck detection lemma with parameters (ϵ',k',L,θ) and use the corresponding values of η_0, G_0.
	
	
	The following lemma, is an immediate consequence of the neck detection lemma. It shows that the shrinking curvature neck given by that lemma can be represented at each time as a hypersurface neck. This can be done at all times, including the initial time which could coincide with a surgery time.
	
	For this we will introduce the following. We say that a point (p,t) lies at the centre of a neck if p∈ℳ lies at the centre of a neck with respect to the immersion F(·,t). We introduce a time-dependent version of the notion of curvature neck. For s≤0 we set,
	
    ρ(r,s)=√(r^2-2(n-1)s)

	so that ρ(r,s) is the radius at time s of a standard n-dimensional cylinder evolving by the fully nonlinear flow having radius r at time s=0. Moreover we have
	
    r≤ρ(r,s)≤2r for all s∈[d^#r^2,0]

	otherwise the cylinder would violate Lemma <ref>.
	
		In Lemma <ref>, we can choose the constants η_0,h_0 so that the additional following property holds. Suppose that L≥10 and that θ≤ d^#. Denote
		
    r_0=(n-1)(n-2)/2G(p,t), B_0=B_g(t_0)(p_0,r_0L).

		Then for any t∈[t_0-θ r_0^2,t_0], the point (p_0,t) lies at the centre of an (ϵ,k_0-1)-hypersurface neck N_t⊂ B_0, satisfying the following properties:
		
			
  * The mean radius r(z) of every cross section of N_t is equal to ρ(r_0,t-t_0)(1+O(ϵ));
			
  * The length of N_t is at least L-2;
			
  * There exists a unit vector ω∈ℝ^n+1 such that |ν(p,t)·ω|≤ϵ for any p∈ N_t.
		
	
	
	
		Proved in the same way as part (i) of Lemma <ref>. By contradiction, we take a suitable η_0,G_0, then our parabolic neighbourhood is as close as we wish to an exact cylinder evolving by the fully nonlinear flow over the same time interval. The cylinder has radius r_0 at the final time, hence it has radius ρ(r_0,t-t_0) at previous times.
		
		At the final time, C_t_0 is a neighbourhood of radius r_0L of p_0. Let B_L⊂ C be the set of points of C having intrinsic distance less than L from p_0. Clearly, B_L cannot be written in the form S^n-1×[a,b] for any a,b,. However, it is easy to see that for L≥(π^2+1)/2, then S^n-1×[-(L-1),(L-1)]⊂ B_L⊂ S^n-1×[-L,L]. Using this logic, we can see C_t_0) contains a sub-cylinder of length 2(L-1). The same sub-cylinder is contained in C_t for t<t_0; however since the scaling factor is given by ρ(r_0,t-t_0) rather than r_0, then length of the sub-cylinder becomes 2r_0(L-1)/ρ(r_0,t-t_0). Recalling (<ref>), we see the sub-cylinder has length at least 2(L-1) for the times under consideration. Since we can make our parabolic neighbourhood as close as we wish in the (k_0-1)-norm to the cylinder C_t we can find a geometric neck parametrizing the part of the neighbourhood corresponding to the sub-cylinder found above, and this neck will satisfy properties (i) and (ii). Property (iii) follows from choosing ω to be the axis of our cylinder C_t.
	
	
	
	
	
	Just as before in the mean curvature flow case, we rely on (ND2) and have seen it is crucial in the proof of the neck detection lemma. The next result shows that (ND2) follows from the other assumptions of the neck detection lemma provided the curvature at (p_0,t_0) is large enough compared to the curvature of the regions changed by the previous surgeries.
	
	
		Consider a flow with surgeries satisfying the same assumptions of Lemma <ref>. Let d^# be as before and let ϵ,k,L,θ be given with θ< d^#. Then we can find η_0,G_0 with the following property. Let (p_0,t_0) be any point satisfying
		
    G(p_0,t_0)≥max{G_0,5k}, λ_1(p_0,t_0)/G(p_0,t_0)≤η_0

		where K is the maximum of the curvature at the points changed in the surgeries at times before t_0. Then (p_0,t_0) satisfies hypothesis (ND2) and the conclusions (i)-(ii) of Lemma <ref>. In addition, the neighbourhood
		
    𝒫(p_0,t_0,(n-1)(n-2)/2G(p_0,t_0)L,(n-1)^2(n-2)^2/4K^2d^#),

		which is larger in time than (ND2) does not contain surgeries. 
	
	
	
		Let ϵ,k,L,θ be given, with θ≤ d^#. The constants η_0,G_0 depend continuously on the parameters L,θ measuring the size of the parabolic neighbourhood. Thus, if L_2>L_1 and θ_2>θ_1>0 it is possible to find η_0,H_0 which apply to any L∈[L_1,L_2] and θ∈[θ_1,θ_2]. Thus we can find values η_0,G_0 such that the conclusions of the neck detection lemma hold for our choice of (ϵ,k,L,θ), and also if we replace L with any L'∈[d^#,L]. In addition, we can also assume that G_0≥ 2G^#. We claim that such values of η_0,G_0 satisfy the conclusions of the present lemma.
		
		By our choice of η_0,G_0, the conclusions don't hold only if (ND2) is not satisfied, i.e.𝒫̂(p_0,t_0,L,θ) contains surgeries.
		
		By Lemma <ref>, at least the neighbourhood 𝒫̂(p_0,t_0,d^#,0) does not contain surgeries. Therefore, if (ND2) s violated, there exists a maximal L'∈[d^#,L] such that 𝒫̂(p_0,t_0,L,θ) does not contain surgeries. We apply the neck detection lemma to this neighbourhood and deduce that it is an (ϵ,k_0-1,L',θ)-shrinking neck. In particular G(p_0,t_0)(1+O(ϵ))≥ 4K for all p such that d_g(t_0)(p_0,p)≤r̂(p_0,t_0)d^#. But then Lemma <ref> shows that the larger neighbourhood 𝒫̂(p_0,t_0,L'+d^#,θ) does not contain surgeries as well, contradicting the maximality of L'. This proves (ND2) holds and that the neck detection lemma can be applied to the whole neighbourhood 𝒫̂(p_0,t_0,L,θ).
		
		To obtain the last claim, take any q such that
		
    d_g(t_0)(q,p_0)≤(n-1)(n-2)/2G(p_0,t_0)L.

		
		By the previous part of the statement, G(q,t_0)=G(p_0,t_0)(1+O(ϵ))>2K. Then, Lemma <ref>(ii) implies that q has not been affected by any surgery between time t_0-(n-1)^2d^#/K^2 and t_0. Since this holds for any q in the neighbourhood, the statement is proved.
	
	
	In the next results we assume that our flow with surgeries satisfies certain properties. 
	
		
  * The surgeries take place in a region where G is approximately equal to some fixed value K^*. More precisely, there is K^*>2G^# such that each surgery is performed at a cross section Σ_z_0 of a normal neck with r(z_0)=r^*, where r^*=n-1/K^*.
		
  * The two portions of a normal neck resulting from surgery have the following properties. One portion belongs to a component of the surface which will be removed after the surgery. In the other portion, the part of the neck that has been left unchanged by the surgery will have the following structure: the first cross section which coincides with the boundary of the region changed by the surgery the mean radius satisfies r(z)≤(1-/11)r*, on the last section r(z)≥ 2r^* and in the sections in between r^*≤ r(z)≤ 2r^*.
		
  * Each surgery is essential for removing a region of the surface with curvature greater than 10K^*. That is, if we consider any of the surgeries performed at a given surgery time, we can find a component removed afterwards which contains some point with G≥ 10K^*, and which would not have been disconnected from the rest of the surface without the surgery.
	
	
	If the neck parameter ϵ_0 is chosen small enough, then (g1) tells us that the areas modified by surgery will have G between K^*/2 and 2K^* after the surgery.
	(g1) also implies that r(z)≤ (11/10)r^* on the first cross section.
	Property (g3) is a natural assumption as we wish the reduce the curvature by a certain amount each time we perform surgery. Whilst (g1) and g(3) together imply that the regions with largest curvature are not the ones affected by surgery but the ones that become disconnected from the surface and removed as they have known topology.
	Lastly, property (g3) tells us that surgeries are actually performed at a certain distance away from the ends of the neck, this will be useful in the next part of the lemma.
	
	
		Consider a flow with surgeries with our usual assumptions, and (g1)-(g3). Let L,θ>0 be such that θ≤ d^# and L≥ 20. Then there exist η_0, G_0 such that the following property holds. Let (p_0,t_0) satisfy (ND1) and (ND2) of the neck detection lemma, and suppose in addition that the parabolic neighbourhood 𝒫̂_G(p_0,t_0,L,θ) is adjacent to a surgery region. Then (p_0,t_0) lies at the centre of a hypersurface neck N of length at least L-3, which is bordered on one side by a disc D. The value of G on N∪ D at time t_0 is less than 5K^*, where K^* is defined above in property (g1).
	
	
	
		Begin by applying part (i) of the neck detection lemma to find η_0 and H_0 such that any point (p_0,t_0) satisfying (ND1) and (ND2) lies at the centre of an (ϵ,k_0-1,L,θ) shrinking curvature neck. By refining the choice of η_0,G_0 we can also obtain that for all times under consideration the neck can be parametrised as a geometric neck. Now let (p_0,t_0) satisfy the assumptions of the lemma for such values of η_0,G_0. Let us also set
		
    r_0=(n-1)(n-2)/2G,  B_0={p∈ℳ | d_g(t_0)(p,p_0)≤ r_0L}

		Our assumptions are that B_0 is not modified by any surgery for t∈[t_0-θ r_0^2,t_0], but that there is a point q_0∈∂ B_0 and a time s_0∈[t_0-θ r_0^2,t_0] such that q_0 lies in the closure of a region modified by surgery at time s_0. Our aim is to now show that the structure is not affected by the other surgeries which may occur between time s_0 and t_0.
		
		Let us denote by D^* the region modified by the surgery which includes q_0 in its closure, and let N^* be the pat of the neck left unchanged with the properties described in (g2). Let us denote by Σ^*_1 and Σ_2^* the two components of ∂ N^* having mean radius less than (11/10)r^* and greater than 2r^* respectively. By (g2), Σ_1^*∂ D^*, and so q_0∈Σ_1^*. It follows that the mean radius of Σ_1^* is equal to (n-1)(n-2)/2G(q_0,s_0) up to an error of order O(ϵ). Then we know that G(p,s_0)≥(n-1)(n-2)/2r^*(10/11+O(ϵ))=K^*(10/11+O(ϵ)) for all p∈ B_0, because the fully non-linear curvature G is constant up to O(ϵ) on B_0 at any fixed time.
		
		We claim that B_0 must be contained in N^*. In fact, we know that B_0 has not been changed by the surgery at time s_0, and so it has no common points with D^*. If B_0 were not contained in N^*, then it would intersect the other component Σ_2^* of ∂ N^*. But this is impossible, since at time s_0 the points in B_0 and in Σ_2^* have a value of G respectively greater than (10/11)K^*. and less than K^*/2 up to O(ϵ).
		
		Let z∈[z_1,z_2] be the parameter describing the cross sections of N^*, where z=z_i corresponds to Σ_i^*. Then we can find a maximal interval [a,b]⊂[z_1,z_2] such that the neck corresponding to z∈[a,b] is centred at p_0 and is contained in B_0. Let us denote by N_0 this neck. Arguing as in Lemma <ref>, we can see that N_0 has a length at least L-2.
		
		Let us now denote with N' the part of N^* corresponding to z∈[z_1,a]. Then we have that p_0 belongs to N_0, which is a normal k_0-hypersurface neck of length at least L-2, and which is bordered on one side by the region N'∪ D^*, which is diffeomorphic to a disc. This is the statement of our theorem, except it holds at time s_0 rather than the final t_0.
		
		It remains to show that, if there are any surgeries between time s_0 and t_0, they do not affect the region N_0∪ N'∪ D^*. Observe that in this region G(p,s_0)≤ 2K^* for any p in this region. By our choice of D^#, G^# we have G(p,t)≤ 4K^* for any p∈ N_0∪ N'∪ D^* and t between s_0 and either t_0 or the first surgery time, if it exists, that affect this region. But this shows that there cannot be any such surgery. In fact, observe that N_0 is contained in B_0, which by assumption is not changed by surgeries in [s_0,t_0]. The neck N_0 disconnects the region  D^*∪ N' from the rest of the manifold. By (g3), if a surgery changes this part, it must disconnect a region contained in N_0∪ N'∪ D^*  here the maximum of the curvature is at least 10K^*. This contradicts the bound on the curvature we just found, which proves that the topology of the region does not change up to time t_0, and that the curvature remains below the value 5K^* in this region. To conclude the proof, it suffices to parametrise the geometric neck N_0 in normal form at the final time t_0, using the property that N_0⊂ B_0 which is an (ϵ,k_0-1) curvature neck at any fixed time.
	
	
	In the neck detection lemma we assume that the the point under consideration λ_1/G is small. The next result will allow us the deal with the case where it is not and can be reduced in some sense to the former one. It is a general property of hypersurfaces and not related to geometric flows, so the proof is exactly as in <cit.>.
	
	
		Let F:ℳ→ℝ^n+1, with n>1 be a smooth connected immersed hypersurface (not necessarily closed). Suppose that there exist c^#,G^#>0 such that |∇ G(p)|≤ c^# G^2(p) for all p∈ℳ such that G(p)≥ G^#. Then, for any η_0>0 we can find α_0>0 and γ_0>1 (depending only on c^# and η_0), such that the following holds. Let p∈ℳ satisfy λ_1(p)>η_0 G(p) and G(p)≥γ_0G^#. Then either ℳ is closed with λ_1>η_0G>0 everywhere, or there exists a point q∈ℳ such that
		
			
  * λ_1(q)≤η_0G(q)
			
  * d(p,q)≤α_0/G(p)
			
  * G(q')≥ G(p)/γ_0 for all q'∈ℳ such that d(p,q')≤α_0/G(p); in particular, G(q)≥ G(p)/γ_0.
		
	
	
	
		Given α_0>0, set γ_0=1+c^#α_0. For a given p∈ℳ, let us set ℳ_p,α_0={q∈ℳ| d(p,q)≤α_0/G(p)}. By <ref>, we obtain that, if G(p)≥γ_0G^#, then
		
    G(q)≥G(p)/1+c^#d(p,q)G(p)≥G(p)/γ_0

		for all q∈ℳ_p,α_0.
		
		To prove the theorem suppose now that p∈ℳ is such that G(p)≥γ_0G^# and that λ_1(q)>η_0G(q) for all q∈ℳ_p,α_0. We claim that is α_0 is suitably large, these properties imply that ℳ coincides with ℳ_p,α_0 and is therefore compact with λ_1>η_0 G everywhere. 
		
		To prove this, we show that the Gauss map ν:ℳ_p,α_0→ S^n is surjective. Take any ω∈ S^n, such that ω≠±ν(p). We consider the curve γ solution of the ODE, γ̇=ω^T(γ)/|ω^T(γ)| with γ(0)=p, where for any q∈ℳ, ω^T(q)=ω-⟨ω,ν(p)⟩ν(q) is the component of ω tangential to ℳ at q. Since γ̇|=1, the curve γ will be parametrized by arc length The curve can be continued until |ω^T(γ)|≠0, i.e. ν(γ)±ω. As long as γ(s) is contained in ℳ_p,α_0, which is the case if s∈[0,α_0/G(p)], we can use the property λ_1>η_0G to derive some estimate.
		
		Namely, if we take an orthonormal basis e_1,… ,e_n of the tangent space to ℳ at point γ(s), we have
		
    d/ds⟨ν,ω⟩   =∑_i=1^n⟨γ̇,e_i⟩⟨∇_e_iν,ω⟩=1/ω^Th_ij⟨ω,e_i⟩⟨ω,e_j⟩
       ≥1/ω^Tη_0H|ω^T|^2=η_0G√((1-⟨ν,ω⟩^2)),

		which implies d/dsarcsin⟨ν,ω⟩≥η_0 G.
		
		Now suppose that γ(s) exists for s∈[0,α_0/G(p)]. Then we have
		
    π   >arcsin⟨ν(γ(α_0/G(p))),ω⟩-arcsin⟨ν(p),ω⟩≥η_0∫_0^α_o/G(p)G(γ(s))ds
       ≥η_0∫_0^α_0/G(p)1/G(p)^-1+c^#s=η_0/c^#ln(1+c^#α_0).

		
		Thus, if α_0 is large enough to have
		

			α_0>1/c^#(exp(c^#π/η_0)-1)
		

		we obtain a contradiction. Therefore there exists s^*∈(0,α_0/G(p)] such that either ν(γ(s))·ω→1 or ν(γ(s))·ω→-1 as s→ s^*. Since arcsin⟨ν,ω/rangle is increasing, only the first possibility can occur. This shows that γ(s) converges, as s→ s*, to some point q^*∈ℳ_p,α_0 such that ∋(q^*)=ω, as desired.
		
		It remains to consider the case when ω=±ν(p), when ω trivially belongs to the image of the Gauss map. If instead we have ω=-ν(p), it suffices to replace p with another point p' sufficiently close to p; by convexity, we have ν(p')≠ν(p)=-ω and the previous argument can be applied. Thus, we have proved that the Gauss map is surjective from ℳ_p,α_0 to S^n. Since λ_1>0 on ℳ_p,α_0, the Gauss map is also a local diffeomorphism. Then, since S^n is simply connected for n>1, it follows that the map is a global diffeomorphism.
	
	
	Putting all this together, we are able to provide a result about the existence of necks before a first singular time is approached.
	
	
		Let ℳ_t be a smooth G-flow of closed 2-convex hypersurfaces. Given neck parameters ϵ,k,L there exists G^* (depending on ℳ_0) such that, if G_max(t_0)≥ G^*, then the hypersurface at time t_0 either contains an (ϵ,k,L)-hypersurface neck or it is convex.
	
	
	
	  Combine Lemma (<ref>) with Theorem (<ref>). Since we assume the flow is smooth, the parabolic neighbourhood in hypothesis (ND2) trivially does not contain surgeries.
	
	
	Therefore, unless the surface becomes convex, we are able to perform surgery before the singular time. Section 8 of <cit.> then describes the process by which after performing surgery, the maximum of the curvature will decrease making our surface less singular, it also goes on to proving the existence of necks after the first surgery, as we have to check that condition (ND2) is still satisfied.
	
	Now that we've obtained all our results regarding neck detection. One can then continue as in <cit.> to prove the following theorem.
	
	
		Let ℳ_0 be a smooth closed two-convex hypersurface immersed in ℝ^n+1, with n≥ 3. Then there exist constants G_1<G_2<G_3 and a G-flow with surgeries starting from ℳ_0 with the following properties:
		
			
  * Each surgery takes place at a time T_i such that max G(·,T_i-)=G_3.
			
  * After the surgery, all the components of the manifold (except for ones diffeomorphic to spheres of to S^n+1× S^1 which are neglected afterwards) satisfy max G(·,T_i+)≤ G_2.
			
  * Each surgery starts from a cross section of a normal hypersurface neck with mean radius r(z_o)=(n-1)(n-2)/2G.
			
  * The flow with surgeries terminates after finitely many steps.
		
		The constants G_i can be any values such that G_1≥ω_1,G_2=ω G_1 and G_3=ω_3G_2, with ω_i>1.
	
	
	 
	 
	
*
plain


	
	
	
