TWO-SIDED NON-COLLAPSING CURVATURE FLOWS BEN ANDREWS AND MAT LANGFORD

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1 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS BEN ANDREWS AND MAT LANGFORD Abstract. It was proved in [ALM2] that embedded solutions of Euclidean hypersurface flows with speeds given by concave convex, degree one homogeneous functions of the Weingarten map are interior exterior non-collapsing. These results were extended to hypersurface flows in the sphere and hyperbolic space in [AHLW]. In the first part of the paper, we show that locally convex solutions are exterior noncollapsing for a larger class of speed functions than previously considered; more precisely, we show that the previous results hold when convexity of the speed function is relaxed to inverse-concavity. We note that inverse-concavity is satisfied by a large class of concave speed functions [An07]. 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 mean curvature flow. In the second part of the paper, we demonstrate the utility of two sided non-collapsing with a 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, in that we show that the exterior ball curvature is a viscosity supersolution of the linearised flow. The new ingredient is the following observation: Since the function which provides 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 flow equation depend only on the first factor, we are privileged with a freedom of choice in which second derivatives from the extra directions to include in the calculation. The optimal choice is closely related to the class of inverse-concave speed functions.. Introduction We consider embedded solutions X : M n [0, T N n+ σ t Xx, t = F x, tνx, t, of curvature flows of the form CF where Nσ n+ is the complete, simply connected Riemannian manifold of constant curvature σ {, 0, } that is, either hyperbolic space H n+, Euclidean space R n+, or the sphere S n+, ν is a choice of normal field for the evolving hypersurface X, and the speed F is given by 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, monotone increasing, degree one homogeneous function of the Weingarten map W of X. Moreover, we will always assume that F is normalised such that F,..., = ; however, this is merely a matter of convenience all of the results hold, up to a recalibration of constants, in the un-normalised case. 200 Mathematics Subject Classification. 53C44, 35K55, 58J35. This research was partially supported by Discovery grant DP of the Australian Research Council. The second author gratefully acknowledges the support of an Australian Postgraduate Award and an Australian National University HDR Supplementary Scholarship, and the support and hospitality of the Mathematical Sciences Center at Tsinghua University, Beijing, and the Department of Mathematics at East China Normal University, Shanghai during the completion of part of this work.

2 2 BEN ANDREWS AND MAT LANGFORD The interior and exterior ball curvatures of a family of embeddings X : M n [0, T R n+ with normal ν were introduced in [ALM2]. These are, respectively, defined by kx, t := sup y x kx, y, t, and kx, t := inf y x kx, y, t, where kx, y, t := 2 Xx, t Xy, t, νx, t Xx, t Xy, t 2.. Equivalently, kx, t resp. kx, t gives the curvature of the largest region in R n+ with totally umbilic boundary that lies on the opposite resp. same side of the hypersurface XM, t as νx, t, and touches it at Xx, t. Therefore, for a compact, convex embedding with outward normal, they are, respectively, the curvature of the largest enclosed, and smallest enclosing spheres which touch the embedding at Xx, t [ALM2, Proposition 4]. It follows that κ max k and κ min k. For flows in Euclidean space, it was proved in [ALM2] that embedded solutions of CF are 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; more precisely, there exist k 0 R, K 0 > 0 such that k k 0 F in the former case, and k K 0 F in the latter. In particular, solutions of the mean curvature flow in which case the speed is the trace of the Weingarten map are both interior and exterior non-collapsing. Two-sided noncollapsing has many useful consequences; for example, for uniformly convex hypersurfaces we obtain uniform pointwise bounds on the ratios of principal curvatures, and a uniform bound on the ratio of circumradius to in-radius. This leads to a neat proof of Huisken s theorem [Hu84] on the convergence of convex hypersurfaces to round points under the mean curvature flow. Moreover, one-sided non-collapsing can also provide useful information; for example, interior non-collapsing rules out certain singularity models, such as products of the Grim Reaper curve with R n. For convex speeds, exterior non-collapsing is sufficient to obtain a bound on the ratio of circumradius to in-radius, and the proof of convergence of locally convex initial hypersurfaces to round points, which appeared in [An94a], is also simplified. More recently, it was shown that the above statements have natural analogues when the ambient space is either the sphere or hyperbolic space [AHLW]: Considering the sphere S n+ as the embedded submanifold {X R n+2 : X, X = } of R n+2, and hyperbolic space H n+ as the embedded submanifold {X R n+, : X, X = } of Minkowski space R n+,, the function k may be formally defined as in., except that now we take, and to be the inner product and induced norm on, in the case of the sphere, R n+2, and, in the case of hyperbolic space, the spacelike vectors in R n+,. Then, if F is a concave function of the curvatures, there exists K 0 > 0 such that k F η K 0e 2σηt, and, if F is a convex function of the curvatures, there exists k 0 R such that k F η k 0e 2σηt, where η > 0 depends on bounds for the derivative of F. Let us recall the following definition from [An07]: Definition.. Let Ω be an open subset of the positive cone Γ + := {z R n : z i > 0 for all 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 z,..., z n := f z,..., z n is concave.

3 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS 3 The main result of this article may now be stated as follows: Theorem.2. Let f : Γ + R be a smooth, symmetric function which is homogeneous of degree one, and monotone increasing in each argument. Let X be a solution of CF with speed given by F = fκ,..., κ n, where κ i are the principal curvatures of X. Then, if f is inverse-concave, X is exterior non-collapsing; that is, If N n+ = R n+, then, for all x, t M [0, T, kx, t F x, t inf k M {0} F. 2 If N n+ = S n+, and tr F η, then, for all x, t M [0, T, kx, t F x, t k η inf M {0} F e 2ηt. η 3 If N n+ = H n+, and tr F η, then, for all x, t M [0, T, kx, t F x, t k η inf M {0} F e 2ηt, η where F is the derivative of F with respect to the Weingarten map. In particular, combining Theorem.2 with the results of [ALM2] and [AHLW], we find that solutions of flows in spaceforms by concave, inverse-concave speed functions are both interior and exterior non-collapsing. We note that concave speed functions satisfy tr F n, so in that case we may take η = n in case 3 of Theorem.2. The reader is referred to [An07] for a satisfactorily large class of admissible speeds which are both concave and inverse-concave. 2. Proof of Theorem.2 As in [ALM2], we first extend k to a continuous function on the compact manifold with boundary M. As a set, M := M M \ D SM, where D := {x, x : x M} is the diagonal submanifold. The manifold-with-boundary structure is defined by the atlas generated by all charts for M M \ D, together with the charts Ŷ defined by Ŷ z, s := expsy z, exp sy z for s sufficiently small, where Y is a chart for SM. The extension of k to M is then given by setting kx, y, t := W x,t y, y for x, y S x,t M. We also recall the notation of [ALM2]; namely, we define dx, y, t := Xx, t Xy, t and wx, y, t := Xx, t Xy, t Xx, t Xy, t, and use scripts x and y to denote quantities pulled back to M M by the respective projections onto the first and second factor. With this notation in place, k may be written as k = 2 d 2 dw, ν x. Theorem.2 is a direct consequence of the following proposition: Proposition 2.. If the flow speed F is inverse-concave, the exterior ball curvature k is a viscosity supersolution of the linearised flow t u = L u + W 2 F σtr F u + 2σu, LF

4 4 BEN ANDREWS AND MAT LANGFORD where L := F ij i j, and W 2 F := F ij W 2 ij. Proof of Proposition 2.. Consider, for an arbitrary point x 0, t 0 M [0, T, an arbitrary lower support funtion φ for k at x 0, t 0 ; that is, φ is C 2 on a backwards parabolic neighbourhood P := U x0 t 0 ε, t 0 ] of x 0, t 0, and φ k with equality at x 0, t 0. Then we need to prove that the differential inequality t φ L φ + W 2 F σtr F φ + 2σφ holds at x 0, t 0. We note that kx, y, t kx, t φx, t for all x, y, t M [0, T such that x, t P, and, since k is continuous and M is compact, we either have kx 0, t 0 = kx 0, y 0, t 0 for some y 0 M \ {x 0 }, or kx 0, t 0 = W x0,t 0 y 0, y 0 for some y 0 S x0,t 0 M. We consider the former case first. The interior case. We first suppose that inf M kx 0,, t 0 is not attained on the boundary of M. In that case, we have κ x 0, t 0 > kx 0, t 0 = kx 0, y 0, t 0 for some y 0 M \ {x 0 }, and kx, y, t kx, t φx, t for all x, t P and all y M \ {x}. In particular, we have the inequalities t k φ 0, and L k φ 0 2. at x 0, y 0, t 0 for any elliptic operator L on M M. We would like L to project to L on the first factor. This leads us to consider operators of the form L = F x ij xi +Λ p i y p x j +Λ q j y q, where Λ is any n n matrix. We note that, in both of the cases σ = ±, the ambient Euclidean/Minkowskian derivative decomposes into tangential and normal components as D = D g ν, where D, g, and ν are, respetively, the induced metric, connection and normal of Nσ n+ with respect to its embedding. Since ν, ν = σ, and the ambient position vector is normal to Nσ n+, straightforward computation yields x i + Λ i p y pk = 2 d 2 x i Λ i p y p, ν x kdw + dw, W x i p x p. 2.2 If we choose the coördinates {x i } n i= and {yi } n i= to be orthonormal coördinates with respect to the induced metric g at time t 0 centred at x 0 and y 0 respectively, then a further straightforward computation using the vanishing of 2.2 at x 0, y 0, t 0 and the Codazzi equation yields x j +Λ q j y q x i+λ p i y p k = 2 { d 2 W x ijν x σδ ij X x + Λ p i Λ q j W y pqν y + σδ pq X y, ν x kdw + i x Λ p i p, y W x j q q x x j + Λ q j y qk i x Λ p i p, y dw k i x Λ p i p, y j x Λ q j q y + x j Λ q j q y, W x i p p x + dw, W x ij σw x ijx x W x i r W x rjν x at the point x 0, y 0, t 0. x i + Λ i p y pk x j Λ j q y q, dw } 2.3

5 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS 5 Next, noting that the normal satisfies D t ν = D t ν + σf X = grad F + σf X, we compute t k = 2 d 2 Fx ν x + F y ν y, ν x kdw + dw, grad F x + σf X x. 2.4 Combining 2.3 and 2.4, we obtain t L k = 2 { d 2 F y F x ij Λ p i Λ q j W y pqν y + σδ pq X y, ν x kdw + kf x ij x i Λ p i p, y j x Λ q j q y 2 F x ij x j Λ q j q y, W x i p p x + σtr F } x X x, ν x kdw + 2σF x X x, dw + 4 d 2 Fx ij x i+λ p i y p k j x Λ q j q y, dw + W x 2 F k at the point x 0, y 0, t 0. We now note that the vanishing of the y-derivatives at an off-diagonal extremum y 0 M of kx 0,, t 0 determines the tangent plane to X at y 0 : Lemma 2.2 [ALM2]. Suppose that a point x, y, t is an off-diagonal extremum of k; that is, y x is an extremum of kx,, t. Then at x, y, t. span{ x i 2 x i, w w} n i= = span{ y i }n i= Proof of Lemma 2.2. We compute at the point x, y, t. Note first that { x i 2 x i, w w}n i= are an orthonormal set, and that ν x kdw =. Next, observe that the vanishing of y ik implies If σ 0, we also have y i, ν x kdw = 0. X y, ν x kdw = 0. Thus span{ i } = span{σx y, ν x kdw}. On the other hand For σ 0, we also have x i 2 x i, w w, ν x kdw = 0. X y, x i 2 x i, w w = 0. Thus span{ x i 2 x i, w w} = span{x y, ν x kdw}. The claim follows. Thus, without loss of generality, we may assume y i = x i 2 x i, w w at x 0, y 0, t 0. Note also that, when σ 0, 2 d 2 X x, ν x kdw x0,y 0,t 0 = 2 d 2 X x X y, ν x kdw x0,y 0,t 0 = kx 0, y 0, t 0. Using these observations, and the vanishing of y ik, we obtain t L k = k + 2σF x + 2F x ij x ikw x ki p j x pk + 2 { d 2 F y F x [ kδij W x ij 2Λ p i kδ pj W x pj + Λ p i Λ q j kδ pq W y pq ]} 2.5 W x 2 F σtr F x + Fx ij

6 6 BEN ANDREWS AND MAT LANGFORD at any off-diagonal extremum x 0, y 0, t 0. Applying the inequalities 2., we obtain 0 t L k φ t L φ + W x 2 F σtr F x + 2 d 2 {F y F x + Fx ij k + 2σF x + 2F x ij i kw x ki jp p k [ kδ ij W x ij 2Λ i p kδ pj W x pj + Λ i p Λ j q kδ pq W y pq Since k < κ, the gradient term is non-negative, and it remains to demonstrate non-negativity of the term on the second line for some choice of the matrix Λ. Since we are free to choose the orthonormal basis at y 0 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 monotone increasing in each variable and inverse-concave, and let F : C + R be the corresponding function on the cone C + of positive definite symmetric matrices defined by F A = fλa, where λ denotes the eigenvalue map. Then for any k R, any diagonal B C +, and any A C + with k < min i {λ i A}, we have 0 F B F A + F [ ] ij A sup kδ ij A ij 2Λ p i kδ pj A pj + Λ p i Λ q j kδ pq B pq. Λ Proof of Proposition 2.3. Since the expression in the square brackets is quadratic in Λ, it is easy to see that the supremum is attained with the choice Λ = A kib ki, where I denotes the identity matrix. Thus, given any A C +, we need to show that 0 QB := F B F A F ij A A ki ij [ A ki B ki A ki ] ij Since B is diagonal, and the expression QB is invariant under similarity transformations with respect to A, we may diagonalise A to obtain QB := fb fa f i a [a i k a i k 2 ], b i k where we have set a = λa and b = λb. We are led to consider the function q defined on Γ + by qz := fz fa f i a [a i k a i k 2 ]. z i k We compute and q i = f i f i a a i k 2 z i k 2, q ij = f ij + 2f i a a i k 2 z i k 3 δij = f ij f + 2 i δ ij z i k 2 qi δ ij z i k. It follows that q ij + 2 qi δ ij z i k = f ij f + 2 i δ ij z i k > f ij f + 2 i δ ij 0, 2.6 z i where the last inequality follows from inverse-concavity of f [An07, Corollary 5.4]. Thus the minimum of q is attained at the point z = a, where it vanishes. This completes the proof. ]}..

7 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS 7 This completes the proof in the interior case. The boundary case Cf. [An07, Theorem 3.2]. We now consider the case that inf M kx 0,, t 0 occurs on the boundary of M; that is, kx 0, t 0 = W x0,t 0 y 0, y 0 for some y 0 S x0,t 0 M. Consider the function K defined on T M [0, T by Kx, y, t = W x,t y, y. Then the function Φx, y, t := φx, tg x,t y, y is a lower support for K at x 0, y 0, t 0. In particular, t K Φ 0 and L K Φ at x 0, y 0, t 0 for any elliptic operator L on T M. We require the operator project to L on the first factor at least at the point x 0, y 0, t 0, which leads us to consider an operator L locally of the form L = F x ij x i Λ p i y p x j Λ q j y q, where {x i, y i } n i= are coördinates for T M such that {x i } n i= are normal coördinates on M with respect to g t 0 based at x 0, and {y i } n i= are the corresponding fibre coördinates defined by x, y = x, yi x i for tangent vectors x, y near x 0, y 0. We may assume that { x i x0 } n i= is a basis of eigenvectors of W x0,t 0 with y 0 = x x0. Writing locally K Φ = y k y l W kl g kl, we find x i Λ i p y pk Φ = y k y l x iw kl x iφ g kl 2Λ i p y k W kp φ g kp. Thus, at the point x 0, y 0, t 0, we obtain We next compute 0 = x i Λ i p y pk Φ = i W i φ. x i Λ i p y p x j Λ j q y qk Φ = y k y l x i x jw kl x i x jφ g kl x iφ x jg kl x jφ x ig kl φ x i x jg kl At the point x 0, y 0, t 0, we obtain L K Φ = L W L φ Finally, we compute the time derivative which at x 0, y 0, t 0 becomes 2Λ j q y k x iw kq x iφ g kq φ x ig kp 2Λ i p y k x jw kp x jφ g kp φ x jg kp + 2Λ i p Λ j q W pq φ g pq. 2 F ij[ 2Λ i p j W p j W δ p Λ i p Λ j q W pq W δ pq ]. 2.8 t K Φ = y k y l t W kl t φ g kl φ t g kl, t K Φ = t W W t g t φ. 2.9 Let us recall the evolution equations for W and g, which are derived in [An94a, An94b]: t W ij = L W ij + F pq,rs i W pq j W rs 2WijF 2 + W ij σtr F W 2 F + 2σW ij, 2.0

8 8 BEN ANDREWS AND MAT LANGFORD and t g ij = 2F W ij. 2. Putting 2.8 and 2.9 together, and applying the evolution equations 2.0 and 2., and the inequalities 2.7, we obtain 0 t L K Φ = t L φ W 2 F σtr F φ + 2σF + F pq,rs W pq W rs + 2F [ ] ij p Λ i 2 j W p j W δ p Λ q j W pq W δ pq 2.2 at the point x 0, y 0, t 0. Note that the term in the last line with p = vanishes. Proceeding as in [Br, Proposition 8] see also [AL, Theorem 7] we also obtain W = 0 at the point x 0, t 0 : Lemma 2.4. W vanishes at x 0, y 0, t 0. Sketch proof of Lemma 2.4. Since κ x 0, t 0 = inf y x0 kx 0, y, t 0, we have 0 Zx 0, y, t 0 := 2 Xx 0, t 0 Xy, t 0, νx 0, t 0 κ x 0, t 0 Xx 0, t 0 Xy, t 0 2 for all y M. In particular, 0 fs := Zx 0, γs for all s, where γs := exp x0 sy 0. It is straightforward to compute 0 = f0 = f 0 = f 0, which, since f 0, implies that f 0 = 0. But f 0 = 2 W. Applying the following proposition to 2.2 completes the proof. Proposition 2.5. Let f : Γ + R be a smooth, symmetric function which is monotone increasing in each variable and inverse-concave, and let F : C + R be the corresponding function on the cone C + of positive definite symmetric matrices defined by F A = fλa, where λ denotes the eigenvalue map. If A C + and y is an eigenvector of A corresponding to its smallest eigenvalue, then, for any totally symmetric 3-tensor T with T y, y, y = 0, we have 0 y i y j F pq,rs T ipq T jrs + 2F [ ] kl sup 2Λ p k y i T ilp y r y s T lrs δ iq Λ p k Λ q l A pq y r y s A rs δ pq at the matrix A. Λ 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 := 2F kl[ ] 2Λ p k y i T ilp y r y s T lrs δ iq Λ p k Λ q l A pq y r y s A rs δ pq is continuous in A, and hence the supremum over Λ is upper semi-continuous in A; so the general 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 } n i= be an orthonormal frame of eigenvectors of A with e = y. Then Q = 2 F kl[ 2Λ k p T lp T l δ q Λ k p Λ l q A pq A δ pq Observe that the supremum over Λ occurs when Λ lq = λ q λ T lq for i, p >. With this choice, we obtain Q = 2F kl R pq T kq T lp, ].

9 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS 9 where R pq := λ p λ δ pq for p, q and zero otherwise. Therefore, it suffices to prove that 0 F pq,rs + 2 F pr R qs B pq B rs for any symmetric B with B = 0. Applying [An07, Theorem 5.], the expression may be written in terms of the function f as: F pq,rs + 2F pr R qs B pq B rs = f pq B pp B qq + f p f q n B 2 f p pq + 2 Bpq 2 λ p λ q λ p q p=, q=2 q λ = f pq B pp B qq + 2 f p δ pq B pp B qq + f p f q Bpq 2 λ p>, q> p λ λ p λ q p q We first estimate f pq B pp B qq + 2 p>,q> + 2 n p=2 f Bp λ p λ p>, q>, p q f p δ pq λ p λ B pp B qq f pq B pp B qq + 2 = f pq f + 2 p δ pq λ p f p Bpq 2. λ q λ p=2,q=2 f p δ pq B pp B qq λ p B pp B qq 0, where the final inequality follows from inverse-concavity of f by [An07, Theorem 2.]. The remaining terms are f p f q n B 2 f pq + 2 B 2 f p p + 2 Bpq 2 λ p λ q λ p λ λ q λ p q = 0. p=2 p>, q>, p q p>, q>, p q p>, q>, p q f p f q f p + 2 Bpq λ p λ q λ q λ f p f q f + p + Bpq λ p λ q λ q f q λ p n f p f f + λ p λ λ p λ p=2 n p=2 f p λ p λ The second term is clearly non-negative. Non-negativity of the first term follows from inverseconcavity of f [An07, Corollary 5.4]. This completes the proof. This completes the proof that k is a viscosity supersolution of LF. Since the speed function satisfies LF, the statement of Theorem.2 follows from a simple comparison argument for viscosity solutions, just as in [ALM2] and [AHLW]: Define ϕt := e 2σηt inf M {t} k/f /η, where η > 0 is such that ση σtr F ; that is, η tr F for flows in the sphere, η tr F for flows in hyperbolic space, and η > 0 for flows in Euclidean space. We claim that ϕ is non-decreasing. It suffices to prove that k, t /η + e 2σηt ϕt 0 εe Lt t 0 F, t > 0 for any t 0 [0, T, t [t 0, T B 2 p B 2 p

10 0 BEN ANDREWS AND MAT LANGFORD and any ε > 0, where we have set L := 2ση. Taking ε 0 then gives k, t /η + e 2σηt ϕt 0 F, t 0; that is, ϕt ϕt 0 for all t t 0 for any t 0. Now, at time t 0 we have kx, t 0 /η + e 2σηt 0 ϕt 0 ε F x, t 0 εf x, t 0 > 0. So suppose, contrary to the claim, that there is a point x, t M [t 0, T and some ε > 0 such that kx, t /η + e 2σηt ϕt 0 εe t t 0 F x, t = 0. Assuming that t is the first such time, this means precisely that the function φx, t := /η + e 2σηt ϕt 0 εe Lt t 0 F x, t is a lower support for k at x, t. But k is a viscosity supersolution of LF, so that, at the point x, t, φ satisfies 0 t L φ + W 2 F σtr F φ + 2σF = 2ησe 2σηt ϕt 0 + Lεe Lt t 0 F η + e 2σηt ϕt 0 εe Lt t 0 W 2 F + σtr F + η + e 2σηt ϕt 0 εe Lt t 0 F W 2 F σtr F + 2σF = 2e 2σηt ϕt 0 F ση σtr F + εe Lt t 0 F L + 2σtr F + 2 η F ση σtr F εe Lt t 0 > 0, where we used L := 2ση, and ση σtr F in the last line. This contradiction proves that φ could not have reached zero on [t 0, T, which, as explained above, proves that ϕ is non-decreasing. Therefore, kx, t F x, t η e 2σηt This proves Theorem.2. k inf M {t} F e 2σηt =: ϕt ϕ0 = η 3. Convex solutions in Euclidean space k inf M {0} F. η We will demonstrate in this section the usefulness of two-sided non-collapsing by proving that convex hypersurfaces contract to round points under the flow CF in Euclidean space if the speed is both concave and inverse-concave. The exposition will remain brief as the only new ingredient is the use of two-sided non-collapsing. The rest of the details appear in the references [Hu84, An94a, An94b, AC]. Theorem 3.. Let X : M n [0, T R n+ 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 uniformly to a constant p R n+ as t T, and the rescaled immersions Xx, t := Xx, t p 2T t converge in C as t T to a limit immersion with image equal to the unit sphere S n. We first note that we can enclose the initial hypersurface by a large sphere, which shrinks to a point in finite time under the flow. The avoidance principle see, for example, [ALM2, Theorem 5] then implies that the maximal time T of existence of our solution must be finite. Global regularity may be obtained by considering the hypersurface locally as a graph. The

11 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS graph height function u satisfies the scalar parabolic equation t u = + grad u 2 grad u F D. + grad u 2 Since F is a concave, the second derivative Hölder estimate of Evans [Ev82] and Krylov [Kr82] applies, and we obtain higher regularity from the Schauder estimate cf. [An94a]. This method also yields bounds on derivatives of curvature in terms of bounds on the curvature, although these estimates are also possible using the maximum principle, as in [An94a, Section 7]. Thus the solution remains smooth until F blows-up, and it follows that max M {t} F as t T. We can now apply Theorem.2 to show that the solution converges uniformly to a point p R n+ : First observe that Xx, t Xx 2, t 2r + t for every x, x 2 M and every t [0, T, where r + t denotes the circumradius of XM, t; that is, the radius of the smallest closed ball in R n+ that contains the hypersurface XM, t. Thus, since X remains in the compact region enclosed by some initial circumsphere, it suffices to show that r + 0 as t T. But this follows directly from Theorem.2: Since kx, t is the curvature of the smallest ball which encloses the hypersurface XM, t, and touches it at Xx, t, Theorem.2 yields max r k k 0 max F. + M {t} M {t} But max M {t} F. We now prove smooth convergence of the rescaled hypersurfaces to the unit sphere: Since max M {t} F, we may choose a sequence of times t i T at each of which a new maximum of F is attained; that is, λ 2 i := max F = F x i, t i M [0,t i ] for some x i M. We now consider the following sequence of rescalings of X: X i x, t := λ i [X x, t i + t ] λ 2 X i x i, t i. i It is easily checked that X i : M n [ λ 2 i t i, λ 2 i T t i R n+ is a solution of the flow CF for each i. Moreover, for each i, we have max M [ λ 2 i t i,0] F i = F i x i, 0 = and X i x i, 0 = 0. The curvature bounds deduced above allow us to deduce that the sequence converges locally uniformly in C along a subsequence to a limit flow X : M, Σ R n+ see, for example, [Ba, Br, Co]. But, since the function k rescales with the same factor as the speed, the exterior non-collapsing estimate yields k i x i, 0 k 0 > 0, which implies that the limit is a flow of compact hypersurfaces. In particular, the convergence is smooth, so that M = M, and Σ <. Moreover, since κmin k k 0 F > 0, the limit must be a flow of convex hypersurfaces. In fact, using the strong maximum principle, we may deduce that the limit flow is a shrinking sphere: Lemma 3.2. The limit flow X : M, Σ R n+ is a shrinking sphere. Proof of Lemma 3.2. We first observe that k/f and also k/f must be constant on the limit flow X ; for if not, by Theorem.2, and the strong maximum principle, its spatial maximum must must decrease monotonically, by an amount L say, on some sub-interval [t, t 2 ]

12 2 BEN ANDREWS AND MAT LANGFORD of, Σ. But then passing to the convergent subsequence there must exist sequences of times t,i, t 2,i [ λ 2 i t i, λ 2 i T t i with t,i t and t 2,i t 2 such that k L ε max, t i + t,i k M F λ 2 max, t i + t 2,i i M F λ 2 3. i for any ε > 0, so long as i is chosen accordingly large. But since λ i, and k/f converges to a finite value as t i T, the right hand side of 3. converges to zero. Thus max M {t} k/f is independent of t. Consequently, the space-time supremum of k/f is attained at an interior space-time point, and we deduce that k/f is constant on the limit flow. Similar considerations lead us to deduce that k/f is also constant on the limit flow. We can now use the differential identities associated to k to show that X is a shrinking sphere: Fix any t 0, Σ and consider the set U := {x M : κ x, t 0 > kx, t 0 }. Then for any x 0 U there is a point y 0 M \ {x 0 } such that kx 0, t 0 = kx 0, y 0, t 0. Since k k 0 F, we have kx, y, t k 0 F x, t 0, with equality at x 0, y 0, t 0. Thus, computing as in 2.5, we have at x 0, y 0, t 0 0 t L k k 0 F F x ij x ikw x ki x jk 0. It follows that 0 = grad k = k 0 grad F at x 0, t 0. Since k 0 > 0, we obtain grad F = 0 at x 0, t 0. But x 0 U was arbitrary, so we find grad F 0 on U. Since U is open, and W > 0, the classification of constant Weingarten hypersurfaces of Ecker-Huisken [EH84] implies that U is a part of a sphere. But then, by the very assumption that U is nonempty, we must have U M. Thus either U is empty or has a non-empty boundary in M. Suppose that U is non-empty, so that there is a point x 0 U. By continuity, x 0 is also umbillic, so that κ x 0, t 0 = k 0 F x 0, t 0 = k 0 κ x 0, t 0. Thus k 0 =. Consider now the set V := {x M : κ n x, t 0 < kx, t 0 }. Similarly as above, V M, and if V then K 0 =. In that case we obtain k k, which is again counter to the assumption that U and V are non-empty. Thus either U or V is empty. Suppose that U is empty, so that κ k k 0 F. Then proceeding as in Lemma 2.4 we obtain 0 grad κ k 0 grad F, so the claim follows again from the Ecker-Huisken theorem. The proof for V = is similar. Lemma 3.2 implies that the ratio of circumradius to in-radius approaches one at the final time on the original flow. The rest of the proof of Theorem 3. is now straightforward: Proof of Theorem 3.. We will first deduce Hausdorff convergence of the rescaled hypersurfaces XM, t to the unit sphere: By Lemma 3.2, for all ε > 0 there is a time t ε [0, T such that r + t + εr t for all t [t ε, T. By the avoidance principle the remaining time of existence at each time t is no less than the lifespan of a shrinking sphere of initial radius r t, and no greater than the lifespan of a shrinking sphere of initial radius r + t. These observations yield r t 2T t r + t + εr t 3.2 for all t [t ε, T. It follows that the circum- and in-radii of the rescaled solution approach unity as t T. We can also control the distance from the final point p to the centre p t of any in-sphere of XM, t: For any t [t, T, the final point p is enclosed by XM, t, which is enclosed by the sphere of radius r + t 2 2t t about p t. Taking t T and applying 3.2 gives p p t r + t 2 2T t + ε 2 2T t 2T t

13 TWO-SIDED NON-COLLAPSING CURVATURE FLOWS 3 for all t [t ε, T. Thus p p t 2T t + ε This yields the desired Hausdorff convergence of X to the unit sphere. Next we obtain universal bounds on the curvature of the rescaled flow X: Non-collapsing and the inequalities r t 2T t r + t derived above yield 2T t r t min kx, t k 0 min F k 0 min x M x M K kx, t k 0 min 0 x M K κ minx, t, 0 x M and 2T t r + t max kx, t K 0 max F k 0 max x M x M K kx, t K 0 max 0 x M k κ maxx, t. 0 x M This yields C 2 -convergence of the rescaled solution to the unit sphere. The rest of the proof is a standard bootstrapping procedure, which is the same as appears in the literature. Namely, using the time-dependent curvature bounds, one obtains time-dependent bounds on the derivatives of the Weingarten map of the underlying solution of the flow to all orders from the curvature derivative estimates. Unfortunately, the resulting estimates do not quite have the right dependence on the remaining time, and it is necessary to employ the interpolation inequality as in [Hu84, Sections 9]. One then proceeds as in [Hu84, Section 0] to obtain exponential C -convergence of the corresponding solution of the normalised flow to the unit sphere, which, by construction, yields C -convergence of the rescaled solution. References [An94a] Ben Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 994, no. 2, 5 7. [An94b] Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Diff. Geom , no. 2, [An07] Ben Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. reine angew. Math , [AC] Ben Andrews and Xuzhong Chen, Curvature flow in hyperbolic spaces, preprint. [AHLW] Ben Andrews, Xiaoli Han, Haizhong Li and Yong Wei, Non-collapsing for hypersurfaces flows in the sphere and hyperbolic space, arxiv: v. [ALM2] Ben Andrews, Mat Langford and James McCoy, Non-collapsing in fully non-linear curvature flows, Ann. Inst. H. Poincaré Anal. Non Lin eare , no., [ALM3] Ben Andrews, Mat Langford and James McCoy, Convexity estimates for hypersurfaces moving by convex curvature functions, to appear in APDE. [AL] Ben Andrews and Haizhong Li, Embedded constant mean curvature tori in the three-sphere, arxiv: [Ba] Charles Baker, A partial classification of type-i singularities of the mean curvature flow in high codimension, arxiv: v. [Br] Simon Brendle, Embedded minimal tori in S 3 and the Lawson conjecture, to appear in Acta Math. [EH84] Klaus Ecker and Gerhard Huisken, Immersed hypersurfaces with constant Weingarten curvature, Math. Ann., , [Br] P. Breuning, Immersions with local Lipschitz representation, PhD thesis, Albert-Ludwigs-Universität, Freiburg 20. [Co] A. A. Cooper, A compactness theorem for the second fundamental form, arxiv: [math.dg], 200.

14 4 BEN ANDREWS AND MAT LANGFORD [Ev82] Lawrence C. Evans. Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math., , no. 3, [HK] Robert Haslhofer and Bruce Kleiner, Mean curvature flow of mean convex hypersurfaces, arxiv: v. [Hu84] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom , no., [Kr82] Nicolai V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat., , no. 3, Mathematical Sciences Institute, Australian National University, ACT 0200 Australia Mathematical Sciences Center, Tsinghua University, Beijing 00084, China Morningside Center for Mathematics, Chinese Academy of Sciences, Beijing 0090, China address: Mathematical Sciences Institute, Australian National University, ACT 0200 Australia address: