Two-sided non-collapsing curvature flows
aa r X i v : . [ m a t h . DG ] J a n TWO-SIDED NON-COLLAPSING CURVATURE FLOWS
BEN ANDREWS AND MAT LANGFORD
Abstract.
It was recently shown that embedded solutions of curvature flows in Euclideanspace with concave (convex), degree one homogeneous speeds are interior (exterior) non-collapsing [6]. These results were subsequently extended to hypersurface flows in the sphereand hyperbolic space [5]. In the first part of the paper, we show that locally convex solutionsare exterior non-collapsing for a larger class of speed functions than previously considered;more precisely, we show that the previous results hold when convexity of the speed functionis relaxed to inverse-concavity. We note that inverse-concavity is satisfied by a large classof concave speed functions [4]. As a consequence, we obtain a large class of two-sided non-collapsing flows, whereas previously two-sided non-collapsing was only known for the meancurvature flow.In the second part of the paper, we demonstrate the utility of two sided non-collapsing witha straightforward proof of convergence of compact, convex hypersurfaces to round points.The proof of the non-collapsing estimate is similar to the previous results mentioned, inthat we show that the exterior ball curvature is a viscosity supersolution of the linearisedflow equation. The new ingredient is the following observation: Since the function whichprovides an upper support in the derivation of the viscosity inequality is defined on M × M (or T M in the ‘boundary case’), whereas the exterior ball curvature and the linearised flowequation depend only on the first factor, we are privileged with a freedom of choice in whichsecond derivatives from the extra directions to include in the calculation. The optimal choiceis closely related to the class of inverse-concave speed functions. Introduction
We consider embedded solutions X : M n × [0 , T ) → N n +1 σ of curvature flows of the form ∂ t X ( x, t ) = − F ( x, t ) ν ( x, t ) , (CF)where N n +1 σ is the complete, simply connected Riemannian manifold of constant curvature σ ∈ {− , , } (that is, either hyperbolic space H n +1 , Euclidean space R n +1 , or the sphere S n +1 ), ν is a choice of normal field for the evolving hypersurface X , and the speed F is givenby a smooth, symmetric, degree one homogeneous function of the principal curvatures κ i of X which is monotone increasing with respect to each κ i . Equivalently, F is a smooth, monotoneincreasing, degree one homogeneous function of the Weingarten map W of X . Moreover, wewill always assume that F is normalised such that F (1 , . . . ,
1) = 1; however, this is merelya matter of convenience–all of the results hold, up to a recalibration of constants, in theun-normalised case.
Mathematics Subject Classification . 53C44, 35K55, 58J35.Research partially supported by Discovery grant DP120100097 of the Australian Research Council.The second author gratefully acknowledges the support of an Australian Postgraduate Award and an Aus-tralian National University HDR Supplementary Scholarship, and the support and hospitality of the Mathe-matical Sciences Center at Tsinghua University, Beijing, and the Department of Mathematics at East ChinaNormal University, Shanghai during the completion of part of this work.
The interior and exterior ball curvatures [6] of a family of embeddings X : M n × [0 , T ) → R n +1 with normal ν are, respectively, defined by k ( x, t ) := sup y = x k ( x, y, t ), and k ( x, t ) :=inf y = x k ( x, y, t ), where k ( x, y, t ) := 2 h X ( x, t ) − X ( y, t ) , ν ( x, t ) i|| X ( x, t ) − X ( y, t ) || . (1.1)Equivalently, k ( x, t ) (resp. k ( x, t )) gives the curvature of the largest region in R n +1 withtotally umbilic boundary that lies on the opposite (resp. same) side of the hypersurface X ( M, t ) as ν ( x, t ), and touches it at X ( x, t ) (with sign determined by ν ) [6, Proposition 4].Therefore, for a compact, convex embedding with outward normal, they are, respectively, thecurvature of the largest enclosed, and smallest enclosing spheres which touch the embeddingat X ( x, t ). It follows that κ max ≤ k and κ min ≥ k .For flows in Euclidean space, embedded, F > interior non-collapsing when the speed is a concave function of the Weingarten map, and exterior non-collapsing when the speed is a convex function of the Weingarten map [6]; more precisely, there exist k ∈ R , K > k ≥ k F in the former case, and k ≤ K F in the latter. Inparticular, solutions of mean convex mean curvature flow (in which case the speed is thetrace of the Weingarten map) are both interior and exterior non-collapsing. Two-sided non-collapsing has many useful consequences; for example, for uniformly convex hypersurfaces weobtain uniform pointwise bounds on the ratios of principal curvatures, and a uniform boundon the ratio of circumradius to in-radius. This leads to a new proof of Huisken’s theorem[19] on the convergence of convex hypersurfaces to round points. Moreover, one-sided non-collapsing can also provide useful information; for example, interior non-collapsing rules outcertain singularity models, such as products of the Grim Reaper curve with R n − . For convexspeeds, exterior non-collapsing is sufficient to obtain a bound on the ratio of circumradius toin-radius, and the proof of convergence of locally convex initial hypersurfaces to round points[2], is also simplified.More recently, it was shown that the above statements have natural analogues when theambient space is either the sphere or hyperbolic space [5]: Considering the sphere S n +1 asthe embedded submanifold { X ∈ R n +2 : h X, X i = 1 } of R n +2 , and hyperbolic space H n +1 as the embedded submanifold { X ∈ R n +1 , : h X, X i = − } of Minkowski space R n +1 , , thefunction k may be formally defined as in (1.1), except that now we take h · , · i and || · || tobe the inner product and induced norm on, in the case of the sphere, R n +2 , and, in the caseof hyperbolic space, the spacelike vectors in R n +1 , . Then, if F is a concave function of thecurvatures, there exists K > kF − η ≤ K e − σηt , and, if F is a convex function of the curvatures, there exists k ∈ R such that kF − η ≥ k e − σηt , where η > F .Let us recall the following definition [4]: Definition 1.1.
Let Ω be an open subset of the positive cone Γ + := { z ∈ R n : z i > i } .Define the set Ω ∗ := { ( z − , . . . , z − n ) } . Then a function f : Ω → R is called inverse-concave if the function f ∗ : Ω ∗ → R defined by f ∗ (cid:0) z − , . . . , z − n (cid:1) := f ( z , . . . , z n ) − is concave. WO-SIDED NON-COLLAPSING CURVATURE FLOWS 3
The main result of this article may now be stated as follows:
Theorem 1.2.
Let f : Γ + → R be a smooth, symmetric function which is homogeneous ofdegree one, and monotone increasing in each argument. Let X be a solution of (CF) withspeed given by F = f ( κ , . . . , κ n ) , where κ i are the principal curvatures of X . Then, if f isinverse-concave, X is exterior non-collapsing ; that is,(1) If N n +1 = R n +1 , then, for all ( x, t ) ∈ M × [0 , T ) , k ( x, t ) F ( x, t ) ≥ inf M ×{ } kF . (2) If N n +1 = S n +1 , and tr ( ˙ F ) ≤ η , then, for all ( x, t ) ∈ M × [0 , T ) , k ( x, t ) F ( x, t ) − η ≥ inf M ×{ } (cid:18) kF − η (cid:19) e − ηt . (3) If N n +1 = H n +1 , and tr ( ˙ F ) ≥ η , then, for all ( x, t ) ∈ M × [0 , T ) , k ( x, t ) F ( x, t ) − η ≥ inf M ×{ } (cid:18) kF − η (cid:19) e ηt , where ˙ F is the derivative of F with respect to the Weingarten map. In particular, combining Theorem 1.2 with the previous non-collapsing results [6, 5], wefind that solutions of flows in spaceforms by concave, inverse-concave speed functions are bothinterior and exterior non-collapsing. We note that concave speed functions satisfy tr ( ˙ F ) ≥ η = 1 in case (3) of Theorem 1.2.We note that the class of admissible speeds which are both concave and inverse-concave issurprisingly large [4], and includes, for example, the degree one homogeneous ratios and rootsof the elementary symmetric polynomials.The authors wish to express their thanks to Chen Xuzhong and Yong Wei for their helpfulcomments and suggestions on earlier versions of this work.2. Proof of Theorem 1.2
We first extend (cf. [6]) k ( · , · , t ) to a continuous function on the compact manifold withboundary c M . As a set, c M := ( M × M \ D ) ⊔ SM , where D := { ( x, x ) : x ∈ M } is thediagonal submanifold and SM is the unit tangent bundle with respect to the metric at time t . The manifold-with-boundary structure is defined by the atlas generated by all charts for( M × M ) \ D , together with the charts b Y defined by b Y ( z, s ) := (cid:0) exp( sY ( z )) , exp( − sY ( z )) (cid:1) for s sufficiently small, where Y is a chart for SM . The extension is then given by setting k ( x, y, t ) := W ( x,t ) ( y, y ) for ( x, y ) ∈ S ( x,t ) M .We also recall some useful notation [6]; namely, we define d ( x, y, t ) := || X ( x, t ) − X ( y, t ) || and w ( x, y, t ) := X ( x, t ) − X ( y, t ) || X ( x, t ) − X ( y, t ) || , and use scripts x and y to denote quantities pulled back to M × M by the respective projectionsonto the first and second factor. With this notation in place, k may be written as k = 2 d h dw, ν x i . Theorem 1.2 is a direct consequence of the following proposition:
BEN ANDREWS AND MAT LANGFORD
Proposition 2.1.
If the flow speed F is inverse-concave, then the exterior ball curvature k is a viscosity supersolution of the equation ∂ t u = L u + (cid:16) |W| F − σ tr ( ˙ F ) (cid:17) u + 2 σF , (2.1) where L := ˙ F ij ∇ i ∇ j , h u, v i F := ˙ F ij v i v j , and |W| F := ˙ F ij W ij . We note that the speed function satisfies the equation ∂ t F = L F + (cid:16) |W| F − σ tr ( ˙ F ) (cid:17) F under the flow [3]. Proof of Proposition 2.1.
Consider, for an arbitrary point ( x , t ) ∈ M × [0 , T ), an arbitrarylower support funtion φ for k at ( x , t ); that is, φ is C , on a backwards parabolic neigh-bourhood P := U x × ( t − ε, t ] of ( x , t ), and φ ≤ k with equality at ( x , t ). Then we needto prove that the differential inequality ∂ t φ ≥ L φ + (cid:16) |W| F − σ tr ( ˙ F ) (cid:17) φ + 2 σF holds at ( x , t ).We note that k ( x, y, t ) ≥ k ( x, t ) ≥ φ ( x, t ) for all ( x, y, t ) ∈ c M × [0 , T ) such that ( x, t ) ∈ P ,and, since k is continuous and c M is compact, we either have k ( x , t ) = k ( x , y , t ) for some y ∈ M \ { x } , or k ( x , t ) = W ( x ,t ) ( y , y ) for some y ∈ S ( x ,t ) M . We consider the formercase first. The interior case.
We first suppose that inf M k ( x , · , t ) < k ( x , y , t ) for all boundary points ( x , y ) of c M . In that case, we have κ ( x , t ) > k ( x , t ) = k ( x , y , t ) for some y ∈ M \ { x } , and k ( x, y, t ) ≥ k ( x, t ) ≥ φ ( x, t ) for all ( x, t ) ∈ P and all y ∈ M \ { x } . In particular, we have theinequalities ∂ t ( k − φ ) ≤ , and c L ( k − φ ) ≥ x , y , t ) for any elliptic operator c L on M × M . We would like c L to project to L on thefirst factor. This leads us to consider operators of the form c L = ˙ F ijx ∇ ∂ xi +Λ ip ∂ yp ∇ ∂ xj +Λ jq ∂ yq ,where Λ is any n × n matrix.We note that, in both of the cases σ = ±
1, the ambient Euclidean/Minkowskian derivativedecomposes into tangential and normal components as D = D − g ⊗ ν , where D , g , and ν are, respectively, the induced connection, metric, and outer/future-pointing normal of N n +1 σ with respect to its embedding. Using the fact that h ν, ν i = σ , and that the ambient positionvector is normal to N n +1 σ , a straightforward computation yields( ∂ x i + Λ ip ∂ y p ) k = 2 d (cid:0)(cid:10) ∂ xi − Λ ip ∂ yp , ν x − kdw (cid:11) + (cid:10) dw, W xip ∂ xp (cid:11)(cid:1) . (2.3)If we choose the co¨ordinates { x i } ni =1 and { y i } ni =1 to be orthonormal co¨ordinates (with respectto the induced metric g at time t ) centred at x and y respectively, then a further straight-forward computation using the vanishing of (2.3) at ( x , y , t ) and the Codazzi equation WO-SIDED NON-COLLAPSING CURVATURE FLOWS 5 yields ∇ ∂ xj +Λ jq ∂ yq ∇ ∂ xi +Λ ip ∂ yp k = 2 d n h−W xij ν x − σδ ij X x + Λ ip Λ jq ( W ypq ν y + σδ pq X y ) , ν x − kdw i + (cid:10) ∂ xi − Λ ip ∂ yp , W xjq ∂ xq (cid:11) − ( ∂ x j + Λ jq ∂ y q ) k (cid:10) ∂ xi − Λ ip ∂ yp , dw (cid:11) − k (cid:10) ∂ xi − Λ ip ∂ yp , ∂ xj − Λ j q ∂ yq (cid:11) + (cid:10) ∂ xj − Λ j q ∂ yq , W xip ∂ xp (cid:11) + h dw, ∇W xij − σ W xij X x − W xir W xrj ν x i− ( ∂ x i + Λ ip ∂ y p ) k (cid:10) ∂ xj − Λ j q ∂ yq , dw (cid:11) o (2.4)at the point ( x , y , t ).Next, noting that the normal satisfies D t ν = D t ν + σF X = grad F + σF X , we compute ∂ t k = 2 d (cid:0) h− F x ν x + F y ν y , ν x − kdw i + h dw, grad F x + σF X x i (cid:1) . (2.5)Combining (2.4) and (2.5), we obtain (cid:16) ∂ t − c L (cid:17) k = 2 d n D F y ν y − ˙ F ijx Λ ip Λ jq ( W ypq ν y + σδ pq X y ) , ν x − kdw E + k ˙ F ijx (cid:10) ∂ xi − Λ ip ∂ yp , ∂ xj − Λ jq ∂ yq (cid:11) − F ijx (cid:10) ∂ xj − Λ j q ∂ yq , W xip ∂ xp (cid:11) + σ tr ( ˙ F x ) h X x , ν x − kdw i + 2 σF x h X x , dw i o + 4 d ˙ F ijx ∇ ∂ xi +Λ ip ∂ yp k (cid:10) ∂ xj − Λ j q ∂ yq , dw (cid:11) + |W x | F k at the point ( x , y , t ).We now note that the vanishing of the y -derivatives at an off-diagonal extremum y ∈ M of k ( x , · , t ) determines the tangent plane to X at y : Lemma 2.2 ([6, 5]) . Suppose that a point ( x, y, t ) is an off-diagonal extremum of k ; that is, y = x is an extremum of k ( x, · , t ) . Then span { ∂ xi − h ∂ xi , w i w } ni =1 = span { ∂ yi } ni =1 at ( x, y, t ) , where { ∂ xi } ni =1 and { ∂ yi } are bases for T x M and T y M respectively.Proof of Lemma 2.2. We may assume that { ∂ xi } ni =1 and { ∂ yi } ni =1 are orthonormal. Then { ∂ xi − h ∂ xi , w i w } ni =1 is also orthonormal; note also that || ν x − kdw || = 1. Next, observe that thevanishing of ∂ y i k implies h ∂ yi , ν x − kdw i = 0for each i . If σ = 0, a short computation, using d = 2( σ − h X x , X y i ), yields h X y , ν x − kdw i = 0 . Thus, the orthogonal compliment of span { ∂ yi } ni =1 is span { σX y , ν x − kdw } . On the other hand,one easily computes h ∂ xi − h ∂ xi , w i w, ν x − kdw i = 0for each i , and, for σ = 0, h ∂ xi − h ∂ xi , w i w, X y i = 0 . Thus, span { ∂ xi − h ∂ xi , w i w } ⊥ = span { σX y , ν x − kdw } . The claim follows. (cid:3) BEN ANDREWS AND MAT LANGFORD
Thus, without loss of generality, we may assume ∂ yi = ∂ xi − h ∂ xi , w i w at ( x , y , t ). Note also that, when σ = 0,2 d h X x , ν x − kdw i| ( x ,y ,t ) = 2 d h X x − X y , ν x − kdw i| ( x ,y ,t ) = − k ( x , y , t ) . Finally, observe that (2.3) implies 2 d h dw, ∂ xi i = R ip ∂ x p k . Using these observations, and the vanishing of ∂ y i k , we obtain (cid:16) ∂ t − c L (cid:17) k = (cid:16) |W x | F − σ tr ( ˙ F x ) (cid:17) k + 2 σF x + 2 ˙ F ijx ∂ x i k R j p ∂ x p k + 2 d n F y − F x + ˙ F ijx (cid:2) ( kδ ij − W xij ) − ip ( kδ pj − W xpj ) + Λ ip Λ jq ( kδ pq − W ypq ) (cid:3)o (2.6)at any off-diagonal extremum ( x , y , t ), where we have defined R := ( W x − kI ) − with I denoting the identity.Applying the inequalities (2.2), we obtain0 ≥ ( ∂ t − c L )( k − φ ) ≥ − ( ∂ t − L ) φ + (cid:16) |W x | F − σ tr ( ˙ F x ) (cid:17) k + 2 σF x + 2 ˙ F ijx ∂ i k R j p ∂ p k + 2 d n F y − F x + ˙ F ijx h ( kδ ij − W xij ) − ip ( kδ pj − W xpj ) + Λ ip Λ jq ( kδ pq − W ypq ) io . It remains to demonstrate non-negativity of the term on the second line for some choice of thematrix Λ. Since we are free to choose the orthonormal basis at y such that W is diagonalised,this follows from the following proposition. Proposition 2.3.
Let f : Γ + → R be a smooth, symmetric function which is monotoneincreasing in each variable and inverse-concave, and let F : C + → R be the function definedon the cone C + of positive definite symmetric matrices by F ( A ) = f ( λ ( A )) , where λ denotesthe eigenvalue map. Then for any k ∈ R , any diagonal B ∈ C + , and any A ∈ C + with k < min i { λ i ( A ) } , we have ≤ F ( B ) − F ( A ) + ˙ F ij ( A ) sup Λ h ( kδ ij − A ij ) − ip ( kδ pj − A pj ) + Λ ip Λ j q ( kδ pq − B pq ) i . Proof of Proposition 2.3.
Since the expression in the square brackets is quadratic in Λ, it iseasy to see that the supremum is attained with the choice Λ = ( A − kI ) · ( B − kI ) − , where I denotes the identity matrix. Thus, given any A ∈ C + , we need to show that0 ≤ Q A ( B ) := F ( B ) − F ( A ) − ˙ F ij ( A ) (cid:16) ( A − kI ) ij − (cid:2) ( A − kI ) · ( B − kI ) − · ( A − kI ) (cid:3) ij (cid:17) . Since B is diagonal, and the expression Q A ( B ) is invariant under similarity transformationswith respect to A , we may diagonalise A to obtain Q A ( B ) := f ( b ) − f ( a ) − ˙ f i ( a ) (cid:20) ( a i − k ) − ( a i − k ) b i − k (cid:21) , WO-SIDED NON-COLLAPSING CURVATURE FLOWS 7 where we have set a = λ ( A ) and b = λ ( B ). We are led to consider the function q a defined onΓ + by q a ( z ) := f ( z ) − f ( a ) − ˙ f i ( a ) (cid:20) ( a i − k ) − ( a i − k ) z i − k (cid:21) . We compute ˙ q ia = ˙ f i − ˙ f i ( a ) ( a i − k ) ( z i − k ) , and ¨ q ija = ¨ f ij + 2 ˙ f i ( a ) ( a i − k ) ( z i − k ) δ ij = ¨ f ij + 2 ˙ f i δ ij z i − k − q ia δ ij z i − k . It follows that ¨ q ija + 2 ˙ q ia δ ij z i − k = ¨ f ij + 2 ˙ f i δ ij z i − k > ¨ f ij + 2 ˙ f i δ ij z i ≥ , (2.7)where the last inequality follows from inverse-concavity of f [4, Corollary 5.4]. Thus theminimum of q is attained at the point z = a , where it vanishes. This completes the proof. (cid:3) This completes the proof in the interior case.
The boundary case (Cf. [4, Theorem 3.2]).We now consider the case that inf M k ( x , · , t ) occurs on the boundary of c M ; that is, k ( x , t ) = W ( x ,t ) ( y , y ) for some y ∈ S ( x ,t ) M . Consider the function K defined on T M × [0 , T ) by K ( x, y, t ) = W ( x,t ) ( y, y ). Then the function Φ( x, y, t ) := φ ( x, t ) g ( x,t ) ( y, y ) is alower support for K at ( x , y , t ). In particular, ∂ t ( K − Φ) ≤ c L ( K − Φ) ≥ x , y , t ) for any elliptic operator c L on T M . We require the operator project to L onthe first factor (at least at the point ( x , y , t )), which leads us to consider an operator c L locally of the form c L = ˙ F ijx ( ∂ ix − Λ ip ∂ y p )( ∂ jx − Λ j q ∂ y q ), where { x i , y i } ni =1 are co¨ordinates for T M near ( x , y ). We choose these co¨ordinates such that { x i } ni =1 are normal co¨ordinateson M (with respect to g t ) based at x , and { y i } ni =1 are the corresponding fibre co¨ordinates(defined by ( x, y ) = ( x, y i ∂ x i ) for tangent vectors ( x, y ) near ( x , y )). Moreover, we mayassume that { ∂ x i | x } ni =1 is a basis of eigenvectors of W ( x ,t ) with y = ∂ x | x .Writing locally K − Φ = y k y l ( W kl − g kl ), we find( ∂ x i − Λ ip ∂ y p )( K − Φ) = y k y l ( ∂ x i W kl − ∂ x i φ g kl ) − ip y k ( W kp − φ g kp ) . Thus, at the point ( x , y , t ), we obtain0 = ( ∂ x i − Λ ip ∂ y p )( K − Φ) = ∇ i W − ∇ i φ . BEN ANDREWS AND MAT LANGFORD
We next compute( ∂ x i − Λ ip ∂ y p )( ∂ x j − Λ jq ∂ y q )( K − Φ) = y k y l (cid:0) ∂ x i ∂ x j W kl − ∂ x i ∂ x j φ g kl − ∂ x i φ ∂ x j g kl − ∂ x j φ ∂ x i g kl − φ ∂ x i ∂ x j g kl (cid:1) − j q y k ( ∂ x i W kq − ∂ x i φ g kq − φ ∂ x i g kp ) − ip y k ( ∂ x j W kp − ∂ x j φ g kp − φ ∂ x j g kp )+ 2Λ ip Λ jq ( W pq − φ g pq ) . At the point ( x , y , t ), we obtain c L ( K − Φ) = L W − L φ − F ij (cid:2) ip ( ∇ j W p − ∇ j W δ p ) − Λ ip Λ j q ( W pq − W δ pq ) (cid:3) . (2.9)Finally, we compute the time derivative ∂ t ( K − Φ) = y k y l (cid:0) ∂ t W kl − ∂ t φ g kl − φ ∂ t g kl (cid:1) , which at ( x , y , t ) becomes ∂ t ( K − Φ) = ∂ t W − W ∂ t g − ∂ t φ . (2.10)Let us recall the evolution equations for W and g [2, 3]: ∂ t W ij = L W ij + ¨ F pq,rs ∇ i W pq ∇ j W rs − W ij F + (cid:16) W ij − σ tr ( ˙ F ) (cid:17) |W| F + 2 σ W ij , (2.11)and ∂ t g ij = − F W ij . (2.12)Putting (2.9) and (2.10) together, and applying the evolution equations (2.11) and (2.12),and the inequalities (2.8), we obtain0 ≥ ( ∂ t − L )( K − Φ) = − ( ∂ t − L ) φ + ( |W| F − σ tr ( ˙ F )) φ + 2 σF + ¨ F pq,rs ∇ W pq ∇ W rs + 2 ˙ F ij Λ ip h ∇ j W p − ∇ j W δ p ) − Λ jq ( W pq − W δ pq ) i (2.13)at the point ( x , y , t ). Note that the term in the last line with p = 1 vanishes.Using a trick of Brendle [12, Proposition 8] (see also [8, Theorem 7]) we also obtain ∇ W =0 at the point ( x , t ): Lemma 2.4. ∇ W vanishes at ( x , y , t ) .Sketch proof of Lemma 2.4. Since κ ( x , t ) = inf y = x k ( x , y, t ), we have0 ≤ Z ( x , y, t ) := 2 h X ( x , t ) − X ( y, t ) , ν ( x , t ) i − κ ( x , t ) || X ( x , t ) − X ( y, t ) || for all y ∈ M . In particular, 0 ≤ f ( s ) := Z ( x , γ ( s ) , t ) for all s , where γ ( s ) := exp x sy .It is straightforward to compute 0 = f (0) = f ′ (0) = f ′′ (0), which, since f ≥
0, implies that f ′′′ (0) = 0. But a further straightforward computation yields f ′′′ (0) = 2 ∇ W . (cid:3) Applying the following proposition to (2.13) completes the proof.
WO-SIDED NON-COLLAPSING CURVATURE FLOWS 9
Proposition 2.5.
Let f : Γ + → R be a smooth, symmetric function which is monotoneincreasing in each variable and inverse-concave, and let F : C + → R be the function definedon the cone C + of positive definite symmetric matrices by F ( A ) = f ( λ ( A )) , where λ denotesthe eigenvalue map. If A ∈ C + and y is an eigenvector of A corresponding to its smallesteigenvalue, then, for any totally symmetric 3-tensor T with T ( y, y, y ) = 0 , we have ≤ y i y j ¨ F pq,rs T ipq T jrs + 2 ˙ F kl sup Λ h kp y i ( T ilp − y r y s T lrs δ iq ) − Λ kp Λ lq ( A pq − y r y s A rs δ pq ) i at the matrix A . Moreover, equality holds only if T ( v, y, y ) = 0 for all v ∈ R n .Proof of Proposition 2.5. We first observe that it suffices to prove the claim for those A ∈ C + having distinct eigenvalues: The expression Q := 2 ˙ F kl h kp y i ( T ilp − y r y s T lrs δ iq ) − Λ kp Λ lq ( A pq − y r y s A rs δ pq ) i is continuous in A , and hence the supremum over Λ is upper semi-continuous in A ; so thegeneral case follows by taking a sequence of matrices A ( k ) ∈ C + approaching A with each A ( k ) having distinct eigenvalues.So suppose that A has distinct eigenvalues and let { e i } ni =1 be an orthonormal frame ofeigenvectors of A with e = y . Then Q = 2 ˙ F kl h kp ( T lp − T l δ q ) − Λ kp Λ lq ( A pq − A δ pq ) i . Observe that the supremum over Λ occurs when Λ lq = ( λ q − λ ) − T lq for i, p >
1. With thischoice, we obtain Q = 2 ˙ F kl R pq T kq T lp , where R pq := ( λ p − λ ) − δ pq for p, q = 1 and zero otherwise. Therefore, it suffices to provethat 0 ≤ (cid:16) ¨ F pq,rs + 2 ˙ F pr R qs (cid:17) B pq B rs for any symmetric B with B = 0 with equality only if B q = 0 for all q . The expression wewant to estimate may be written in terms of the function f as follows (see, for example, [4,Theorem 5.1]): (cid:16) ¨ F pq,rs + 2 ˙ F pr R qs (cid:17) B pq B rs = ¨ f pq B pp B qq + X p = q ˙ f p − ˙ f q λ p − λ q B pq + 2 n X p =1 , q =2 ˙ f p λ q − λ B pq = ¨ f pq B pp B qq + 2 X p> , q> ˙ f p δ pq λ p − λ B pp B qq + X p = q ˙ f p − ˙ f q λ p − λ q B pq + 2 n X p =2 ˙ f λ p − λ B p + 2 X p> , q> ,p = q ˙ f p λ q − λ B pq . We first estimate¨ f pq B pp B qq + 2 X p> ,q> ˙ f p δ pq λ p − λ B pp B qq ≥ ¨ f pq B pp B qq + 2 n X p =2 ,q =2 ˙ f p λ p δ pq B pp B qq = ¨ f pq + 2 ˙ f p λ p δ pq ! B pp B qq ≥ , where the final inequality follows from inverse-concavity of f [4, Theorem 2.1]. The remainingterms are X p = q ˙ f p − ˙ f q λ p − λ q B pq + 2 n X p =2 ˙ f λ p − λ B p + 2 X p> , q> ,p = q ˙ f p λ q − λ B pq = X p> , q> ,p = q ˙ f p − ˙ f q λ p − λ q + 2 ˙ f p λ q − λ ! B pq + 2 n X p =2 ˙ f p − ˙ f λ p − λ + ˙ f λ p − λ ! B p ≥ X p> , q> ,p = q ˙ f p − ˙ f q λ p − λ q + ˙ f p λ q + ˙ f q λ p ! B pq + 2 n X p =2 ˙ f p λ p − λ ! B p . The first term is non-negative by inverse-concavity of f [4, Corollary 5.4] and the secondterm is clearly non-negative and vanishes only if B q = 0 for all q >
1. This completes theproof. (cid:3)
This completes the proof that k is a viscosity supersolution of (2.1). (cid:3) Since the speed function satisfies (2.1), the statement of Theorem 1.2 follows from a simplecomparison argument for viscosity solutions (cf. [6, 5]):Define ϕ ( t ) := e σηt (cid:0) inf M ×{ t } k/F − /η (cid:1) , where η > ση ≥ σ tr ( ˙ F ); thatis, η ≥ tr ( ˙ F ) for flows in the sphere, η ≤ tr ( ˙ F ) for flows in hyperbolic space, and η > ϕ is non-decreasing. It suffices to provethat k ( · , t ) − (cid:0) /η + e − σηt ϕ ( t ) − ε e L ( t − t ) (cid:1) F ( · , t ) > t ∈ [0 , T ) , t ∈ [ t , T )and any ε >
0, where we have set L := 1 − ση . Taking ε → k ( · , t ) − (cid:0) /η + e − σηt ϕ ( t ) (cid:1) F ( · , t ) ≥
0; that is, ϕ ( t ) ≥ ϕ ( t ) for all t ≥ t for any t . Now, attime t we have k ( x, t ) − (cid:0) /η + e − σηt ϕ ( t ) − ε (cid:1) F ( x, t ) ≥ εF ( x, t ) >
0. So suppose,contrary to the claim, that there is a point ( x , t ) ∈ M × [ t , T ) and some ε > k ( x , t ) − (cid:0) /η + e − σηt ϕ ( t ) − ε e t − t (cid:1) F ( x , t ) = 0. Assuming that t is the first suchtime, this means precisely that the function φ ( x, t ) := (cid:0) /η + e − σηt ϕ ( t ) − ε e L ( t − t ) (cid:1) F ( x, t )is a lower support for k at ( x , t ). But k is a viscosity supersolution of (2.1), so that, at thepoint ( x , t ), φ satisfies0 ≥ − ( ∂ t − L ) φ + (cid:16) |W| F − σ tr ( ˙ F ) (cid:17) φ + 2 σF = (cid:16) ησ e − σηt ϕ ( t ) + Lε e L ( t − t ) (cid:17) F − (cid:18) η + e − σηt ϕ ( t ) − ε e L ( t − t ) (cid:19) (cid:16) |W| F + σ tr ( ˙ F ) (cid:17) + (cid:18) η + e − σηt ϕ ( t ) − ε e L ( t − t ) (cid:19) F (cid:16) |W| F − σ tr ( ˙ F ) (cid:17) + 2 σF = 2e − σηt ϕ ( t ) F (cid:16) ση − σ tr ( ˙ F ) (cid:17) + ε e L ( t − t ) F (cid:16) L + 2 σ tr ( ˙ F ) (cid:17) + 2 η F (cid:16) ση − σ tr ( ˙ F ) (cid:17) ≥ ε e L ( t − t ) > , where we used L := 1 − ση , and ση ≥ σ tr ( ˙ F ) in the last line. This contradiction provesthat φ could not have reached zero on [ t , T ), which, as explained above, proves that ϕ is WO-SIDED NON-COLLAPSING CURVATURE FLOWS 11 non-decreasing. Therefore, (cid:18) k ( x, t ) F ( x, t ) − η (cid:19) e σηt ≥ inf M ×{ t } (cid:18) kF − η (cid:19) e σηt =: ϕ ( t ) ≥ ϕ (0) = inf M ×{ } (cid:18) kF − η (cid:19) . This proves Theorem 1.2.3.
Convex solutions in Euclidean space
We now give an application of non-collapsing to flows of convex hypersurfaces; namely, wegive a new proof that convex hypersurfaces contract to round points under the flow (CF) inEuclidean space when the speed is both concave and inverse-concave [4].We begin with some background results on fully non-linear curvature flows (CF). Givensmooth, compact initial data on which F is defined, we obtain unique solutions for a shorttime [2]. Since we can enclose the initial hypersurface by a large sphere, which shrinks toa point in finite time, the avoidance principle (see, for example, [6, Theorem 5]) impliesthat the maximal time T of existence of the solution must be finite. For inverse-concavespeeds, the non-collapsing estimate yields a preserved cone of curvatures for the flow, since κ min ≥ k ≥ k F . This implies that the flow is uniformly parabolic, since positive bounds for˙ F on the intersection of the preserved cone with the unit sphere {|W| = 1 } extend to boundson the entire cone. If F is also a concave function, then global regularity of solutions maybe obtained by appealing to the scalar parabolic theory of Krylov-Safanov [21] and Evansand Krylov [17, 20] (cf. [7, 9]). We conclude that the solution will remain smooth untilmax M ×{ t } F → ∞ as t → T .The key to our proof of the convergence theorem is showing that the normalised interiorand exterior ball curvatures improve to unity at a singularity. This is achieved using a blow-upargument and applying the strong maximum principle. Theorem 3.1.
Suppose F is concave and inverse-concave. Then along any convex solution X : M n × [0 , T ) → R n +1 of (CF) the following estimates hold:(1) For every ε > there exists F ε < ∞ such that F > F ε ⇒ k ≤ (1 + ε ) F . (2) For every δ > there exists F δ < ∞ such that F > F δ ⇒ k ≥ (1 − δ ) F .
Proof.
We will blow the solution up at a point where F is becoming large. Applying thestrong maximum principle, and making use of the gradient term appearing in (2.1), we findthat this limit must be a shrinking sphere, from which the claims follow. We note that theonly auxillary result we require is the fact that the only closed, convex hypersurfaces of R n with F constant are spheres [13]. When F is the mean curvature, this is a well-known theoremof Alexandrov [1].Suppose the first estimate were false. Then there exists a sequence ( x i , t i ) ∈ M × [0 , T )such that F ( x i , t i ) → ∞ but kF ( x i , t i ) → (1 + ε ), where ε >
0. By Theorem 1.2, ε < ∞ .Set λ i := F ( x i , t i ) and consider the blow-up sequence X i ( x, t ) := λ i (cid:0) X (cid:0) x, λ − i t + t i (cid:1) − X i ( x i , t i ) (cid:1) . It is easily checked that X i : M n × (cid:2) − λ i t i , → R n +1 is a solution of the flow (CF) for each i .Moreover, for each i , we have max M × [ − λ i t i , F i = F i ( x i ,
0) = 1 and X i ( x i ,
0) = 0. It follows that the sequence X i converges locally uniformly along a subsequence to a smooth limit flow X ∞ : M ∞ × ( −∞ , → R n +1 (cf. [11, 14, 15]).Since the ratio k/F is invariant under rescaling, we have k i F i ( x i ,
0) = kF ( x i , t i ) ≥ k > , which implies that the image of each X i is contained in a compact set. It follows that theconvergence is global, so that M ∞ ∼ = M .We now show that k/F must be constant on the limit flow X ∞ ; for if not, by Proposition2.1 and the strong maximum principle (see, for example, [16]), its spatial maximum must mustdecrease monotonically, by an amount L say, on some sub-interval [ t , t ] of ( −∞ , t ,i , t ,i ∈ (cid:2) − λ i t i ,
0] with t ,i → t and t ,i → t such that L − ε ≤ max M kF (cid:0) · , λ − i t ,i + t i (cid:1) − max M kF (cid:0) · , λ − i t ,i + t i (cid:1) (3.1)for any ε >
0, so long as i is chosen accordingly large. But since λ i → ∞ , the right hand side of(3.1) converges to zero. It follows that sup M ×{ t } k/F is independent of t . Since M is compact,the space-time supremum of k/F is attained at an interior space-time point, and we deducethat k/F is constant. Since there is a sequence of points x i for which k i F i ( x i , → (1 + ε ), wemust have k ≡ (1 + ε ) F on the limit. In particular, we have 0 ≡ ( ∂ t − L ) ( k − (1 + ε ) F ).But then, computing as in Proposition 2.1, we find 0 ≡ ∇ k ≡ (1 + ε ) ∇ F due to Propositions2.3 and 2.5. But the only closed, convex hypersurfaces of R n with F constant are spheres[13], which satisfy k ≡ F . This contradicts ε > (cid:3) Remark.
We note that, for flows by convex speed functions, where exterior non-collapsingholds, the proof of the exerior ball estimate goes through. However, for flows by concave speedfunctions, where interior non-collapsing holds, the proof of the interior ball estimate does notgo through without some additional condition (such as a pinching condition) to ensure thatthe blow-up limit is convex. In fact, due to the examples constructed by Andrews-McCoy-Zheng [10, § Theorem 3.2.
Let X : M n × [0 , T ) → R n +1 be a maximal solution of the curvature flow (CF) such that the speed is a concave, inverse-concave function of the Weingarten map. Then X converges to a constant p ∈ R n +1 as t → T , and the rescaled embeddings e X ( x, t ) := X ( x, t ) − p p T − t ) converge in C as t → T to a limit embedding with image equal to the unit sphere S n .Proof of Theorem 3.2. We first apply Theorem 1.2 to show that the solution converges uni-formly to a point p ∈ R n +1 in the Hausdorff metric: Observe that | X ( x , t ) − X ( x , t ) | ≤ r + ( t )for every x , x ∈ M and every t ∈ [0 , T ), where r + ( t ) denotes the circumradius of X ( M, t )(this is the radius of the smallest ball in R n +1 that contains the hypersurface X ( M, t )). Since X remains in the compact region enclosed by some initial circumsphere, it suffices to show WO-SIDED NON-COLLAPSING CURVATURE FLOWS 13 that r + → t → T . But this follows directly from Theorem 1.2: Since k ( x, t ) is the curva-ture of the smallest ball which encloses the hypersurface X ( M, t ), and touches it at X ( x, t ),we have 1 r + ≥ max M ×{ t } k ≥ k max M ×{ t } F .
But max M ×{ t } F → ∞ .We now deduce Hausdorff convergence of the rescaled hypersurfaces e X ( M, t ) to the unitsphere: By Theorem (3.1), for all ε > t ε ∈ [0 , T ) such that r + ( t ) ≤ (1+ ε ) r − ( t )for all t ∈ [ t ε , T ), where r − ( t ) is the in-radius of X ( M, t ) (the radius of the largest ball enclosedby X ( M, t )). By the avoidance principle the remaining time of existence at each time t is noless than the lifespan of the shrinking sphere of initial radius r − ( t ), and no greater than thelifespan of the shrinking sphere of initial radius r + ( t ). These observations yield r − ( t ) ≤ p T − t ) ≤ r + ( t ) ≤ (1 + ε ) r − ( t ) (3.2)for all t ∈ [ t ε , T ). It follows that the circum- and in-radii of the rescaled solution approachunity as t → T . We can also control the distance from the final point p to the centre p t ofany in-sphere of X ( M, t ): For any t ′ ∈ [ t, T ), the final point p is enclosed by X ( M, t ′ ), whichis enclosed by the sphere of radius p r + ( t ) − t ′ − t ) about p t . Taking t ′ → T and applying(3.2) gives | p − p t | ≤ p r + ( t ) − T − t ) ≤ p (1 + ε ) · T − t ) − T − t )for all t ∈ [ t ε , T ). Thus | p − p t | p T − t ) ≤ p (1 + ε ) − . (3.3)This yields the desired Hausdorff convergence of e X to the unit sphere.Next we obtain bounds on the curvature of the rescaled flow e X : Non-collapsing and theinequalities r − ( t ) ≤ p T − t ) ≤ r + ( t ) derived above yield1 p T − t ) ≤ r − ( t ) ≤ min x ∈ M k ( x, t ) ≤ k min x ∈ M F ≤ k K min x ∈ M k ( x, t ) ≤ k K min x ∈ M κ min ( x, t ) , and 1 p T − t ) ≥ r + ( t ) ≥ max x ∈ M k ( x, t ) ≥ K max x ∈ M F ≥ k K max x ∈ M k ( x, t ) ≥ K k max x ∈ M κ max ( x, t ) . By a well-known result of Hamilton [18, Lemma 14.2], this also implies convergence of therescaled metrics, and we obtain the desired C -convergence. (cid:3) Remark.
One can obtain C ∞ -convergence in the above theorem by a standard bootstrap-ping procedure [19]. Namely, using the time-dependent curvature bounds, one obtains time-dependent bounds on the derivatives of the Weingarten map (of the underlying solution of theflow) to all orders from the curvature derivative estimates. Unfortunately, the resulting esti-mates do not quite have the right dependence on the remaining time. The correct dependencecan be obtained using the interpolation inequality (cf. [19, § C ∞ -convergence of the corresponding solution of the normalised flow equation to the unit sphere(cf. [19, § C ∞ -convergence of the rescaled solution. References [1] A. Alexandrov,
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