Cylindrical estimates for hypersurfaces moving by convex curvature functions
aa r X i v : . [ m a t h . DG ] O c t CYLINDRICAL ESTIMATES FOR HYPERSURFACES MOVING BYCONVEX CURVATURE FUNCTIONS
BEN ANDREWS AND MAT LANGFORD
Abstract.
We prove a complete family of ‘cylindrical estimates’ for solutions of a class offully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari[HS09, Section 5] for the mean curvature flow. More precisely, we show that, for the classof flows considered, an ( m + 1)-convex (0 ≤ m ≤ n −
2) solution becomes either strictly m -convex, or its Weingarten map approaches that of a cylinder R m × S n − m at points wherethe curvature is becoming large. This result complements the convexity estimate proved in[ALM13] for the same class of flows. Introduction
Let M be a smooth, closed manifold of dimension n , and X : M → R n +1 a smoothhypersurface immersion. We are interested in smooth families X : M × [0 , T ) → R n +1 ofsmooth immersions X ( · , t ) solving the initial value problem ( ∂ t X ( x, t ) = − F ( W ( x, t )) ν ( x, t ) ,X ( · ,
0) = X . (CF)where ν is the outer normal field of the evolving hypersurface X and W the correspondingWeingarten curvature. In order that the problem (CF) be well posed, we require that F ( W )be given by a smooth, symmetric, degree one homogeneous function f : Γ → R of the principalcurvatures κ i which is monotone increasing in each argument. The symmetry of f ensuresthat F is a smooth, basis invariant function of the components of the Weingarten map (oran orthonormal frame invariant function of the components of the second fundamental form)[Gl63]. Monotonicity implies monotonicity with respect to the Weingarten curvature, whichensures that the flow is (weakly) parabolic. This guarantees local existence of solutions of(CF), as long as the principal curvature n -tuple of the initial data lies in Γ [Ba, Main Theorem5].For technical reasons, we require the following additional conditions Conditions. (i) that Γ is a convex cone , and f is homogeneous of degree one; and(ii) that f is convex. Mathematics Subject Classification . 53C44, 35K55, 58J35.Research partially supported by Discovery grant DP120102462 of the Australian Research Council.The second author gratefully acknowledges the support of an Australian Postgraduate Award during thecompletion of this work. We remark that this condition can be slightly weakened. See [ALM13].
Then, since the normal points out of the region enclosed by the solution, we may assumethat (1 , . . . , ∈ Γ, and we we lose no generality in assuming that f is normalised such that f (1 , . . . ,
1) = 1.The additional conditions (i)-(ii) have several consequences. Most importantly, they allowus to obtain a preserved cone Γ ⊂ Γ of curvatures for the flow [ALM13, Lemma 2.4]. Thisallows us to obtain uniform estimates on any degree zero homogeneous function of curva-ture along the flow (Lemma 2.2); in particular, we deduce uniform parabolicity of the flow(Corollary 2.3). The convexity condition then allows us to apply the second derivative H¨olderestimate of Evans [Ev82] and Krylov [Kr82] to deduce that the solution exists on maximaltime interval [0 , T ), T < ∞ , such that max M ×{ t } F → ∞ as t → T , as in [ALM, Proposition2.6]. This paper addresses the behaviour of solutions as F → ∞ . Let us recall the followingcurvature estmate [ALM13] (cf. [HS99a, HS99b]): Theorem 1.1 (Convexity Estimate) . Let X : M × [0 , T ) → R n +1 be a solution of (CF) suchthat f satisfies Conditions (i)–(ii). Then for all ε > there is a constant C ε < ∞ such that G ( x, t ) ≤ εF ( x, t ) + C ε for all ( x, t ) ∈ M × [0 , T ) , where G is given by a smooth, non-negative, degree one homogeneous function of the principalcurvatures of the evolving hypersurface that vanishes at a point ( x, t ) if and only if W ( x,t ) ≥ . Theorem 1.1 implies that the ratio of the smallest principal curvature to the speed isalmost positive wherever the curvature is large. Combining it with the differential Harnackinequality of [An94b] and the strong maximum principle [Ha84] yields useful informationabout the geometry of solutions of (CF) near singularities [ALM13] (cf. [HS99a, HS99b]):
Corollary 1.2.
Any blow-up limit of a solution of (CF) is weakly convex. In particular, anytype-II blow-up limit of a solution of (CF) about a type-II singularity is a translation solutionof (CF) of the form X ∞ : ( R k × Γ n − k ) × R → R n +1 , k ∈ { , , . . . , n − } , such that X ∞ | Γ n − k is a strictly convex translation solution of (CF) in R n − k +1 . Motivated by [HS09, Section 5], we apply Theorem 1.1 to obtain the following family ofcylindrical estimates for solutions of (CF):
Theorem 1.3 (Cylindrical Estimate) . Let X be a solution of (CF) such that Conditions(i)–(ii) hold. Suppose also that X is ( m + 1) -convex for some m ∈ { , , . . . , n − } . That is, κ + · · · + κ m +1 ≥ βF for some β > . Then for all ε > there is a constant C ε > suchthat G m ( x, t ) ≤ εF ( x, t ) + C ε for all ( x, t ) ∈ M × [0 , T ) , where G m : M × [0 , T ) → R is given by a smooth, non-negative, degree one homogeneousfunction of the principal curvatures that vanishes at a point ( x, t ) if and only if κ ( x, t ) + · · · + κ m +1 ( x, t ) ≥ c m f ( κ ( x, t ) , . . . , κ n ( x, t )) , where c m is the value F takes on the unit radius cylinder, R m × S n − m . Theorem 1.3 implies that the ratio of the quantity K m := κ + · · · + κ m +1 − c m F to the speed is almost positive wherever the curvature is large. Observe that this quantityis non-negative on a weakly convex hypersurface Σ only if either Σ is strictly m − convex, YLINDRICAL ESTIMATES 3 or Σ = R m × S n − m . In particular, we find that whenever κ ( x, t ) + · · · + κ m ( x, t ) is smallcompared to the speed, the Weingarten curvature is close to that of a thin cylinder R m × S n − m .We obtain the following refinement of Corollary 1.2: Corollary 1.4.
Any blow-up limit of an ( m + 1) -convex, ≤ m ≤ n − , solution of (CF) iseither strictly m -convex, or a shrinking cylinder R m × S n − m . In particular, if the blow-up is oftype-II, then this limit is a translation solution of (CF) of the form X ∞ : (cid:0) R k × Γ n − k (cid:1) × R → R n +1 for k ∈ { , , . . . , m − } , such that X ∞ | Γ n − k is a strictly convex translation solution of (CF) in R n − k +1 . Huisken-Sinestrari obtained Theorem 1.3 for the mean curvature flow in the case m = 1,making spectacular use of it through their surgery program [HS09], which yielded a classifi-cation of 2-convex hypersurfaces.Moreover, the m = 0 case produces an analogue of Huisken’s curvature estimate for convexsolutions of the mean curvature flow [Hu84, Theorem 5.1]. This estimate implies that a convexsolution of (CF) becomes round at points of large curvature, which is crucial in proving thatsolutions contract to round points. This result was proved by different means for the class offlows considered here [An94a]. 2. Preliminaries
We will follow the notation used in [ALM13]. In particular, we recall that a smooth,symmetric function g of the principal curvatures gives rise to a smooth function G of thecomponents of the Weingarten map. Equivalently, G is an orthonormal frame invariantfunction of the components h ij of the second fundamental form. To simplify notation, wedenote G ( x, t ) ≡ G ( h ( x, t )) = g ( κ ( x, t )) and use dots to denote derivatives of functions ofcurvature as follows:˙ g k ( z ) v k = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 g ( z + sv ) ˙ G kl ( A ) B kl = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 G ( A + sB )¨ g pq ( z ) v p v q = d ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 g ( z + sv ) ¨ G pq,rs ( A ) B pq B rs = d ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 G ( A + sB ) . The derivatives of g and G are related in the following way (cf. [Ge90, An94a, An07]): Lemma 2.1.
Let g : Γ → R be a smooth, symmetric function. Define the function G : S Γ : → R by G ( A ) := g ( λ ( A )) , where λ ( A ) denotes the eigenvalues of A (up to order). Then for anydiagonal A with eigenvalues in Γ we have ˙ G kl ( A ) = ˙ g k ( λ ( A )) δ kl , (2.1) and for any diagonal A with distinct eigenvalues lying in Γ , and any symmetric B ∈ GL ( n ) ,we have ¨ G pq,rs ( A ) B pq B rs = ¨ g pq ( λ ( A )) B pp B qq + 2 X p>q ˙ g p ( λ ( A )) − ˙ g q ( λ ( A )) λ p ( A ) − λ q ( A ) (cid:0) B pq (cid:1) . (2.2)In particular, in an orthonormal frame of eigenvectors of W we have˙ G kl = ˙ g k δ kl ¨ G pq,rs B pq B rs = ¨ g pq B pp B qq + 2 X p>q ˙ g p − ˙ g q κ p − κ q (cid:0) B pq (cid:1) , BEN ANDREWS AND MAT LANGFORD where we are denoting ˙ G ≡ ˙ G ◦ h , etc.We note that ¨ g ≥ g p − ˙ g q )( z p − z q ) ≥ G is convex if and only if g is convex. Lemma 2.2.
Let X : M × [0 , T ) → R n +1 be a solution of (CF) such that f satisfies Conditions(i)–(ii). Let g : Γ → R be a smooth, degree zero homogeneous symmetric function. Then thereexists c > such that − c ≤ g ( κ ( x, t ) , . . . , κ n ( x, t )) ≤ c . for all ( x, t ) ∈ M × [0 , T ) .If g > , then there exists c > such that c ≤ g ( κ ( x, t ) , . . . , κ n ( x, t )) ≤ c . Proof.
Let Γ be a preserved cone for the solution X . Then K := Γ ∩ S n is compact. Since g is continuous, the required bounds hold on K . But these extend to Γ \ { } by homogeneity.The claim follows since κ ( x, t ) ∈ Γ \ { } for all ( x, t ) ∈ M × [0 , T ). (cid:3) By Condition (i), the derivative ˙ f of f is homogeneous of degree zero. Since ˙ f k > k , we obtain uniform parabolicity of the flow: Corollary 2.3.
There exists a constant c > such that for any v ∈ T ∗ M it holds that c | v | ≤ ˙ F ij v i v j ≤ c | v | , where | · | is the (time-dependent) norm on M corresponding to the (time-dependent) metricinduced by the flow. We now recall the following evolution equation (see for example [AMZ13]):
Lemma 2.4.
Let X : M × [0 , T ) → R n +1 be a solution of (CF) such that f satisfies Conditions(i)–(ii). Let G : M × [0 , T ) → R be given by a smooth, symmetric, degree one homogeneousfunction g of the principal curvatures. Then G satisfies the following evolution equation: ( ∂ t − L ) G = ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) ∇ h pq ∇ h rs + G |W| F (2.3) where L := ˙ F kl ∇ k ∇ l is the linearisation of F , and |W| F := ˙ F kl h kr h rl . In particular, the speed function F satisfies( ∂ t − L ) F = F |W| F . As we shall see, in order to obtain Theorem 1.3, it is crucial to obtain a good upper boundon the term Q ( ∇W , ∇W ) := ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) ∇ k h pq ∇ l h rs for the pinching functions G m which we construct in the following section. The followingdecomposition of Q is crucial in obtaining this bound. YLINDRICAL ESTIMATES 5
Lemma 2.5.
For any totally symmetric T ∈ R n ⊗ R n ⊗ R n , we have ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) (cid:12)(cid:12) B T kpq T lrs = ( ˙ g k ¨ f pq − ˙ f k ¨ g pq ) (cid:12)(cid:12) z T kpp T kqq + 2 X p>q ( ˙ f p ˙ g q − ˙ g p ˙ f q ) (cid:12)(cid:12) z z p − z q (cid:16) ( T pqq ) + ( T qpp ) (cid:17) + 2 X k>p>q ( ~g kpq × ~f kpq ) (cid:12)(cid:12) z · ~z kpq ( T kpq ) (2.4) at any diagonal matrix B with distinct eigenvalues z i , where ‘ × ’ and ‘ · ’ are the three dimen-sional cross and dot product respectively, and we have defined the vectors ~f kpq := ( ˙ f k , ˙ f p , ˙ f q ) , ~g kpq := ( ˙ g k , ˙ g p , ˙ g q ) , and ~z kpq := (cid:18) z p − z q ( z k − z p )( z k − z q ) , z k − z q ( z k − z p )( z p − z q ) , z k − z p ( z p − z q )( z k − z q ) (cid:19) . Proof.
Since B is diagonal, Lemma 2.1 yields (supressing the dependence on B )( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) T kpq T lrs = X k,p,q ( ˙ g k ¨ f pq − ˙ f k ¨ g pq ) T kpp T kqq + 2 X k X p>q ˙ g k ˙ f p − ˙ f q z p − z q − ˙ f k ˙ g p − ˙ g q z p − z q ! ( T kpq ) . BEN ANDREWS AND MAT LANGFORD
We now decompose the second term into the terms satisfying k = p , k = q , k > p , p > k > q ,and q > k respectively: X k X p>q ˙ g k ˙ f p − ˙ f q z p − z q − ˙ f k ˙ g p − ˙ g q z p − z q ! ( T kpq ) = X p>q ˙ g p ˙ f p − ˙ f q z p − z q − ˙ f p ˙ g p − ˙ g q z p − z q ! ( T ppq ) + X p>q ˙ g q ˙ f p − ˙ f q z p − z q − ˙ f q ˙ g p − ˙ g q z p − z q ! ( T qpq ) + X k>p>q + X p>k>q + X p>q>k ˙ g k ˙ f p − ˙ f q z p − z q − ˙ f k ˙ g p − ˙ g q z p − z q ! ( T kpq ) = X p>q ˙ f p ˙ g q − ˙ g p ˙ f q z p − z q (cid:16) ( T pqq ) + ( T qpp ) (cid:17) + X k>p>q ˙ g k ˙ f p − ˙ f q z p − z q − ˙ f k ˙ g p − ˙ g q z p − z q + ˙ g p ˙ f k − ˙ f q z k − z q − ˙ f p ˙ g k − ˙ g q z k − z q + ˙ g q ˙ f k − ˙ f p z k − z p − ˙ f q ˙ g k − ˙ g p z k − z p ! ( T kpq ) = X p>q ˙ f p ˙ g q − ˙ g p ˙ f q z p − z q (cid:16) ( T pqq ) + ( T qpp ) (cid:17) + X k>p>q (cid:18) ( ˙ g p ˙ f q − ˙ f q ˙ g p ) (cid:18) z k − z p − z k − z q (cid:19) − ( ˙ g k ˙ f q − ˙ f k ˙ g q ) (cid:18) z p − z q + 1 z k − z p (cid:19) + ( ˙ g k ˙ f p − ˙ f k ˙ g p ) (cid:18) z p − z q − z k − z q (cid:19)(cid:19) ( T kpq ) = X p>q ˙ f p ˙ g q − ˙ g p ˙ f q z p − z q (cid:16) ( T pqq ) + ( T qpp ) (cid:17) + X k>p>q ( ~g kpq × ~f kpq ) · ~z kpq ( T kpq ) . (cid:3) We complete this section by proving that ( m + 1)-convexity is preserved by the flow (CF),so that this assumption need only be made on initial data: Proposition 2.6.
Let X be a solution of (CF) such that Conditions (i)–(ii) are satisfied.Suppose that there is some m ∈ { , . . . , n − } and some β > such that κ σ (1) ( x,
0) + · · · + κ σ ( m ) ( x, ≥ βF ( x, for all x ∈ M and all permutations σ ∈ P n . Then this estimate persists at all later times.Proof. Denote by SM the unit tangent bundle over M × [0 , T ) and consider the function Z defined on ⊕ m SM by Z ( x, t, ξ , . . . ξ m ) = m X α =1 h ( ξ α , ξ α ) − βF ( x, t ) . Since we have inf ξ ,...,ξ m ∈ S ( x,t ) M Z ( x, t, ξ , . . . , ξ m ) = κ σ (1) ( x, t ) + · · · + κ σ ( m ) ( x, t ) − βF ( x, t )for some σ ∈ P n , it suffices to show that Z remains non-negative. First fix any t ∈ [0 , T )and consider the function Z ε ( x, t, ξ , . . . ξ m ) := Z ( x, t, ξ , . . . ξ m ) + ε e (1+ C ) t , where C :=sup M × [0 ,t ] |W| F . Note that C is finite since M is compact and ˙ F is bounded. Observe YLINDRICAL ESTIMATES 7 that Z ε is positive when t = 0. We will show that Z ε remains positive on M × [0 , t ] forall ε >
0. So suppose to the contrary that Z ε vanishes at some point ( x , t , ξ , . . . ξ m ). Wemay assume that t is the first such time. Now extend the vector ξ := ( ξ , . . . ξ m ) to a field ξ := ( ξ , . . . , ξ n ) near ( x , t ) by parallel translation in space and solving ∂ξ iα ∂t = F ξ jα h j i . Since the metric evolves according to ∂ t g ij = − h ij the resulting fields have unit length. Now recall (see for example [ALM13]) the followingevolution equation for the second fundamental form: ∂ t h ij = L h ij + ¨ F pq,rs ∇ i h pq ∇ j h rs + |W| F h ij − F h ij , where L := ˙ F kl ∇ k ∇ l and |W| F := ˙ F kl h kl . It follows that( ∂ t − L ) ( Z ε ( x, t, ξ )) = ε (1 + C )e (1+ C ) t + m X α =1 ¨ F pq,rs ∇ ξ α h pq ∇ ξ α h rs + |W ( x, t ) | F Z ( x, t, ξ ) ≥ ε (1 + C )e (1+ C ) t + |W ( x, t ) | F Z ( x, t, ξ ) . Since the point ( x , t , ξ t = t ) is a minimum of Z ε , we obtain0 ≥ ( ∂ t − L ) (cid:12)(cid:12) ( x ,t ) ( Z ε ( x, t, ξ )) ≥ ε (1 + C )e (1+ C ) t − Cε e (1+ C ) t = ε e (1+ C ) t > . This is a contradiction, implying that Z ε cannot vanish at any time in the interval [0 , t ].Since ε > Z ≥ , t ]. Since t ∈ [0 , T )was arbitrary, we obtain Z ≥ (cid:3) Constructing the pinching function.
In this section we construct the pinching functions G m satisfying the conditions in Theorem1.3. Let us first introduce the ‘pinching cones’Γ m := { z ∈ Γ : z σ (1) + · · · + z σ ( m +1) > c − m f ( z ) for all σ ∈ H m } , where H m is the quotient of P n , the group of permutations of the set { , . . . , n } , by theequivalence relation σ ∼ ω if σ ( { , . . . , m + 1 } ) = ω ( { , . . . , m + 1 } ) . Using the methods of [Hu84], and their adaptations to two-convex flows in [HS09] and fullynon-linear flows in [ALM13], we will see that, in order to prove Theorem 1.3, it suffices toconstruct a smooth function g m : Γ → R satisfying the following properties: Properties. (i) g m ( z ) ≥ for all z ∈ Γ with equality if and only if z ∈ Γ m ∩ Γ ;(ii) g m is smooth an homogeneous of degree one; BEN ANDREWS AND MAT LANGFORD (iii) for every ε > there exists c ε > such that for all diagonal matrices B and totallysymmetric 3-tensors T , it holds that ( ˙ G klm ¨ F pq,rs − ˙ F kl ¨ G pq,rsm ) (cid:12)(cid:12) B T kpq T lrs ≤ − c ε | T | F for all symmetric matrices B satisfying λ ( B ) ∈ Γ , and G m ( B ) ≥ εF ( B ) , where G m isthe matrix function corresponding to g m as described in Section 2, and Γ is a preservedcone for the flow; and(iv) for every δ > , ε > , and C > , there exist γ ε > and γ δ > such that ( G m ˙ F kl − F ˙ G klm ) (cid:12)(cid:12) B B kl ≤ − γ ε F ( G m − δF ) (cid:12)(cid:12) B + γ δ F (cid:12)(cid:12) B for all symmetric, ( m + 1) -positive matrices B satisfying λ ( B ) ∈ Γ , G m ( B ) ≥ εF ( B ) ,and λ min ( B ) ≥ − δF ( B ) − C . Our construction of the pinching function g m will be independent of the choice of m . Solet us fix m ∈ { , , . . . , n − } and assume that the flow is ( m + 1)-convex. We first considerthe preliminary function g : Γ → R defined by g ( z ) := f ( z ) X σ ∈ H m ϕ P m +1 i =1 z σ ( i ) − c m f ( z ) f ( z ) ! , (3.1)where ϕ : R → R is a smooth function which is strictly convex and positive, except on R + ∪ { } , where it vanishes identically. Such a function is readily constructed; for example,we could take ϕ ( r ) = ( r e − r if r <
00 if r ≥ . We note that such a function necessarily satisfies ϕ ( r ) − rϕ ′ ( r ) ≤ ϕ ′ ( r ) ≤ r ≥ G : M × [0 , T ) → R by G ( x, t ) := g ( κ ( x, t ) , . . . , κ n ( x, t )). Then G is a smooth, degree one homogeneous function of the components of the Weingarten mapwhich is invariant under a change of basis. Moreover, G is non-negative and vanishes at, andonly at, points for which the sum of the smallest ( m + 1)-principal curvatures is not less than c − m F . Thus Properties (i) and (ii) are satisfied by g .We now show that property (iii) is satisfied weakly by g : Lemma 3.1.
Let G be the matrix function corresponding to the function g defined by (3.1) .Then for any diagonal matrix B and totally symmetric 3-tensor T , it holds that ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) (cid:12)(cid:12) B T kpq T lrs ≤ Proof.
We will show that each of the terms in the decomposition (2.4) in Lemma 2.5 is non-positive. Note that it suffices to compute at matrices having distinct eigenvalues, since theresult at an arbitrary symmetric matrix B may be obtained by taking a limit B ( k ) → B suchthat each matrix B ( k ) has distinct eigenvalues. Thus we may assume that the eigenvalues In fact, ϕ need only be twice continuously dfferentiable. YLINDRICAL ESTIMATES 9 satisfy z < · · · < z n . We first compute,˙ g k = ˙ f k X σ ∈ H m ϕ ( r σ ) + X σ ∈ H m ϕ ′ ( r σ ) m +1 X i =1 (cid:18) δ σ ( i ) k − z σ ( i ) f ˙ f k (cid:19) = ˙ f k X σ ∈ H m ϕ ( r σ ) − ϕ ′ ( r σ ) P m +1 i =1 z σ ( i ) f ! + X σ ∈ H m m +1 X i =1 ϕ ′ ( r σ ) δ σ ( i ) k , ¨ g pq = X σ ∈ H m ϕ ( r σ ) − X σ ∈ H m ϕ ′ ( r σ ) P m +1 i =1 z σ ( i ) f ! ¨ f pq + X σ ∈ H m ϕ ′′ ( r σ ) f m +1 X i =1 (cid:18) δ σ ( i ) p − z σ ( i ) f ˙ f p (cid:19) m +1 X i =1 (cid:18) δ σ ( i ) q − z σ ( i ) f ˙ f q (cid:19) , where we are denoting r σ ( z ) := P m +1 i =1 z σ ( i ) − c − m f ( z ) f ( z ) . It follows that˙ g k ¨ f pq − ˙ f k ¨ g pq = X σ ∈ H m m +1 X i =1 ϕ ′ ( r σ ) δ σ ( i ) k ¨ f pq − ˙ f k X σ ∈ H m ϕ ′′ ( r σ ) f m +1 X i =1 (cid:18) δ σ ( i ) p − z σ ( i ) f ˙ f p (cid:19) m +1 X i =1 (cid:18) δ σ ( i ) q − z σ ( i ) f ˙ f q (cid:19) . If we fix the index k and set ξ p = T kpp , then, by convexity of ϕ and positivity of ˙ f k , we have − ˙ f k X σ ∈ H m ϕ ′′ ( r σ ) f m +1 X i =1 (cid:18) δ σ ( i ) p − z σ ( i ) f ˙ f p (cid:19) m +1 X i =1 (cid:18) δ σ ( i ) q − z σ ( i ) f ˙ f q (cid:19) ξ p ξ q = − ˙ f k X σ ∈ H m ϕ ′′ ( r σ ) f m +1 X i =1 (cid:18) δ σ ( i ) p − z σ ( i ) f ˙ f p (cid:19) ξ p ! ≤ . On the other hand, since ϕ is monotone non-increasing, and f is convex, we have ϕ ′ ( r σ ) m +1 X i =1 δ σ ( i ) k ¨ f pq ξ p ξ q ≤ σ . Since both inequalities hold for all k , we deduce that X k,p,q (cid:0) ˙ g k ¨ f pq − ˙ f k ¨ g pq (cid:1) T kpp T kqq ≤ . We next consider ˙ f p ˙ g q − ˙ g p ˙ f q = X σ ∈ H m m +1 X i =1 ϕ ′ ( r σ ) (cid:16) δ σ ( i ) q ˙ f p − δ σ ( i ) p ˙ f q (cid:17) = X σ ∈ O q ϕ ′ ( r σ ) ˙ f p − X σ ∈ O p ϕ ′ ( r σ ) ˙ f q Thus, if z p > z q , we obtain˙ f p ˙ g q − ˙ g p ˙ f q ≤ ˙ f p X σ ∈ O q ϕ ′ ( r σ ) − X σ ∈ O p ϕ ′ ( r σ ) . where we have introduced the sets O a := { σ ∈ H m : a ∈ σ ( { , . . . , m + 1 } ) } . We now showthat the term in brackets is non-positive whenever z p > z q : Lemma 3.2. If z p > z q , then X σ ∈ O p ϕ ′ ( r σ ) − X σ ∈ O q ϕ ′ ( r σ ) ≥ . Proof of Lemma 3.2.
First note that X σ ∈ O p ϕ ′ ( r σ ) − X σ ∈ O q ϕ ′ ( r σ ) = X σ ∈ O p,q ϕ ′ ( r σ ) − X σ ∈ O q,p ϕ ′ ( r σ ) , where O a,b := O a \ O b . Next observe that, if σ ∈ O p,q , then z σ (1) + · · · + z σ ( m +1) = z p + z ˆ σ ( i ) · · · + z ˆ σ ( i m ) (3.2)for some ˆ σ ∈ H m − ( p, q ) := P n − ( p, q ) / ∼ , where P n − ( p, q ) is the set of permutations of { , . . . , n } \ { p, q } , { i , . . . , i m } are a choice of m elements of { , . . . , n } \ { p, q } , and ∼ isdefined by ˆ σ ∼ ˆ ω if ˆ σ ( { i , . . . , i m } ) = ˆ ω ( { i , . . . , i m } ) . Observe also that the converse holds (that is, (3.2) defines a bijection), so that X σ ∈ O q,p ϕ ′ ( r σ ) − X σ ∈ O p,q ϕ ′ ( r σ ) = X ˆ σ ∈ H m − ( p,q ) (cid:20) ϕ ′ z p + P mk =1 z ˆ σ ( i k ) − c − m ff ! − ϕ ′ z q + P mk =1 z ˆ σ ( i k ) − c − m ff ! (cid:21) . Since z p > z q the claim follows from convexity of ϕ . (cid:3) Thus, X p>q ˙ f p ˙ g q − ˙ g p ˙ f q z p − z q (cid:16) ( T pqq ) + ( T qpp ) (cid:17) ≤ . We now compute ~g kpq = gf − X σ ∈ H m ϕ ′ ( r σ ) m +1 X i =1 z σ ( i ) f ! ~f kpq + X σ ∈ H m ϕ ′ ( r σ ) m +1 X i =1 (cid:0) δ σ ( i ) k , δ σ ( i ) p , δ σ ( i ) q (cid:1) , YLINDRICAL ESTIMATES 11 so that (cid:16) ~g kpq × ~f kpq (cid:17) · ~z kpq = X σ ∈ H m m +1 X i =1 ϕ ′ ( r σ ) h(cid:0) δ σ ( i ) k , δ σ ( i ) p , δ σ ( i ) q (cid:1) × ~f kpq i · ~z kpq = X σ ∈ H m m +1 X i =1 ϕ ′ ( r σ ) " ( δ σ ( i ) p ˙ f q − δ σ ( i ) q ˙ f p )( z p − z q )( z k − z p )( z k − z q )+ ( δ σ ( i ) q ˙ f k − δ σ ( i ) k ˙ f q )( z k − z q )( z k − z p )( z p − z q )+ ( δ σ ( i ) k ˙ f p − δ σ ( i ) p ˙ f k )( z k − z p )( z k − z q )( z p − z q ) . Removing the positive factor α kpq := [( z k − z p )( z z − z q )( z p − z q )] − and setting P a := P σ ∈ O a ϕ ′ ( r σ ), we obtain (cid:16) ~g kpq × ~f kpq (cid:17) · ~z kpq = α kpq h ( P p ˙ f q − P q ˙ f p )( z p − z q ) + ( P q ˙ f k − P k ˙ f q )( z p − z q ) + ( P k ˙ f p − P p ˙ f k )( z p − z q ) i . Applying Lemma 3.2 yields (cid:16) ~g kpq × ~f kpq (cid:17) · ~z kpq ≤ α kpq (cid:16) P q ˙ f k − P k ˙ f q (cid:17) (cid:2) ( z k − z q ) − ( z k − z p ) − ( z p − z q ) (cid:3) . Since the term in square brackets is non-negative, applying Lemma 3.2 once more yields (cid:16) ~g kpq × ~f kpq (cid:17) · ~z kpq ≤ . This completes the proof of the lemma. (cid:3)
In particular, Lemma 3.1 yields an upper bound for
G/F along the flow:
Corollary 3.3.
There exists C < ∞ such that G/F ≤ C along the flow.Proof. In view of Lemma 3.1 and the evolution equation (2.3) this is a simple application ofthe maximum principle. (cid:3)
In order to obtain the uniform estimate required by property (iii), we modify G in order toobtain a function with a strictly positive term in Q . A well-known trick (cf. [HS99b, Theorem2.14], [ALM13, Lemma 3.3]) then allows us to extract the required uniform estimate. First,we relabel the preliminary pinching funtion g → g ( G → G ), and consider the new pinchingfunction g defined by: g := K ( g , g ) := g g , (3.3)where g ( z ) = M P ni =1 z i − | z | for some large constant M >>
1, for which g is positive alongthe flow. That there is such a constant follows from applying the maximum principle to theevolution equation (2.3) for the function G ( x, t ) := g ( κ ( x, t )) as in [ALM13, Lemma 3.1].Note that ˙ K >
0, ˙ K < K > g > g to g . Wenow show that the estimates listed in Properties (iii) and (iv) are satisfied by the curvaturefunction defined in (3.3). Proposition 3.4.
Let g be the pinching function defined by (3.3) and G its correspondingmatrix function. Then, for every ε > there exists c ε > such that for all diagonal matrices B and totally symmetric 3-tensors T , it holds that ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) (cid:12)(cid:12) B T kpq T lrs ≤ − c ε | T | F whenever G ( B ) ≥ εF ( B ) .Proof. First note that (supressing dependence on B )( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) T kpq T lrs = ˙ K α ( ˙ G klα ¨ F pq,rs − ˙ F kl ¨ G pq,rsα ) T kpq T lrs − ˙ F kl ¨ K αβ ˙ G pqα ˙ G rsβ T kpq T lrs ≤ ˙ K ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) T kpq T lrs ≤ − ˙ K ˙ F kl ¨ G pq,rs T kpq T lrs , where we used Lemma 3.1, convexity of K , and the inequalities ˙ K ≥ F ≥ G ≥ K ≤
0, and convexity of F in the second.Since ˙ K < G > G is strictly concave in non-radial directions, the claimfollows from a well-known trick, exactly as in [ALM13, Lemma 3.3]. (cid:3) The uniform estimate of Proposition 3.4 yields a good bound for the term Q ( ∇W , ∇W ) inthe evolution equation for the pinching functions G . This is a crucial component in obtainingthe L p -estimates of the follwing section. These are the starting point for the Stampacchia-De Giorgi iteration argument. The second crucial estimate is the Poincar´e-type inequality,Lemma 4.2 (see also sections 4 and 5 of [HS09]; in particular, Lemma 5.5), which we can obtainwith the help of property (iv). This estimate (corresponding to Lemma 5.2 of [HS09]) providesan estimate on the zero order term that occurs in contracting the Simons-type identity for˙ F pq ∇ p ∇ q h ij with ˙ G ij (cf. [ALM13, Proposition 4.4]). Proposition 3.5.
Let g be the pinching function defined by (3.3) and G its correspondingmatrix function. Then, for every δ > , ε > , and C δ > there exist γ > and γ > suchthat ( F ˙ G kl − G ˙ F kl ) (cid:12)(cid:12) B B kl ≥ γ ε F ( G − δF ) (cid:12)(cid:12) B − γ δ F (cid:12)(cid:12) B for all symmetric, ( m + 1) -positive matrices B satisfying λ ( B ) ∈ Γ , G m ( B ) ≥ εF ( B ) , and λ min ( B ) ≥ − δF ( B ) − C δ .Proof. So let B be a symmetric, ( m + 1)-positive matrix with eigenvalues z ≤ · · · ≤ z n .Define Z ( B ) := F ˙ G ( B ) − G ˙ F ( B ). Then Z ( B ) = f ˙ g p z p − g ˙ f p z p = X p>q (cid:0) ˙ g p ˙ f q − ˙ g q ˙ f p (cid:1) z p z q ( z p − z q )= X p>q (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q )= X p>q>l + X p>l ≥ q + X l ≥ p>q (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q ) , YLINDRICAL ESTIMATES 13 where we recall the notation P a := P σ ∈ O a ϕ ′ ( r σ ) and we have defined l ≤ m as the numberof non-positive eigenvalues z i . Recalling that P p ˙ f q − P q ˙ f p ≥ z p ≥ z q , we discardthe final sum and part of the first to obtain Z ( B ) ≥ n X p = m +2 m +1 X q = l +1 (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q ) + n X p = l +1 l X q =1 (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q ) . Observe that when a ≤ m + 1, we have P a ≤ ϕ ′ (cid:18) z + · · · + z m +1 − c − m ff (cid:19) , which is strictly negative: for it can only vanish if z + · · · + z m +1 − c − m f ≥
0, in which case G ( B ) = 0, which contradicts G ( B ) ≥ εF ( B ) >
0. It follows that, for q ≤ m + 1, the term P p ˙ f q − P q ˙ f p ≥ ˙ f p ( P p − P q ) can only vanish if P p = P q , which will only occur if z p = z q since ϕ is strictly convex where it is positive (cf. Lemma 3.2). Since P p ˙ f q − P q ˙ f p is homogeneousof degree zero with respect to z , we obtain the uniform bound n X p = m +2 m +1 X q = l +1 (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q ) ≥ c n X p = m +2 m +1 X q = l +1 z p z q ( z p − z q )for some c >
0. On the other hand, again by homogeneity, the term P p ˙ f q − P q ˙ f p is alsobounded above (for all p , q ), in which case we obtain n X p = l +1 l X q =1 (cid:0) P p ˙ f q − P q ˙ f p (cid:1) z p z q ( z p − z q ) ≥ C n X p = l +1 l X q =1 z p z q ( z p − z q )for some C < ∞ . Agreeing to denote positive constants simply by c , we deduce Z ( B ) ≥ c n X p = l +1 l X q =1 z p z q ( z p − z q ) + n X p = m +2 m +1 X q = l +1 z p z q ( z p − z q ) (3.4)We control the first sum using the ‘convexity estimate’ z ≥ − δF − C δ as follows: n X p = l +1 l X q =1 z p z q ( z p − z q ) ≥ ( n − l ) z n l X q =1 z q ( z n − z q ) (3.5) ≥ n − l ) c F l X q =1 z q ≥ − n − l ) c F ( δF + C δ ) ≡ − cF ( δF + C δ ) , (3.6)where we estimated − c ≤ z i /F ≤ c for each i .Recall m ≤ n −
2. Then we may decompose the good second term in the brackets on theright hand side of (3.4) as n X p = m +2 m +1 X q = l +1 z p z q ( z p − z q ) = n X p = m +2 m +1 X q = l +1 z p z q ( z p − z q ) − F l X k =1 z k + F l X k =1 z k where l is again the number of non-positive eigenvalues. Consider first the term in thebrackets, S := P np = m +2 P m +1 q = l +1 z p z q ( z p − z q ) − F P lk =1 z k . Since each of the terms is non-negative, S can only vanish if z k = 0 for all k ≤ l and z p ( z p − z q ) = 0 for all p > q > l .That is, if there are no negative eigenvalues, and the positive ones (of which there are at least n − m ) are all equal. But this implies ( z + · · · + z k +1 ) − c − m f ≥
0, which in turn implies g = 0 < εf , a contradiction. We thus obtain a positive lower bound for the degree zerohomogeneous quantity S / ( F G ): S ≥ cF G for some c >
0. The remaining term is again easily estimated using the convexity estimate: S := F l X k =1 z k ≥ − cF ( δF + C δ ) . The claim follows. (cid:3)
We note that the above estimate is only useful in the presence of the convexity estimate,Theorem 1.1, since in that case, for any δ >
0, there is a constant C δ > δ,C δ := { z ∈ Γ : z i > − δf ( z ) − C δ for all i } is preserved by the flow.4. Proof of Theorem 1.3
In order to prove Theorem 1.3 it suffices to obtain for any ε > G ε,σ := (cid:18) GF − ε (cid:19) F σ for some σ >
0. We will use the estimates of Propositions 3.5 and 3.4 to obtain bounds onthe space-time L p -norms of the positive part of G ε,σ , so long as p is sufficiently large and σ sufficiently small, just as in [HS99a, HS99b, HS09] (see also [ALM13] where these techniquesare applied in the fully non-linear setting). The Stampacchia-De Giorgi iteration procedureintroduced in [Hu84] (see also [HS99a, ALM13]) then allows us to extract a supremum boundon G ε,σ .We recall the following evolution equation from [ALM13]: Lemma 4.1.
The function G ε,σ satisfies the following evolution equation: ( ∂ t − L ) G ε,σ = F σ − ( ˙ G kl ¨ F pq,rs − ˙ F kl ¨ G pq,rs ) ∇ k h pq ∇ l h rs + 2(1 − σ ) F h∇ G ε,σ , ∇ F i F − σ (1 − σ ) F |∇ F | F + σG ε,σ |W| F , (4.1) where h u, v i F := ˙ F kl u k u l . Now set E := max { G ε,σ , } . We need to obtain space-time L p -estimates for E . Let us firstobserve that integration by parts and application of Young’s inequality, in conjunction withLemma 2.2 and Proposition 3.4, yields the estimate (cf. [ALM13]) ddt Z E p dµ ≤ − (cid:16) A p ( p − − A p (cid:17) Z E p − |∇ G ε,σ | dµ − (cid:16) B p − B p (cid:17) Z E p |∇W| F dµ + C σp Z E p |W| dµ (4.2) YLINDRICAL ESTIMATES 15 for some positive constants A , A , B , B , C which are independent of σ and p .To estimate the final term, we use Proposition 3.5 in a similar manner to [HS09, Section5]. We first observe: Lemma 4.2.
There are positive constants A , A , A , B , B , C which are independent of p and σ such that: Z E p Z ( W ) F dµ ≤ (cid:0) A p + A p + A (cid:1) Z E p − |∇ G ε,σ | dµ + (cid:0) B p + B (cid:1) Z E p |∇W| F dµ . Proof.
As in [ALM13, Section 4], contraction of the commutation formula for ∇ W with ˙ F and ˙ G yields the identity L G ε,σ = − F σ − Q ( ∇W , ∇W ) + F σ − Z ( W ) + F σ − ( F ˙ G kl − G ˙ F kl ) ∇ k ∇ l F + σF G ε,σ L F − − σ ) F h∇ F, ∇ G ε,σ i F + σ (1 − σ ) F G ε,σ |∇ F | F . The claim is now proved using integration by parts and Young’s inequality, with the help ofLemma 2.2 and Propostion 3.4 (cf. [ALM13, Lemma 4.2]). (cid:3)
Corollary 4.3.
For all ε > there exist constants ℓ > and L > such that for all p > L and < σ < ℓp − there is a constant K = K ε,σ,p for which the following estimate holds: Z ( G ε,σ ) p + dµ ≤ Z ( G ε,σ ( · , p + dµ + tKµ ( M ) , where µ is the measure induced on M by the initial immersion.Proof. Recall Proposition 3.5. Setting δ = ε/ Z ( W ) F ≥ ε γ F − γ F whenever G − εF >
0. By Young’s inequality, for all σp > K σ,p suchthat F ≤ σpF + K σ,p F − σp , so that (cid:16) ε γ − σpγ (cid:17) F ≤ K σ,p F − σp + Z ( W ) F .
If we are careful to ensure σpγ ≤ εγ /
4, we obtain εγ F ≤ K σ,p F − σp + Z ( W ) F .
Since G ε,σ is bounded by F σ , and |W| is bounded by F , we obtain E p |W| ≤ K ε,σ,p + c ε E p Z ( W ) F , for some constants K ε,σ,p > ε , σ and p , and c ε > ε (butindependent of σ and p ). Combining Lemma 4.2 and inequality (4.2) now yields ddt Z E p dµ ≤ K ε,σ,p µ ( M ) − (cid:16) α p − α σp − α p − α p (cid:17) Z E p − | G ε,σ | dµ − (cid:16) β p − β σp − β σp − β p (cid:17) Z E p |∇W| F dµ . for some positive constants α i and β i , which depend on ε but not on σ or p , and K ε,σ,p , whichdepends on ε , σ and p .It is clear that L > ℓ > (cid:16) α p − α σp − α p − α p (cid:17) ≥ (cid:16) β p − β σp − β σp − β p (cid:17) ≥ p > L and 0 < σ < ℓp − . The claim then follows by integrating with respect to thetime variable. (cid:3) The proof of Theorem 1.3 is completed by proceeding with Huisken’s Stampacchia-DeGiorgi iteration scheme. We omit these details as the arguments required already appear in[ALM13, Section 5] with no significant changes necessary.
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E-mail address : [email protected] Mathematical Sciences Institute, Australian National University, ACT 0200 Australia
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