Convexity estimates for mean curvature flow with free boundary
aa r X i v : . [ m a t h . DG ] J un CONVEXITY ESTIMATES FOR MEAN CURVATURE FLOWWITH FREE BOUNDARY
NICK EDELEN
Abstract.
We prove the estimates of [HS99b] and [HS99a] for finite-timesingularities of mean-convex, mean curvature flow with free boundary in abarrier S . Here S can be any embedded, oriented surface in R n +1 of boundedgeometry and positive inscribed radius. We also prove the estimate [Hui84]in the case of convex flows and S = S n , which gives an alternative proof to[Sta96a]. Introduction
We are interested in immersed, mean-convex, mean curvature flow with freeboundary in a surface S . We reprove the estimates in [HS99b], and [HS99a] for thisclass of flows. These provide very direct, general pinching results for limit flows atsingularities, and require no embeddedness or curvature assumptions. We furtherprove the estimates in [Hui84] when S is the sphere.Consider a smooth, embedded, oriented hypersurface S ⊂ R n +1 , with choice ofnormal ν S , having bounded geometry and positive inscribed radius. We refer to S as the barrier surface . If Σ n ⊂ R n +1 is a compact, mean-convex hypersurfacewith boundary, we say Σ meets S orthogonally if ∂ Σ ⊂ S , and the outer normal of ∂ Σ ⊂ Σ coincides with ν S .Let Σ = Σ meet the barrier S orthogonally. Then the mean curvature flow ofΣ , with free-boundary in S , is a family of immersions F t : Σ × [0 , T ) → R n +1 such that ∂ t F t = − Hν, for all p ∈ Σ , t > F t (Σ) meets S orthogonally for all t ≥ F ≡ Id Σ . Here H is the mean curvature, and ν the outer normal, oriented so that H = − Hν is the mean curvature vector. We often write Σ t = F t (Σ), and will identify andsurface and its immersion.It was shown by Stahl [Sta96b] that the mean curvature flow with free-boundaryin S always exists on some maximal time interval [0 , T ), for T ≤ ∞ , such that if T < ∞ then necessarily max Σ t | A | → ∞ as t → T . Here | A | is the norm of thesecond fundamental form A .Type-I tangent flows of mean curvature flow with free boundary have been classi-fied by Buckland [Buc05]. Our convexity estimates work towards classifying type-IIlimit flows with free boundary. Stahl [Sta96a] has shown Theorem 1.6 using a dif-ferent method. Key words and phrases. mean curvature flow; convexity estimates; free boundary; limit flows.
We prove the following theorems concerning the mean curvature flow of Σ withfree-boundary in S . Throughout the duration of this paper we assume Σ is com-pact, mean-convex. Theorem 1.1.
There are constants α = α ( S ) ≥ and C = C ( S, Σ ) so that (1) max Σ t | A | H ≤ Ce αt for all time of existence. In particular, if T < ∞ , then | A | ≤ C ( S, Σ , T ) H. Definition 1.1.1.
Given a vector µ ∈ R n , and k ∈ { , . . . , n } , we let s k ( µ ) = X ≤ i <...
1. If s k − ( µ ) = 0, we let q k ( µ ) = s k ( µ ) s k − ( µ ) . Given a real symmetric n × n matrix M , define s k ( M ) = s k ( µ ) where µ ∈ R n isthe vector of eigenvalues of M . Similarly, where possible set q k ( M ) = q k ( µ ). Noticethat s k is a polynomial in the entries of M .Given a surface Σ, define the smooth function S k by S k ( p ) = s k ( A ( p )) = s k ( λ ( p ))where λ the vector of principle curvatures. Similarly where possible set Q k = q k ( A ).We have that H ≡ S , and | A | ≡ S − S . Theorem 1.2 (Convexity pinching) . If T < ∞ , then for any k ∈ { , . . . , n } , η > ,there is a constant C = C ( S, Σ , T, η, n ) such that S k ≥ − ηH k − C at all points in spacetime. For
T < ∞ , by rescaling Σ t along an essential blow-up sequence (c.f. Section 4of [HS99b]), we obtain an eternal limit flow ˜Σ τ with free boundary in a hyperplane.This can be reflected to a mean curvature flow without boundary. Theorem 4.1 of[HS99a] therefore proves the following Corollary of Theorem 1.2. Corollary 1.3. If T < ∞ , then any limit flow of Σ t at a type-II singularity isa weakly convex flow ˜Σ τ with free boundary in a hyperplane. After reflection to aflow without boundary, ˜Σ τ is a convex translating soliton. Further, we can write ˜Σ τ = R n − k × ˜Σ kτ , where ˜Σ kτ is strictly convex. Remark 1.4.
By our assumptions on S , Σ t can only move a finite distance infinite time (see Proposition 7.3). In other words, if T < ∞ , Σ t approaches a set-theoretic limit. One can probably construct examples with free boundary in abarrier of unbounded geometry/zero inscribed radius which shoot off to infinity infinite time.If T = ∞ and Σ t stays in a bounded region, then either | Σ t | → t approaches a minimal surface. ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 3
Remark 1.5.
Theorems 1.1 and 1.2 also hold in a Riemannian manifold of boundedgeometry. In fact the error terms introduced are almost entirely subsumed by theperturbations we already make.
Theorem 1.6 (Umbilic pinching, [Sta96a]) . If Σ is convex and S = S n , then Σ t shrinks to a point in finite time, and any limit flow at the singularity is umbilic. Inparticular, there is a sequence of rescalings which converge to a shrinking half-spherewith free-boundary in a hyperplane. The following notation is used extensively.
Definition 1.6.1.
Writing f = O ( g ) means there is a constant c = c ( n, S ) suchthat | f | ≤ c | g | .We outline our approach. The main obstruction to analyzing free boundarybehavior in a general barrier S is obtaining boundary conditions on | A | , or S k when k >
1. We perturb the second fundamental form so that the normal ν S isan eigenvector, which allows us to obtain boundary conditions on the perturbedprinciple curvatures.This introduces relatively large error terms into the evolution equations of theperturbed | ¯ A | and ¯ H . The error is too large to naively give exponential behaviorof the quantity | ¯ A | /H . To handle this, and to correct the boundary behavior, inproving Theorem 1.1 we must consider instead the evolution of(2) | ¯ A | + aH φ, for some large constant a , and barrier function φ .The evolution equation for (2) will have the right form except for a gradient termresulting from φ . To control bad gradient terms we observe that by restricting topoints where | ¯ A | ≥ H , we can squeeze a term out of Cauchy’s inequality: |∇ ¯ A | − |∇| ¯ A || ≥ c |∇ ¯ A | + O ( | ¯ A | ) . Given Theorem 1.1, we can adapt the Stampacchia iteration scheme used by[HS99a] to prove Theorems 1.2 and 1.6. The key step is proving a trace-like formulafor free boundary surfaces. The argument is sufficiently robust to handle withoutproblem the perturbation terms.I am very grateful to my advisor Simon Brendle for his guidance and encour-agement, Brian White for many illuminating discussions, and Otis Chodosh forhis support and advice. I also thank Robert Haslhofer and Gerhard Huisken forhelpful conversations. This work was partially supported by the Royden fellowship.Some of this work was also completed while visiting Columbia University, and I’mgrateful for their hospitality.2.
Michael-Simon with (free) boundary
We adapt the Michael-Simon inequality [MS73] to surfaces with smooth bound-ary, and surfaces meeting a barrier surface orthogonally.
Lemma 2.1.
There is a constant c = c ( n, S ) such that for any Σ meeting S orthogonally, and any v ∈ C ( ¯Σ) , (3) 1 c Z ∂ Σ | v | ≤ Z Σ |∇ v | + Z Σ | Hv | + Z Σ | v | . NICK EDELEN
Proof.
Choose (and fix, for the duration of the paper) a smooth vector field X on R n +1 which is 0 outside a neighborhood of S , and X ≡ ν S on S . Then Z ∂ Σ | v | = Z ∂ Σ | v | X · ν = Z Σ div Σ ( | v | X T )= Z Σ ∇| v | · X + | v | div Σ ( X ) − | v | X · νH ≤ max | X | Z |∇ v | + n max |∇ X | Z | v | + max | X | Z | vH | (cid:3) Theorem 2.2.
There is a constant c = c ( n ) such that for any v ∈ C c ( ¯Σ) , we have (4) 1 c (cid:18)Z Σ | v | nn − (cid:19) n − n ≤ Z Σ |∇ v | + Z Σ | Hv | + Z ∂ Σ | v | . Proof.
By replacing v with | v | we can without loss of generality suppose v ≥ x ∈ ∂ Σ, let γ x ( t ) be the unit speed geodesic in Σ with initial conditions γ x (0) = x and γ ′ x (0) ⊥ ∂ Σ. For sufficiently small ǫ , depending only on the curva-tures of Σ and ∂ Σ ⊂ Σ, the function φ : [0 , ǫ ] × ∂ Σ → Σ mapping ( t, x ) γ x ( t ) isa diffeomorphism, with its Jacobian bounded like | Jφ | ∈ [ , ǫ sufficiently small, Z dist( · ,∂ Σ) ≤ ǫ v = Z ǫ Z ∂ Σ v | Jφ |≤ Z ǫ Z ∂ Σ v (0 , x ) + 2 Z ǫ Z ∂ Σ t ∂v∂t ( t ∗ ( x ) , x ) ≤ ǫ Z ∂ Σ v + ǫ | ∂ Σ | sup Σ |∇ v | (5)here t ∗ ( x ) ∈ (0 , ǫ ).Now take η a function which is ≡ · , ∂ Σ) ≥ ǫ and ≡ ∂ Σ, and suchthat |∇ η | ≤ /ǫ . From (5) Z Σ ((1 − η ) v ) nn − ≤ Z dist( · ,∂ Σ) ≤ ǫ v nn − ≤ ǫ Z ∂ Σ v nn − + ǫ | ∂ Σ | sup Σ |∇ ( v nn − ) |≤ ǫC for C independant of ǫ .Therefore, using the Michel-Simon inequality and (5) again, || v || nn − ≤ || ηv || nn − + || (1 − η ) v || nn − ≤ c Z Σ η |∇ v | + c Z Σ | H | ηv + c Z Σ |∇ η | v + ǫ n − n C ≤ c Z Σ |∇ v | + c Z Σ | H | v + 2 c/ǫ Z dist( · ,∂ Σ) ≤ ǫ v + ǫ / C ≤ c Z Σ |∇ v | + c Z Σ | H | v + 4 c Z ∂ Σ v + ǫ | ∂ Σ | sup Σ |∇ v | + ǫ / C ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 5 for c = c ( n ) and all ǫ sufficiently small. Taking ǫ to 0 proves the lemma. (cid:3) Theorem 2.3. If Σ meets S orthogonally, and v ∈ C ( ¯Σ) , then for any p < n , (6) || v || npn − p ;Σ ≤ c ( ||∇ v || p ;Σ + || Hv || p ;Σ + || v || p ;Σ ) where c = c ( n, p, S ) .Proof. Combine Lemma 2.1 and Theorem 2.2 to obtain the desired inequality with p = 1. Then set v = w γ to obtain (cid:18)Z w γ nn − (cid:19) n − n ≤ cγ Z w γ − |∇ w | + c Z w γ − Hw + c Z w γ − w ≤ c (cid:16) w ( γ − pp − (cid:17) p − p ( ||∇ w || p + || Hw || p + || w || p ) . Now choose γ such that γ nn − γ − pp − (cid:3) Corollary 2.4. If n = 2 , then for any q ∈ (1 , ∞ ) , (7) || v || q ;Σ ≤ c | Σ | q ( ||∇ v || + || Hv || + || v || ) where c = c ( q, S ) .Proof. Take n = 2 and p = 2 − δ in Theorem 2.3, for δ ∈ (0 , q = − δδ . Thenwe have for any r ∈ (1 , ∞ ), || v || q ≤ c ( ||∇ v || − δ + || Hv || − δ + || v || − δ ) ≤ c | Σ | (1 − /r ) − δ ( ||∇ v || r (2 − δ ) + || Hv || r (2 − δ ) + || v || r (2 − δ ) ) . Then set r = − δ . (cid:3) Remark 2.5.
Since | Σ t | is monotone decreasing (Remark 4.2), c ( q, S ) | Σ t | / q ≤ c ( q, S, | Σ | ) . General inequalities and Stampacchia iteration
Each pinching result uses a Stampacchia iteration scheme to obtain pointwisebounds from L p -bounds. All cases can be handled by the following general principle.Take (Σ t ) t ∈ [0 ,T ) a mean curvature flow with free boundary in S , and assume T < ∞ . Let f α be some non-negative function on Σ t , depending on some parameters α = α ( S, Σ , T, n ). Let ˜ G ≥ H > t such that H = O ( ˜ H ) , ∇ ˜ H = O ( ˜ G ) . Let f = f α ˜ H σ , and f k = ( f − k ) + , where σ > k > A ( k ) = { f ≥ k } , and A ( k, t ) = A ( k ) ∩ Σ t .We say f satisfies ( ⋆ ) if there are constants c = c ( S, Σ , T, n, α ), and C = C ( S, Σ , T, n, α, p, σ ), such that for any p > p ( n, α, c ), σ < / k > β > NICK EDELEN (POINCARE-LIKE)1 c Z Σ t f p ˜ H ≤ ( p + p/β ) Z Σ t f p − |∇ f | + (1 + βp ) Z Σ t ˜ G ˜ H − σ f p − + Z Σ t f p + Z ∂ Σ t f p − ˜ H σ (EVOLUTION-LIKE) ∂ t Z Σ t f pk ≤ − p Z Σ t f p − k |∇ f | − p/c Z Σ t ˜ G ˜ H − σ f p − k + cpσ Z A ( k,t ) ˜ H f p − / Z Σ t ˜ H f pk + C Z A ( k,t ) f p + C | A ( k ) | + cp Z ∂ Σ t f p − k ˜ H σ This section culminates in proving
Theorem 3.1. If f satisfies ( ⋆ ) , then for p sufficiently big, and σ sufficiently small(depending on p ), f is uniformly bounded in spacetime. The bound will depend on ( S, Σ , T, n, α, p, σ ) . The following Lemma is the key step in handling the free boundary behavior.We first make a useful observation.
Remark 3.2.
Let g be an arbitrary non-negative function on Σ t . If r ∈ (0 , q ∈ (0 , p ) with rp/q <
2, then for any µ > Z g q ˜ H r ≤ Z g p ˜ H rp/q + | spt g |≤ µ Z g p ˜ H + C ( µ, r, q, p ) Z g p + | spt g | Lemma 3.3.
For any µ > and p > , σ < / , we can pick constants c = c ( n, S ) and C = ( n, S, µ, p ) such that Z ∂ Σ t f p − k ˜ H σ ≤ c Z Σ t |∇ f | f p − k + cσ Z Σ t ˜ G ˜ H − σ f p − k + cp µ Z A ( k,t ) f p ˜ H C Z A ( k,t ) f p + C | A ( k, t ) | Proof.
Using the trace formula of 2.1, and Peter-Paul, we have (all integrals on theright-hand-side are over Σ t ) Z ∂ Σ t f p − k ˜ H σ ≤ cp Z f p − k |∇ f | ˜ H σ + cσ Z f p − k ˜ H σ − |∇ ˜ H | + c Z f p − k ˜ H σ + c Z f p − k ˜ H σ ≤ c Z f p − k |∇ f | + cp Z f p − k ˜ H σ + cσ Z f p − k ˜ G ˜ H − σ + c Z f p − k ( ˜ H σ + ˜ H σ ) . The Lemma follows by Remark 3.2. (cid:3)
ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 7
The hard part of Theorem 3.1 is establishing L p bounds for appropriately large σ . In particular, we establish spacetime L p bounds for σ ∼ p − / , rather than thenaive σ ∼ p − , and thereby have the following wiggle room. Lemma 3.4.
Suppose there is a p and c σ , independent of p, σ , such that whenever p > p and σ < c σ √ p , Z T Z Σ t f p < ∞ . Then for m > , Z T Z Σ t ˜ H m f p < ∞ provided p > m /c σ + p and σ < c σ √ p .Proof. Follows directly from ˜ H m f p = ( f α ˜ H σ + m/p ) p . (cid:3) Lemma 3.5.
Given ( ⋆ ) , then Z T Z Σ t f p < ∞ for p > p ( c ) , and σ < c σ ( c ) p − / .Proof. Combining equations (POINCARE-LIKE), (EVOLUTION-LIKE), and Lemma3.3, we have the following inequalities. We adhere to the convention c = c ( S, Σ , T, n, α ) NICK EDELEN and C = C ( S, Σ , T, n, α, p, σ, µ ). Unless stated otherwise all integrals are on Σ t . ∂ t Z f p ≤ − p / Z |∇ f | f p − − p/c Z ˜ G ˜ H − σ f p − + cpσ " p (1 + 1 /β ) Z |∇ f | f p − + (1 + βp ) Z ˜ G ˜ H − σ f p − + Z f p + Z ∂ Σ t f p − ˜ H σ (cid:21) − / Z f p ˜ H + C Z f p + C | Σ t | + cp Z ∂ Σ t f p − ˜ H σ ≤ ( − p / cp σ (1 + 1 /β )) Z |∇ f | f p − + ( − p/c + cpσ (1 + βp )) Z ˜ G ˜ H − σ f p − + cp "Z |∇ f | f p − + σ Z ˜ G ˜ H − σ f p − + p /µ Z f p ˜ H + C Z f p + C | Σ t | + C Z f p + C | Σ t | − / Z f p ˜ H ≤ ( − p / cp σ (1 + 1 /β ) + cp ) Z |∇ f | f p − + ( − p/c + cpσ (1 + βp ) + cpσ ) Z ˜ G ˜ H − σ f p − + ( cp /µ − / Z f p ˜ H + C Z f p + C | Σ t | Choose σ = ( c p ) − / , β = ( cp ) − / and µ = 10 cp , then for p > c we havethat R Σ t f p increases at most exponentially. (cid:3) Now for arbitrary k , we can combine equation (EVOLUTION-LIKE) with Lemma3.3 in an identical manner to obtain ∂ t Z Σ t f pk ≤ − p / Z Σ t |∇ f | f p − k + C Z A ( k,t ) f p ˜ H + C Z A ( k,t ) f p + C | A ( k, t ) | for σ , and p satisfying the same bounds as Lemma 3.5. Here, as in Lemma 3.5, c and C are both independent of k .The following Theorem will complete the proof of Theorem 3.1. Theorem 3.6.
Suppose there is a p and c σ , independent of p, σ, k , such thatwhenever p > p and σ < c σ √ p , we have Z T Z Σ t f p < ∞ and (8) ∂ t Z Σ t f pk + 1 /c Z Σ t |∇ f p/ k | ≤ C Z A ( k,t ) ˜ H f p + C Z A ( k,t ) f p + C | A ( k, t ) | ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 9 for any k > . Here c and C can depend on any quantity except k . Then for p sufficiently large, and σ sufficiently small, f p is uniformly bounded in spacetime.The bound will depend on ( S, Σ , T, n, p, σ, α ) .Proof. By Theorem 2.3 and Corollary 2.4, for each n ≥ q >
1, and c = c ( n, q, | Σ | ), such that (cid:18)Z Σ v q (cid:19) /q ≤ c Z Σ | Dv | + c Z Σ v H + c Z Σ v . So take v = f p/ k and integrate (8) to obtain (for possibly larger C )max ( sup [0 ,T ) Z Σ t f pk , Z T (cid:18)Z Σ t f pqk (cid:19) /q ) ≤ C Z Z A ( k ) f p + C Z Z A ( k ) ˜ H f p + C | A ( k ) | . provided k ≥ k (Σ , n, p, σ, α ). All terms on the right are bounded by virtue ofLemma 3.4, and the monotonicity of | Σ t | . Therefore Z T Z f p q − q k ≤ Z T (cid:18)Z f pqk (cid:19) /q (cid:18)Z f pk (cid:19) q − q ≤ C Z Z A ( k ) f p + Z Z A ( k ) ˜ H f p + | A ( k ) | ! q − q ≤ C | A ( k ) | q − q (1 − /r ) Z Z A ( k ) f pr ! /r + Z Z A ( k ) ˜ H r f pr ! /r + | A ( k ) | /r q − q ≤ C ( S, α, p, σ, T, c σ , Σ ) | A ( k ) | α for any r , provided p > r/c σ + p and σ < c σ √ p . If we fix r sufficiently large, then α = q − q (1 − /r ) >
1. Fix p , σ , then for any ℓ > k , we have the inequality(9) | ℓ − k | β | A ( ℓ ) | ≤ C | A ( k ) | α where β = p q − q >
0, and C is independent of ℓ, k . It follows by a standardargument that A ( k ) = 0 for k > k ( α, β, C ), C as in (9). (cid:3) Mean curvature flow with free boundary preliminaries
Let (Σ t ) t ∈ [0 ,T ) be the mean curvature flow of Σ , with free-boundary in S . Here,as always in this paper, T is the maximal time of existence.Write g = ( g ij ) and A = ( h ij ) for the induced metric and second fundamentalform on Σ t . We follow the usual convention that g ij is the matrix inverse to g ij ,and a raised index such as h ij means P k g ik h kj . We denote dV the volume formon Σ t , and take N for the outward normal of ∂ Σ ⊂ Σ.We write ∇ for covariant differentiation in Σ, and ¯ ∇ for covariant differentiationin R n +1 . We write ( k ij ) for the second fundamental form of the barrier surface S . Proposition 4.1.
We have the following evolution equations, using summationconvention. ∂ t g ij = − Hh ij ∂ t h ij = ∆ h ij − Hh im h mj + | A | h ij and ∂ t H = ∆ H + | A | H∂ t dV = − H dV∂ t ν = ∇ H Proof.
See [Hui84]. (cid:3)
Remark 4.2.
Since the boundary ∂ Σ t is always orthogonal to the direction ofmotion, ∂ t | Σ t | = − Z Σ t H ≤ . Specifying other angles of contact would add a boundary term to ∂ t | Σ t | , and couldeven cause area increase. Proposition 4.3.
We have N ( H ) = k νν H. In particular, positivity of H is preserved for all time. If S is convex, then H isnon-decreasing, and in fact must blow up in finite time.Proof. Differentiate the relation < N, ν > = 0 in time. Evolution behavior followsfrom Proposition 4.1. (cid:3)
Remark 4.4.
Notice that H may still decrease. We will show later that H de-creases at worst exponentially in time. Proposition 4.5.
For any X ∈ T p ∂ Σ , h N,X = − k ν,X . Proof.
Since N ≡ ν S along T p ∂ Σ, h ( N, X ) = − < ν, ¯ ∇ X ν S > = − k ( ν, X ) (cid:3) As mentioned in the Introduction the key technical issue in extending the esti-mates to general barrier surfaces is in calculating ∇ N h N,X = ∇ X h N,N , for X ∈ T p ∂ Σ. To avoid the issue we perturb h so that h N,X = 0.
Definition 4.5.1.
Extend and fix k and ν S to be defined on R n +1 . Define the perturbed second fundamental form ¯ A of Σ to be(10) ¯ h ij = h ij + T ijν + D g ij where T is a 3-tensor defined on the ambient space by T ( X, Y, Z ) = k ( X, Z ) g ( Y, ν S ) + k ( Y, Z ) g ( X, ν S ) . We choose and fix the constant D so that T ( X, X, ν ) + D ≥ X . From henceforth when a constant depends on D or theextensions of k or ν S , we will only say it depends on the barrier surface S .Our choice of D and Proposition 4.3 imply that(11) ¯ H ≥ H + 1 ≥ , | ¯ A | ≥ . ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 11 Evolution of tensors
Proposition 5.1.
Let T be a 3-tensor defined on the ambient space. If T ijν is the2-tensor T ( · , · , ν ) restricted to T Σ , then ∇ T ijν = O (1 + | A | ) ∇ T ijν = O (1 + | A | + |∇ A | )( ∂ t − ∆) T ijν = O (1 + | A | )( ∂ t − ∆) T ijν = O (1 + | A | ) . Proof.
Choose orthonormal geodesic coordinates ∂ i at a fixed point p . We use thesummation convention, excepting of course on ν . We have ∇ p T ijν = ¯ ∇ p T ijν + T ij ¯ ∇ p ν + T ∇ ⊥ p ijν + T i ∇ ⊥ p jν = ¯ ∇ p T ijν + h pk T ijk − h pi T νjν − h pj T iνν = O (1 + | A | )We work towards calculating ∇ T and ∆ T . We have ∇ q ( h pi T νjν ) = ( ∇ q h pi ) T νjν + h pi ∇ q T νjν = ∇ i h pq T νjν + h pi ( ¯ ∇ q T νjν + h qk T νjk + h qk T kjν − h qj T ννν )= ∇ i h pq T νjν + O (1 + | A | )and ∇ q ( h pk T ijk ) = ∇ k h pq T ijk + h pk ( ¯ ∇ q T ijk − h qi T νjk − h qj T iνk − h qk T ijν )= ∇ k h pq T ijk + O (1 + | A | )and ∇ q ¯ ∇ p T ijν = ¯ ∇ q,p T ijν + ¯ ∇ ∇ ⊥ q p T ijν + ¯ ∇ p T ∇ ⊥ q ijν + ¯ ∇ p T i ∇ ⊥ q jν + ¯ ∇ p T ij ∇ q ν = ¯ ∇ q,p T ijν − h qp ¯ ∇ ν T ijν − h qi ¯ ∇ p T νjν − h qj ¯ ∇ p T iνν + h qk ¯ ∇ p T ijk = O (1 + | A | ) . We therefore have ∇ q,p T ijν = ∇ q ( ¯ ∇ p T ijν + h pk T ijk − h pi T νjν − h pj T iνν )= ∇ k h pq T ijk − ∇ i h pq T νjν − ∇ j h pq T iνν + O (1 + | A | )= O (1 + | A | + |∇ A | )and ∆ T ijν = ∂ k HT ijk − ∂ i HT νjν − ∂ j HT iνν + O (1 + | A | )= O (1 + | A | + |∇ H | ) . We calculate the time derivative. Here (¯ x α ) are standard coordinates in R n +1 . ∂ t T ijν = ∂ t (cid:18) T αβγ ( F ( x )) ∂F α ∂x i ∂F β ∂x j ν γ (cid:19) = (cid:18) ∂T αβγ ¯ x δ ∂ t F δ (cid:19) ∂ αi F ∂ βj F ν γ + T αβγ (cid:18) ∂∂x i ∂F α ∂t (cid:19) ∂ βj F ν γ + T αβγ ∂ αi F (cid:18) ∂∂x j ∂F β ∂t (cid:19) ν γ + T αβγ ∂ αi F ∂ βj F ν γ ∂t = − H ¯ ∇ ν T ijν + T ( ¯ ∇ i ( − Hν ) , ∂ j , ν ) + T ( ∂ i , ¯ ∇ j ( − Hν ) , ν ) + T ( ∂ i , ∂ j , ∇ H )= − H ¯ ∇ ν T ijν − ∂ i HT νjν − Hh ik T kjν − ∂ j HT iνν − Hh jk T ikν + ∂ k HT ijk = − ∂ i HT νjν − ∂ j HT iνν + ∂ k HT ijk + O (1 + | A | )which proves the penultimate formula. The last formula follows by observing that ∂ t g ij = O ( | A | ). (cid:3) Corollary 5.2.
We have O ( | ¯ A | ) , | A | = O ( | ¯ A | ) , |∇ A | = O ( |∇ ¯ A | + | ¯ A | ) . Proof.
The first formula follows trivially from | ¯ A | ≥
1. The second because ¯ A = A + O (1). The third since ∇ A = ∇ A + O ( | A | + 1). (cid:3) Theorem 5.3.
We have the evolution equations ∂ t ¯ h ij = ∆¯ h ij + | ¯ A | ¯ h ij + O ( | ¯ A | ) ∂ t | ¯ A | = ∆ | ¯ A | + 2 | ¯ A | − |∇ ¯ A | + O ( | ¯ A | ) ∂ t H = ∆ H + | ¯ A | H + O ( | ¯ A | ) H Proof.
We deduce the first formula by Propositions 4.1 and 5.1.We have12 ( ∂ t − ∆) | ¯ A | = 12 ∂ t ( g ik g kl ¯ h ij ¯ h kl ) − < ∆ ¯ A, ¯ A > −|∇ ¯ A | = 2 Hh ik g jl ¯ h ij ¯ h kl + g ik g jl ( ∂ t − ∆)( h ij + T ijν + Dg ij )¯ h kl − |∇ ¯ A | = 2 Hh ik g jl (¯ h ij ¯ h kl − h ij ¯ h kl ) + | A | < A, ¯ A > + O (1 + | ¯ A | ) − D Hg ik g jl h ij ¯ h kl − |∇ ¯ A | = | ¯ A | − |∇ ¯ A | + O ( | ¯ A | ) . The third formula is an immediate consequence of Proposition 4.1 and Corollary5.2. (cid:3)
Lemma 5.4.
Let M be a symmetric matrix, and η > . If | M | > (1 + η )tr( M ) ,then | M | − max i | λ i | ≥ c | M | . Here { λ i } are the eigenvalues of M , and c = c ( n, η ) .Proof. Otherwise, we can pick a sequence of counterexamples M ( j ) with | M ( j ) | = 1and | M ( j ) | − max i | λ ( j ) i | ≤ | M ( j ) | /j = 1 /j. ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 13
Since each entry lies in the interval [ − , M ( j ′ ) con-verging to M . Then all but one eigenvalue of M is zero, contradicting | M | ≥ (1 + η )tr( M ). (cid:3) Proposition 5.5. If | ¯ A | > H , then (12) |∇ ¯ A | − |∇| ¯ A || ≥ c |∇ ¯ A | + O ( | ¯ A | ) where c = c ( n ) .Proof. We have that ∇ i ¯ h jk = ∇ j ¯ h ik + O ( | ¯ A | )and therefore, if we pick orthonormal coordinates so that ∂ = ∇| ¯ A | / |∇| ¯ A || at thepoint in question, | ¯ A | ( |∇ ¯ A | − |∇| ¯ A || ) = (cid:12)(cid:12) | ¯ A |∇ i ¯ h jk − ∇ i | ¯ A | ¯ h jk (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | ¯ A |∇ i ¯ h jk −
12 ( ∇ i | ¯ A | ¯ h jk + ∇ j | ¯ A | ¯ h ik ) + 12 ( ∇ j | ¯ A | ¯ h ik − ∇ i | ¯ A | ¯ h jk ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12) ∇ j | ¯ A | ¯ h ik − ∇ i | ¯ A | ¯ h jk (cid:12)(cid:12) − c | ¯ A | |∇| ¯ A ||≥ |∇| ¯ A || ( | ¯ A | − X k ¯ h k ) − c | ¯ A | |∇| ¯ A ||≥ |∇| ¯ A || ( | ¯ A | − max i | ¯ λ i | ) − c | ¯ A | |∇| ¯ A || . Here c = c ( n, S ), and ¯ λ i are the eigenvalues of ¯ A .By Lemma 5.4 there is a c n depending only on n so that | ¯ A | − max i | ¯ λ i | ≥ c n | ¯ A | and hence by Peter-Paul we deduce that |∇ ¯ A | − |∇| ¯ A || > c n |∇| ¯ A || − c | A | . This can be rearranged to deduce |∇ ¯ A | − |∇| ¯ A || > c n + 1 |∇ ¯ A | − c | A | (cid:3) Corollary 5.6.
Whenever | ¯ A | > H , (13) ( ∂ t − ∆) | ¯ A | ≤ | ¯ A | − c |∇ ¯ A | | ¯ A | + O ( | ¯ A | ) where c = c ( n ) .Proof. We have (recalling | ¯ A | ≥ ∂ t − ∆) | ¯ A | = ( ∂ t − ∆) q | ¯ A | = 12 ( ∂ t − ∆) | ¯ A || ¯ A | + 14 |∇| ¯ A | | | ¯ A | = | ¯ A | + |∇| ¯ A || − |∇ ¯ A | | ¯ A | + O ( | ¯ A | ) . Now apply Proposition 5.5. (cid:3) Boundary derivatives
Fix a p ∈ ∂ Σ. Choose coordinates so that ∂ ≡ N along ∂ Σ, ( ∂ i ) i> are or-thonormal geodesic normal coordinates on ∂ Σ at p , and the integral curves of ∂ are geodesics. Lemma 6.1. At p we have, for i, j > , ∇ h ij = h ij k νν + h k ij − k iα h jα − k jα h iα − ∇ Sν k ij ∇ h = 2( k αβ h αβ + h k νν ) − Kh + ν ( K ) − ∇ Sν k νν where α, β are summed over , . . . , n , and K is the mean curvature of the barrier S .Proof. We calculate for i, j > ∂ h ij = − < ∂ i ∂ j N, ν > − < ∂ i ∂ j F, ∂ ν > = − < ∂ i ( k αj ∂ α + k νj ν ) , ν > + h k ij = k jα h iα − ∇ Si k νj − k ν ∇ Si ∂j − k ∇ Si νj + h k ij = −∇ Sν k ij + h ij k νν + h k ij and hence ∇ h ij = ∂ h ij − h (( ∂ i ∂ F ) T , ∂ j ) − h ( ∂ i , ( ∂ j ∂ ) T )= − k iα h jα − k jα h iα − ∇ Sν k ij + h ij k νν + h k ij We calculate, using Proposition 4.3, N ( H ) = k νν H = ∇ h + tr ∂ Σ ( ∇ h )= ∇ h − k αβ h αβ − tr ∂ Σ ( ∇ Sν k ij ) + ( H − h ) k νν + h ( K − k νν )= ∇ h − k αβ h αβ − h k νν − ν ( K ) + ∇ Sν k νν + Hk νν + Kh and the Lemma follows. (cid:3) Theorem 6.2. At p , for i, j > , ∇ ¯ h ij = O ( | ¯ A | ) ∇ ¯ h = O ( | ¯ A | ) . Proof.
Follows directly from Lemma 6.1 using Theorem 5.1. (cid:3)
Theorem 6.3.
We have that N | ¯ A | = O ( | ¯ A | ) . Proof.
Immediate from Theorem 6.2 and that ¯ h N,X = 0 when X ∈ T p ∂ Σ. (cid:3) ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 15 Controlling | ¯ A | In this section we prove the following Theorem, which will imply Theorem 1.1.
Theorem 7.1.
There are constants α = α ( S, n ) ≥ and C = C ( S, Σ ) so that (14) max Σ t | ¯ A | H ≤ Ce αt for all time of existence. Remark 7.2. If S is convex, then H is non-decreasing, and by carefully calculatingthe normal derivative N | ¯ A | one can take α = 0 in (14).For arbitrary function f and g , recall the useful formula(15) ( ∂ t − ∆) fg = ( ∂ t − ∆) fg − fg ( ∂ t − ∆) g + 2 g < ∇ fg , ∇ g > . Proof of Theorem 7.1.
Recall that | N H | ≤ bH | N | ¯ A || ≤ b | ¯ A | for some constant b = b ( S ).Let d : R n +1 → [ − ,
1] be a smooth function such that d ≡ S , and ν S ( d ) ≥ d we will only say it depends on S . Let φ : R n +1 → R + be the smooth function φ ( x ) = e − αt − bd so that ν S ( φ ) ≤ − bφ .We have, in geodesic orthonormal coordinates,( ∂ t − ∆) φ = − αφ + ¯ ∇ φ · ( ∂ t F − ∆ F ) − X i ¯ ∇ φ ( ∂ i F, ∂ i F )= − αφ − tr T Σ ( ¯ ∇ φ )= ( − α + O (1)) φ. Choose α = α ( S, n ) so that ( ∂ t − ∆) φ < Σ t H/φ is non-decreasing. First calculate(16)
N Hφ ≥ b Hφ , so any spatial minimum is interior. And by our choice of α we obtain(17) ( ∂ t − ∆) Hφ ≥ | A | Hφ + 2 φ < ∇ Hφ , ∇ φ > . In particular, at any spatial minimum p of H/φ , we must have ∂ t Hφ | p ≥ | A | Hφ ≥ . We now consider the quantity f = | ¯ A | + aH/φ for some positive constant a to be determined. We show max Σ t f is non-increasingwhen f is sufficiently big. At the boundary we have by (16)(18) N f ≤ b | ¯ A | H/φ − bH/φ | ¯ A | ( H/φ ) ≤ . So any spatial maximum of f is interior.From Corollary 5.6 and equation (17), whereever | ¯ A | > H we have the evolutionequations ( ∂ t − ∆) | ¯ A | ≤ | ¯ A | − c n |∇| ¯ A || | ¯ A | + c | ¯ A | ( ∂ t − ∆) Hφ ≥ | ¯ A | Hφ − c | ¯ A | Hφ + 2 φ < ∇ Hφ , ∇ φ > . Here c = c ( S, n ) and c n = c n ( n ).We calculate( ∂ t − ∆) f ≤ H/φ (cid:18) | ¯ A | − c n |∇| ¯ A || | ¯ A | + c | ¯ A | (cid:19) − f ( | ¯ A | − c | ¯ A | ) − fH < ∇ Hφ , ∇ φ > + 2 H/φ < ∇ f, ∇ Hφ > ≤ φH (cid:26) | ¯ A | − c n |∇| ¯ A || | ¯ A | + c | ¯ A | − ( | ¯ A | + a )( | ¯ A | − c | ¯ A | ) (cid:27) + 2 H |∇| ¯ A |||∇ φ | + < ∇ f, φ ∇ φ + 2 φH ∇ Hφ > ≤ φH (cid:26) (2 c − a ) | ¯ A | + ac | ¯ A | − c n |∇| ¯ A || | ¯ A | + 2 c n |∇ φ | φ | ¯ A | (cid:27) + < ∇ f, φ ∇ φ + 2 φH ∇ Hφ > .
Notice that |∇ φ | φ = O (1). By the above calculations and equation (18), if f attains its spatial maximum at a point p , and | ¯ A | > H at this point, then ∂ t f | p ≤ φH (cid:8) ( c − a ) | ¯ A | + ca | ¯ A | (cid:9) ≤ a = 2 c and ensure | ¯ A | > c .We still need to prove this implies Theorem 7.1. Recall that ¯ H = H + O (1).Using that min Σ t H/φ is non-decreasing, we have¯ H ≤ H + c ≤ c Hφ (cid:18) Σ H/φ (cid:19) . Define the constant C = 4 c min Σ H/φ + 2 c (cid:18) Σ H/φ (cid:19) . ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 17
Then if f ≥ C , we have | ¯ A | ≥ C Hφ − c ≥ c (1 + (min H/φ ) − ) Hφ + 4 c − c ≥ H + 2 c. We deduce that | ¯ A | H ≤ fφ ≤ φ − max { C, max Σ f } , which proves the Theorem. (cid:3) Proposition 7.3.
There are constants α = α ( S, n ) , C = C ( S, Σ ) so that max x ∈ Σ t dist( x, ≤ Ce αt for all time of existence. In particular, if T < ∞ , then we can find a radius R satisfying [ t ∈ [0 ,T ) Σ t ⊂ B R (0) . Proof.
We consider the quantity f = φ | F | = φ X β F β , where φ is the cutoff function from Theorem 7.1.At the boundary N | F | = 2 X β F β N β ≤ {| F | , } . And in the interior( ∂ t − ∆) | F | = 2 X β F β ( ∂ t − ∆) F β − X β |∇ F β | ≤ . Whenever | F | ≥
1, we obtain (provided b ≥ N f ≤ | F | φ − bφ | F | ≤ , and hence any spatial maximum on {| F | ≥ } is interior.We calculate ( ∂ t − ∆) f ≤ − φ < ∇ f, ∇ φ > +2 | F | φ |∇ φ | φ ≤ − φ < ∇ f, ∇ φ > + cf for c = c ( S, n ). We deduce that | F | ≤ φ − e ct max { , max Σ f } . (cid:3) Convexity pinching
We prove Theorem 1.2. Recall that we wish to show that if
T < ∞ , then for any k ∈ { , . . . , n } and any η > S k ≥ − ηH k − C with C = C ( S, Σ , T, η, n ). Here S k is the k -th symmetric polynomial of the prin-ciple curvatures. We following [HS99a] and prove (19) by induction on k . Noticethis is trivially true for k = 1.From henceforth assume (19) holds up to a fixed k , i.e. S l ≥ − ηH l − C for every l = 1 , . . . , k . We will now prove (19) for k + 1. Of course we also from now onassume T < ∞ .In spirit we would like to consider the function − S k +1 /S k − ηHH H σ and show this is bounded above in spacetime. However for general k we have nopositivity control over the denominator S k . We require a further perturbation ofthe second fundamental form. Definition 8.0.1.
Let ˜ A = ( b ij ) be the twice-perturbed second fundamental form b ij = ¯ h ij + ( ǫH + D − D ) g ij = h ij + T ijν + ( ǫH + D ) g ij . Here D ≥ D + 1 and ǫ ∈ (0 , n ] are constants to be fixed later.We write ˜ λ i for the eigenvalues of b ij , so that if ¯ λ i are the eigenvalues of thefirst-perturbed ¯ h ij , then ˜ λ i = ¯ λ i + ( ǫH + D − D ) . Correspondingly | ˜ A | is the norm of the twice-perturbed second fundamental form,˜ H the mean curvature, and ˜ S k = s k (˜ λ ), ˜ Q k = q k (˜ λ ) where defined.Recall we had fixed D = D ( S ) so that T ( X, X, ν ) + D ≥ X . So we still have the conditions(20) ˜ H ≥ H + 1 ≥ , | ˜ A | ≥ | A | ≤ c ( S, Σ , T ) H , we have(21) | ˜ A | ≤ c ( S, Σ , T ) ˜ H. Remark 8.1.
Since | ˜ A | ≥ ǫ ≤ n , we have1 = O ( | ˜ A | ) , | A | = O ( | ˜ A | ) , |∇ A | = O ( |∇ ˜ A | + | ˜ A | ) . Lemma 8.2. If ¯ h ij = h ij + O (1) , and S l ≥ − θH l − C for any θ > , then we also have ¯ S l ≥ − θ ¯ H l − ¯ C for any θ > . Here both C, ¯ C depend on S, Σ , T, θ, n . ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 19
Proof.
Given θ > C , we have for c = c ( S, Σ , T, n, l ),¯ S l ≥ S l − cH l − ≥ − θH l − C − cH l − ≥ − θ ¯ H l − C − c ¯ H l − − c ≥ − θ ¯ H l − C − c − c (cid:16) cθ (cid:17) l − . (cid:3) Lemma 8.3.
Suppose for any l = 1 , . . . , k and any θ > , we have S l ≥ − θH l − C. Then for any ǫ ∈ (0 , n ] , there is a D ǫ ≥ D + 1 such that (22) ˜ S k ≥ ǫ nǫ n − k + 1 k ˜ S k − ˜ H whenever D ≥ D ǫ .Proof. Lemma 8.2 implies the hypothesis holds for ¯ S l ( l = 1 , . . . , k ). Since b ij =¯ h ij +( ǫH + D − D ) g ij , by Lemma 2.7 of [HS99a] there exists a D = D ( ǫ, S, Σ , T )such that (22) holds whenever D − D ≥ D . Now set D ǫ = D + D + 1. (cid:3) Although we will fix ǫ ∈ (0 , n ] later, for the duration of the paper we take D = D ǫ as in Lemma 8.3. Remark 8.4.
Our inductive hypothesis and our choice of D implies that, for l = 1 , . . . , k (23) ˜ S l ≥ c ( n ) ǫ ˜ H ˜ S l − ≥ c ( n ) ǫ l − ˜ H l . Remark 8.5 (Derivatives of ˜ S l ) . ˜ S l is a homogeneous degree l polynomial in theentries b ij . If ∂ denotes differentiation in the entries of b ij , we have for any d + s ≤ l and any l = 1 , . . . , n (cid:12)(cid:12)(cid:12) ∂ d ˜ S l (cid:12)(cid:12)(cid:12) ≤ c ( S, T, Σ , n ) ˜ H l − d and (cid:12)(cid:12)(cid:12) ∇ s ∂ d ˜ S l (cid:12)(cid:12)(cid:12) ≤ c ( S, T, Σ , n ) ˜ H l − d − s |∇ ˜ A | s . Using Remark 8.4, we also get that, for l = 1 , . . . , k + 1 (cid:12)(cid:12)(cid:12) ∂ d ˜ Q l (cid:12)(cid:12)(cid:12) ≤ c ( S, Σ , T, n, ǫ ) ˜ H − d . Definition 8.5.1.
For η, σ ∈ (0 , f = − ˜ Q k +1 − η ˜ H ˜ H − σ . We see that f is well-defined by Remark 8.4 and f ≥ Q k +1 ≤ − η ˜ H. By Remark 8.5 we have that(24) | ∂ d f | ≤ c ( S, Σ , T, ǫ, n ) ˜ H σ − d . Lemma 8.6.
Suppose for every ǫ ∈ (0 , n ] and η ∈ (0 , , there exists σ ∈ (0 , and C = C ( S, Σ , T, n, ǫ, σ ) such that f + < C. Then for any θ > there is a ¯ C = ¯ C ( S, Σ , T, n, θ ) such that S k +1 ≥ − θH k +1 − ¯ C. Proof.
Recall we have fixed D = D ǫ . The proof of Lemma 2.8 in [HS99a] showsthe hypotheses imply that ¯ S k +1 ≥ − θ ¯ H k +1 − ¯ C for any θ >
0, and ¯ C = ¯ C ( S, Σ , T, θ, n ). Now use Lemma 8.2. (cid:3) We work towards bounding f + , for a given η >
0. We first calculate the orderof boundary derivatives. Choose orthonormal coordinates at a fixed p ∈ ∂ Σ suchthat ∂ ≡ N . Theorem 8.7. At p we have, for i, j > , ∇ b ij = O ( | ˜ A | ) ∇ b = O ( | ˜ A | ) Proof.
By Theorem 6.2 and Proposition 4.3, we calcuate ∇ b = ∇ (¯ h + ( ǫH + D ) g )= O ( | A | + 1) + ǫ∂ Hg = O ( | ¯ A | )and the proof for i, j > (cid:3) Corollary 8.8.
For every l = 1 , . . . , n , (25) N ˜ S l = O ( | ˜ A | l ) . Proof.
For l = 1 this follows from the boundary condition N H = O ( H ) and Propo-sition 5.1. For l >
1, write S l as the sum of l -by- l minors of b ji , and use that b N,X = 0 for X ∈ T p ∂ Σ. (cid:3) Theorem 8.9.
We have (26) | N f | ≤ c ( S, Σ , T, n, ǫ ) ˜ H σ . Proof.
Immediate from Corollary 8.8 and Remark 8.4. (cid:3)
We obtain an (EVOLUTION-LIKE) equation for f . Proposition 8.10. (27) ∂ t b ij = ∆ b ij + | ˜ A | b ij + O ( D | ˜ A | ) Proof.
By Propositions 4.1 and 5.1,( ∂ t − ∆) b ij = | A | ( h ij + ǫHg ij ) + O (1 + | A | )= | A | b ij − ( D + T ijν ) | A | + O (1 + | A | )= | ˜ A | b ij + O ( D | ˜ A | )recalling that D ≥ (cid:3) ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 21
Lemma 8.11.
Let
B > η > . There are constants c = c ( n, η, B ) and c = c ( c , S, Σ , T, n, ǫ ) , such that whenever − B ˜ S k ˜ S ≤ ˜ S k +1 ≤ − η ˜ S ˜ S k we have (28) ∂ ˜ Q k +1 ∂b ij ∂b pq ∇ l b ij ∇ l b pq ≤ − c |∇ ˜ A | | ˜ A | + c ˜ H Proof.
Choose orthonormal coordinates which diagonalize b ij . We have, using thenotation of Lemma 2.13 of [HS99a], ∂ ˜ Q k +1 ∂b ij ∂b pq ∇ l b ij ∇ l b pq = J (˜ λ, ∇ l ( b ij − T ijν ) , ǫ )(29) + 2 ∂ q k +1 ∂θ ij ∂θ pq ( ˜ A ) ∇ l ( b ij − T ijν ) ∇ l T ijν (30) + ∂ q k +1 ∂θ ij ∂θ pq ( ˜ A ) ∇ l T ijν ∇ l T pqν (31)By this same Lemma 2.13, J (˜ λ, ∇ l ( b ij − T ijν ) , ǫ ) ≤ − c |∇ ( ˜ A − T ) | | ˜ A |≤ − c |∇ ˜ A | | ˜ A | + 1 c |∇ T | | ˜ A | for c = c ( B, n, η ).By Theorem 2.5 and Lemma 2.12 of [HS99a], term (31) is non-positive. Webound term (30). Recall that |∇ T | = O ( H + 1) = O ( ˜ H ). Using Remark 8.52 ∂ q k +1 ∂θ ij ∂θ pq ( ˜ A ) ∇ l ( b ij − T ijν ) ∇ l T ijν ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ q k +1 ∂θ ij ∂θ pq ( ˜ A ) (cid:12)(cid:12)(cid:12)(cid:12) ( |∇ ˜ A | + |∇ T | ) |∇ T |≤ c |∇ ˜ A | + c ˜ H where c = c ( S, Σ , T, n, ǫ ).We deduce ∂ ˜ Q k +1 ∂b ij ∂b pq ∇ l b ij ∇ l b pq ≤ − c |∇ ˜ A | | ˜ A | + c ˜ H for c = c ( S, Σ , T, n, ǫ, c ). (cid:3) Since f is a homogeneous, degree σ , symmetric function of the eigenvalues ˜ λ i of b ij , we obtain that( ∂ t − ∆) f = ∂f∂b ij ( | ˜ A | b ij + O ( | ˜ A | D )) − ∂ f∂b ij ∂b pq ∇ l b ij ∇ l b pq ≤ − ∂ f∂b ij ∂b pq ∇ l b ij ∇ l b pq + σ | ˜ A | f + cD | ˜ A | X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f∂b ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − ∂ f∂b ij ∂b pq ∇ l b ij ∇ l b pq + σ | ˜ A | f + cD ˜ H σ for c = c ( S, Σ , T, n, ǫ ). In the last line we used the inequality (24). Lemma 8.11 allows us to crucially obtain a gradient term wherever f is non-negative: on spt f + we have( ∂ t − ∆) f ≤ − σ )˜ H < ∇ ˜ H, ∇ f > − σ (1 − σ )˜ H f |∇ ˜ H | + 1˜ H − σ ∂ ˜ Q k +1 ∂b ij ∂b pq ∇ m b ij ∇ m b pq + σ | ˜ A | f + cD ˜ H σ ≤ |∇ ˜ H ||∇ f | ˜ H − c |∇ ˜ A | ˜ H − σ + c ˜ H σ + σ | ˜ A | f + cD ˜ H σ . (32) Lemma 8.12.
There are constants c = c ( S, Σ , T, n, k, ǫ, η ) and C = C ( c, p, σ, D ) such that whenever p > p ( c, n ) , we have ∂ t Z Σ t f pk ≤ − p / Z Σ t f p − k |∇ f | − p/c Z Σ t f p − k |∇ ˜ A | ˜ H − σ + c Z ∂ Σ t f p − k ˜ H σ + 2 pσ Z A ( k,t ) f p ˜ H + C Z A ( k,t ) f p + C | A ( k, t ) | − / Z Σ t f pk ˜ H Proof.
We have by equation (32) (all integrals over Σ t unless stated), ∂ t Z f pk = p Z f pk ∆ f − Z f pk H ≤ − p ( p − Z f p − k |∇ f | + p Z ∂ Σ f p − k | N f | + p / Z f p − k |∇ f | + 3 c Z f p − k |∇ ˜ H | ˜ H − σ − p/c Z f p − k |∇ ˜ A | ˜ H − σ + pc Z f p − k ˜ H σ + pσ Z A ( k,t ) f p | ˜ A | + cD Z f p − k ˜ H σ − Z f pk H provided p > c n . Here c = c ( S, Σ , T, n, ǫ, η ) and C = C ( c, p, σ, D ). The lastterm results from H = (cid:18)
11 + nǫ ˜ H + O (1) (cid:19) ≥
14 ˜ H + O (1) . The boundary term is handled by Theorem 8.9. And the other terms are handledby Peter-Paul and/or Remark 3.2. (cid:3)
We obtain a (POINCARE-LIKE) equation for f . Lemma 8.13.
For ǫ < ǫ ( n, k, η ) we have on spt f + ∂ ˜ S k ∂b ij ∇ i ∇ j ˜ S k +1 ≥ ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm ∂b pq ∇ i b lm ∇ j b pq + ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm ∇ l ∇ m b ij + ǫ nǫ ( n − k ) ˜ S k ∂ ˜ S k ∂b ij − ( n − k + 1) ˜ S k − ∂ ˜ S k +1 ∂b ij ! ∇ i ∇ j ˜ H + 12 η ˜ H ˜ S k − cD | ˜ A | k ( D + | ˜ A | ) − c | ˜ A | k − |∇ ˜ A | Proof.
We follow the proof of Lemma 2.15 and Corollary 2.16 in [HS99a]. Recallwe fixed D = D ǫ . From Proposition 5.1 and Remark 8.1 we have ∇ p ∇ q T ij = O ( | ˜ A | + |∇ ˜ A | ). In particular, ∇ p ∇ q ˜ H = (1 + nǫ ) ∇ p ∇ q H + O ( | ˜ A | + |∇ ˜ A | ) ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 23 and ∇ i ∇ j ¯ h lm − ∇ l ∇ m ¯ h ij = ¯ h ij ¯ h lr ¯ h rm − ¯ h lm ¯ h ir ¯ h rj + ¯ h im ¯ h lr ¯ h rj − ¯ h lj ¯ h mr ¯ h ri + O ( | ˜ A | + |∇ ˜ A | ) . We therefore calculate ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm ( ∇ i ∇ j b lm − ∇ l ∇ m b ij )= ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm (cid:2) ∇ i ∇ j ¯ h lm − ∇ l ∇ m ¯ h ij + ǫ nǫ (cid:16) δ lm ∇ i ∇ j ˜ H − δ ij ∇ l ∇ m ˜ H + O ( | ˜ A | + |∇ ˜ A | ) (cid:17)(cid:21) ≥ ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm (cid:20) ∇ i ∇ j ¯ h lm − ∇ l ∇ m ¯ h ij + ǫ nǫ (cid:16) δ lm ∇ i ∇ j ˜ H − δ ij ∇ l ∇ m ˜ H (cid:17)(cid:21) − c | ˜ A | k +1 − c | ˜ A | k − |∇ ˜ A | . Choose a frame which diagonalizes ¯ h ij , and hence b ij , then ∂ ˜ S k ∂b ij ˜ S k +1 ∂b lm ( ∇ i ∇ j ¯ h lm − ∇ l ∇ m ¯ h ij ) = ∂ ˜ S k ∂ ˜ λ i ∂ ˜ S k +1 ∂ ˜ λ m h ¯ λ i ¯ λ m − ¯ λ i ¯ λ m + O ( | ˜ A | + |∇ ˜ A | ) i ≥ ∂ ˜ S k ∂ ˜ λ i ∂ ˜ S k +1 ∂ ˜ λ m h ˜ λ i ˜ λ m − ˜ λ i ˜ λ m + O ( | ˜ A | + |∇ ˜ A | )+ (cid:18) ǫ nǫ ˜ H + O ( D ) (cid:19) (˜ λ m − ˜ λ i )+ (cid:18) ǫ nǫ ˜ H + O ( D ) (cid:19) (˜ λ i − ˜ λ m ) (cid:21) ≥ ∂ ˜ S k ∂ ˜ λ i ∂ ˜ S k +1 ∂ ˜ λ m h ˜ λ i ˜ λ m − ˜ λ i ˜ λ m + ǫ ˜ H nǫ ! (˜ λ m − ˜ λ i ) + ǫ ˜ H nǫ ! (˜ λ i − ˜ λ m ) − cD | ˜ A | k ( D + | ˜ A | ) − c | ˜ A | k − |∇ ˜ A | . Therefore, by precisely the same arguments at in Lemma 2.15 of [HS99a], wehave for any ǫ > ∂ ˜ S k ∂b ij ∇ i ∇ j ˜ S k +1 ≥ ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm ∂b pq ∇ i b lm ∇ j b pq + ∂ ˜ S k ∂b ij ∂ ˜ S k +1 ∂b lm ∇ l ∇ m b ij + ǫ nǫ ( n − k ) ˜ S k ∂ ˜ S k ∂b ij − ( n − k + 1) ˜ S k − ∂ ˜ S k +1 ∂b ij ! ∇ i ∇ j ˜ H − ˜ H ˜ S k ˜ S k +1 + ( k + 1) ˜ S k + k (( k + 1) ˜ S k +1 − ( k + 2) ˜ S k ˜ S k +2 )+ ǫ ˜ H nǫ ! h ( k + 1)( n − k + 1) ˜ S k +1 ˜ S k − − k ( n − k ) ˜ S k i + ǫ ˜ H nǫ ! h ( n − k ) ˜ S k ( ˜ H ˜ S k − ( k + 1) ˜ S k +1 )+( n − k + 1) ˜ S k − (( k + 2)( ˜ S k +2 − ˜ H ˜ S k +1 ) i − cD | ˜ A | k ( D + | ˜ A | ) − c | ˜ A | k − |∇ ˜ A | . And the Lemma follows by the same argument as in Corollary 2.16 of [HS99a]. (cid:3)
Lemma 8.14.
There is a constant c = c ( S, Σ , T, n, ǫ, η, D ) such that for any p > and β > , we have c Z Σ t f p + ˜ H ≤ ( p + p/β ) Z Σ t f p − |∇ f | + (1 + βp ) Z Σ t f p − |∇ ˜ A | ˜ H − σ + Z Σ t f p + + Z ∂ Σ t f p − ˜ H σ Proof.
Fix ǫ = ǫ ( η, n, k ) as in Lemma 8.13. Using inequalities of Remarks 8.4 and8.5, we have for c = c ( S, Σ , T, n, k, ǫ ), ∂ ˜ S k ∂b ij ∇ i ∇ j f ≤ − ˜ H σ − ˜ S − k ∂ ˜ S k ∂b ij ∇ i ∇ j ˜ S k +1 + c ˜ H k + σ − |∇ ˜ A | + c ˜ H k − |∇ f ||∇ ˜ A | + ∂ ˜ S k ∂b ij h ˜ H σ − ˜ S − k ˜ S k +1 ∇ i ∇ j ˜ S k − ( η ˜ H σ − − ( σ −
1) ˜ H − f ) ∇ i ∇ j ˜ H i ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 25
Multiply by f p + ˜ H − k +1 − σ , integrate, and use Lemma 8.13 to obtain η cǫ k − Z f p + ˜ H ≤ η Z ˜ S k ˜ H − k f p + ≤ − Z f p + ˜ H − k +1 − σ ∂ ˜ S k ∂b ij ∇ i ∇ j f − Z f p + ˜ H − k ˜ S − k ∂ ˜ S k ∂b ij n − ˜ S − k ˜ S k +1 ∇ i ∇ j ˜ S k + ∂ ˜ S k ∂b lm ∂b pq ∇ i b lm ∇ j b pq + ∂ ˜ S k +1 ∂b lm ∇ l ∇ m b ij ) + Z f p + ˜ H − k ∂ ˜ S k ∂b ij (cid:18) − η − ǫ nǫ ( n − k ) + ( σ −
1) ˜ H − σ f (cid:19) ∇ i ∇ j ˜ H + ǫ nǫ ( n − k + 1) Z f p + ˜ H − k ˜ S k − ˜ S − k ∂ ˜ S k +1 ∂b ij ∇ i ∇ j ˜ H + cD Z f p + + cD Z f p + ˜ H + c Z f p + ˜ H − |∇ ˜ A | + c Z f p + ˜ H − |∇ ˜ A | + c Z f p + ˜ H − − σ |∇ ˜ A ||∇ f | where c = c ( S, Σ , T, n, ǫ ). As usual all integrals are over Σ t unless otherwise stated.Integrate by parts all double covariant derivatives, using Theorem 8.9 and equa-tion (25) to handle boundary terms. After applying remarks 8.4 and 8.5, we obtainthat ηc Z f p + ˜ H ≤ c Z ∂ Σ t f p − f + c Z f p − |∇ ˜ A | ˜ H − σ + c Z f p + |∇ f ||∇ ˜ A | ˜ H σ + cp Z f p − |∇ f ||∇ ˜ A | ˜ H + cp Z f p − |∇ f | + c Z f p + |∇ ˜ A | ˜ H + 1 µ Z f p + ˜ H + C ( c, D, µ ) Z f p + where µ > µ = 2 c/η . Recalling that f ≤ c ˜ H σ , the Lemmafollows by using Peter-Paul on the remaining terms. (cid:3) In view of Lemma 8.6 and Theorem 3.1, to finish proving Theorem 1.2 we merelyneed to show f + satisfies ( ⋆ ) of Section 3. In the language of Section 3, let ˜ H beitself (the twice-perturbed mean curvature), and ˜ G = |∇ ˜ A | . Then Lemmas 8.12and 8.14 imply f + satisfies ( ⋆ ). We are done.9. Umbilic pinching when S = S n We consider the case when Σ is strictly convex and S is the sphere S n . Weprove the umbilic pinching Theorem 1.6. By Proposition 4.3 we know that T ≤ T (Σ ) < ∞ .Notice in this case h N,X ≡ X ∈ T p ∂ Σ, so the estimates of Lemma 6.1give us directly boundary conditions on the principle curvatures. We can thereforework with the unperturbed second fundamental form. In conjunction with the following Remark we have(33)
N H = H, N | A | = O ( H ) . Remark 9.1.
By Theorem 9.7 in [Sta96a], there is an ǫ = ǫ (Σ , n ) such that(34) h ij ≥ ǫHg ij for all t ∈ [0 , T ). Hence the pointwise estimates of Lemma 2.3 in [Hui84] continueto hold in the spherical-free-boundary case.Arguing as in [Hui84], to prove Theorem 1.6 it will suffice to show the followingTheorems. Theorem 9.2.
For any η > , we have | A | − n H ≤ ηH + C ( η, Σ , n ) . Theorem 9.3.
For any η > , we have |∇ H | ≤ ηH + C ( η, Σ , n ) . We first prove Theorem 9.2. For σ > f = | A | − n H H − σ = H σ n X i,j ( λ i − λ j ) H . By Remark 9.1 and equations 33, we have(36) f = O ( H σ ) , N f = O ( H σ ) . Clearly to prove Theorem 9.2 it suffices to show f is bounded in spacetime forsome choice of σ >
0. We shall demonstrate in the next two Lemmas that f satisfiesthe (EVOLUTION-LIKE) and (POINCARE-LIKE) equations of Section 3. Lemma 9.4.
There is a constant c = c ( n, ǫ ) such that for every η > we have c Z Σ t f p H ≤ ( ηp + 1) Z Σ t |∇ H | H − σ f p − + pη Z Σ t |∇ f | f p − + Z ∂ Σ f p − H σ Proof.
We follow the proof of Lemma 5.4 in [Hui84]. In consideration of Remark9.1, we have 2 nǫ f p H ≤ H − σ f p − Z ≤ f p − ∆ f (37) − H − σ f p − < h ij , ∇ i ∇ j H > (38) + 2(1 − σ ) H f p − < ∇ H, ∇ f > + 2 − σH f p ∆ H. (39)Here h ij is the trace-free second fundamental form, and Z = H tr( A ) − | A | . Weintegrate the above relation, and integrate by parts terms (37), (38) and (39).The resulting interior terms are handled by Peter-Paul, and the inequality | h ij | ≤ ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 27 f H − σ ≤ H . To handle the boundary term use (33) and (36), and that h N,X vanishes when X ∈ T p ∂ Σ. (cid:3) Recall that f satisfies the evolution inequality(40) ∂ t f ≤ ∆ f + 2(1 − σ ) H < ∇ H, ∇ f > − ǫ |∇ H | H − σ + σ | A | f Lemma 9.5.
We have, for c = c ( n, ǫ ) , ∂ t Z Σ t f pk ≤ − p / Z Σ t |∇ f | f p − k − p/c Z Σ t |∇ H | H − σ f p − k + ( σp − Z Σ t H f p + cp Z ∂ Σ t f p − k H σ Proof.
Follows directly by (40), and Proposition 4.1. Use Peter-Paul to handle theinner product term, and equation (36) to handle the boundary term obtained uponintegration by parts. (cid:3)
In view of Lemmas 9.4 and 9.5, we can take ˜ H = H and ˜ G = |∇ H | in Section3. Theorem 9.2 now follows by Theorem 3.1.We prove Theorem 9.3. We cannot obtain a boundary condition on |∇ H | , andtherefore we work instead with the quantity |∇ H − HV | . Here we define V ≡ ¯ V T ,with ¯ V a fixed vector field on R n +1 extending ν S , such that ¯ ∇ ν S ¯ V ≡ S n .Fix η >
0, and choose σ = σ (Σ , n ) and C = C (Σ , n ) so that f ≤ C . Define g = |∇ H − HV | H + bH ( | A | − H /n ) + ba | A | − ηH + D for a, b, D positive constants to be determined, depending only on ( η, Σ , n ). Wewill show g is bounded in spacetime, which clearly suffices to prove Theorem 9.3since |∇ H − HV | ≥ |∇ H | − c n H . Lemma 9.6.
We have the evolution equations ( ∂ t − ∆) |∇ H − HV | ≤ cH |∇ A | + cH − |∇ ( ∇ H − HV ) | ( ∂ t − ∆) |∇ H − HV | H ≤ cH |∇ A | + cH for c = c (Σ , n ) .Proof. By direct calculation we have that ∇ i V = ( ¯ ∇ i ¯ V ) T = O (1)∆ V = − H ( ¯ ∇ ν ¯ V ) T + (tr T Σ ¯ ∇ ¯ V ) T = O ( H + 1) , and ∂ t V i = ∂ t < ¯ V , ∂ i > = − H ¯ ∇ ν ¯ V i − Hh ij V j − ∂ i H < ¯ V , ν > .
We have:12 ∆ |∇ H − HV | = {∇ i ∆ H + ∇ j H ( Hh ij − h ik h kj ) − ∆ HV i + O ( H + H + |∇ H | ) (cid:9) ( ∇ i H − HV i )+ |∇ ( ∇ H − HV ) | and 12 ∂ t |∇ H − HV | = (cid:8) Hh ij ∇ j H + ∇ i (∆ H + | A | H ) − (∆ H + | A | H ) V i + O ( |∇ H | + H ) (cid:9) ( ∇ i H − HV i ) . The first equation follows directly, recalling that H is non-increasing. To prove thesecond formula, use equation (15) to obtain( ∂ t − ∆) |∇ H − HV | H ≤ cH |∇ H | + cH − |∇ ( ∇ H − HV ) | H − |∇ H − HV | | A | H + 4 |∇ H − HV | H |∇|∇ H − HV |||∇ H | − |∇ H − HV | H |∇ H | ≤ cH |∇ H | + cH . (cid:3) Lemma 9.7.
At any point on the boundary ∂ Σ , we have N |∇ H − HV | = 0 N ( | A | − H /n ) ≤ c n H p | A | − H /n. Proof.
Choose orthonormal coordinates at p , such that ∂ ≡ N along ∂ Σ, theintegral curves of ∂ are geodesics. We calculate12 ∂ |∇ H − HV | = − X i,j> g ij ( ∂ i H − HV i )( ∂ j H − HV j )+ X i ( ∂ ∂ i H − ∂ ( HV i ))( ∂ i H − HV i )= −|∇ H − HV | + X i> ( ∂ i ∂ H )( ∂ i H )= −|∇ H − HV | + |∇ H | − |∇ H | . We prove the second formula. Write λ i for the principle curvatures, and λ N forthe curvature in direction N . Using Lemma 6.1, we obtain12 ∂ ( | A | − H /n ) = 3 Hλ N − nλ N − | A | − H /n = ( | A | − H /n ) + 3 Hλ N − nλ N − | A | . ONVEXITY ESTIMATES FOR MEAN CURVATURE FLOW WITH FREE BOUNDARY 29
We calculate3 Hλ N − nλ N − | A | = X i (3 λ N λ i − λ N − λ i )= X i ( λ i − λ N )( λ N − λ i )= − X i ( λ i − λ N ) − λ N X i ( λ i − λ N ) ≤ − n − X i,j ( λ i − λ j ) + H s n X i,j ( λ i − λ j ) = − nn − | A | − H /n ) + √ nH p | A | − H /n. (cid:3) Using Lemma 9.7 and equation (33), we then have
N g ≤ bH ( | A | − H /n ) + c n bH p | A | − H /n + c n ba | A | − ηH ≤ ( g − D ) + c n bH p C H − σ + c n baH − ηH ≤ g provided D = D ( η, C , σ, a, b, n ) is sufficiently big.By Theorem 9.2 (see Lemma 6.5 of [Hui84]), we can choose a = a ( C , σ, n )sufficiently large so that,(41) ( ∂ t − ∆)( | A | − H /n ) ≤ − c n H |∇ A | + a |∇ A | + 3 H ( | A | − H /n ) . Using Lemma 9.6 and equation (41), we have for b = b (Σ , n ) sufficiently large( ∂ t − ∆) g ≤ cH |∇ A | + cH + 6 nH |∇ A | − bc n H |∇ A | + 2 baH + 3 bH ( | A | − H /n ) − ηn H ≤ baH + 3 bC H − σ − ηn H ≤ C ( η, σ, C , Σ , n )using that c = c (Σ , n ).Take φ the cutoff function of Section 7, with constants chosen so that( ∂ t − ∆) φ ≤ , N φ ≤ − φ Then for a, b, D chosen as above, we have N ( gφ ) ≤ ∂ t − ∆) gφ ≤ Cφ + 2 < ∇ g, ∇ φ > . Since φ is uniformly bounded in time, we deduce that max Σ t gφ increases at worstlinearly, and therefore g ≤ ˜ C ( η, Σ , n ) . This completes the proof of the umbilic pinching Theorem 1.6.
References [Buc05] J. Buckland. Mean curvature flow with free boundary on smooth hypersurfaces.
J. reineangew. Math. , 586:71–90, 2005.[HS99a] G. Huisken and C. Sinestrari. Convexity estimates for mean curvature flow and singu-larities of mean convex surfaces.
Acta Math. , 183:45–70, 1999.[HS99b] G. Huisken and C. Sinestrari. Mean curvature flow singularities for mean convex surfaces.
Calc. Var. , 8:1–14, 1999.[Hui84] G. Huisken. Flow by mean curvature of convex surfaces into spheres.
J. DifferentialGeometry , 20:237–266, 1984.[MS73] J. H. Michael and L. M. Simon. Sobolev and mean-value inequalities on generalizedsubmanifolds on R n . Comm. Pure Appl. Math. , 26:361–379, 1973.[Sta96a] A. Stahl. Convergence of solutions to the mean curvauture flow with a neumann boundarycondition.
Calc. Var. , 4:421–441, 1996.[Sta96b] A. Stahl. Regularity estimates for solutions to the mean curvature flow with a neumannboundary condition.
Calc. Var. , 4:385–407, 1996.
Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, CA94305
E-mail address ::