Łojasiewicz inequalities for mean convex self-shrinkers
aa r X i v : . [ m a t h . DG ] J a n LOJASIEWICZ INEQUALITIES FOR MEAN CONVEXSELF-SHRINKERS
JONATHAN J. ZHU
Abstract.
We prove Lojasiewicz inequalities for round cylinders and cylinders over Abresch-Langer curves, using perturbative analysis of a quantity introduced by Colding-Minicozzi.A feature is that this auxiliary quantity allows us to work essentially at first order. This newmethod interpolates between the higher order perturbative analysis used by the author forcertain shrinking cylinders, and the differential geometric method used by Colding-Minicozzifor the round case. Introduction
Self-shrinkers are submanifolds Σ n ⊂ R N satisfying the elliptic PDE φ := − H + x ⊥ = 0;they serve as singularity models for the mean curvature flow. Lojasiewicz inequalities havebeen successful for proving the uniqueness of tangent flows for a variety of model shrinkers[8, 3, 5, 2], and ‘explicit’ forms can also be used to establish rigidity in the class of shrinkers[6, 9]. Explicit Lojasiewicz inequalities for a class of shrinking cylinders were proven by theauthor in [10], and previously by Colding and Minicozzi for the case of round cylinders [3, 5].The purpose of this note is to provide a bridge between these two approaches.Specifically, in [10] we used Taylor expansion of the shrinker quantity φ while in [3, 5] apointwise differential geometric method is used, relying on an auxiliary quantity τ = A | H | . Inthis note we show that Taylor expansion of τ , combined with the techniques in [10], yieldsanother Lojasiewicz inequality for round cylinders and Abresch-Langer cylinders. Denotingby C n (˚Γ) the set of all rotations of ˚Γ k × R n − k ⊂ R N about the origin, we prove: Theorem 1.1 ( Lojasiewicz inequality of the first kind) . Let ˚Γ be a round shrinking sphereor an Abresch-Langer curve. There exists ǫ > ǫ , λ , C j there exist R , l such that if l ≥ l , Σ n ⊂ R N has λ (Σ) ≤ λ and:(1) For some R > R , we have that B R ∩ Σ is the graph of a normal field U over somecylinder in C n (˚Γ) with k U k C ( B R ) ≤ ǫ and k U k L ( B R ) ≤ e − R / ;(2) |∇ j A | ≤ C j on B R ∩ Σ for all j ≤ l ;then there is a cylinder Γ ∈ C n (˚Γ) and a compactly supported normal vector field V over Γwith k V k C ,α ≤ ǫ , such that Σ ∩ B R − is contained in the graph of V , and k V k L ≤ C ( k φ k a l L ( B R ) + k φ k a l L ( B R ) + ( R − a l n e − a l ( R − / ) , where C = C ( n, l, C l , λ , ǫ ) and a l ր l → ∞ . Date : January 25, 2021.
Theorem 1.2 ( Lojasiewicz inequality of the second kind) . Let ˚Γ be a round shrinking sphereor an Abresch-Langer curve. There exists ǫ > λ , C j there exist R , l such that if l ≥ l , Σ n ⊂ R N has λ (Σ) ≤ λ and:(1) For some R > R , we have that B R ∩ Σ is the graph of a normal field U over somecylinder in C n (˚Γ) with k U k C ( B R ) ≤ ǫ and k U k L ( B R ) ≤ e − R / ;(2) |∇ j A | ≤ C j on B R ∩ Σ for all j ≤ l ;then for C = C ( n, β, l, C l , λ ) we have | F (Σ) − F (˚Γ) | ≤ C ( k φ k al L + ( R − n e − ( R − ) . Note that by the work of Colding-Minicozzi [4], the only codimension one, mean convexself-shrinkers are precisely the cylinders over round spheres S k √ k and Abresch-Langer [1]curves Γ a,b . The inequalities above differ slightly from those in [10] by the exponents on theright, but morally they are equivalent since a l may be taken arbitrarily close to 1. Indeed,the above estimates suffice to give alternative proofs of the uniqueness of tangent flows andrigidity for these mean convex shrinkers (see Remark 5.2; cf. [10, Theorems 1.1 and 1.2]).The advantage of the τ quantity is that it allows us to perform the variational analysisonly at first order, whereas in [10] we needed the second order expansion of φ . This is asignificant reduction as the complexity of the method increases quickly with the order ofexpansion. Applying the perturbative method to τ also explains and quantifies the successof the method used by Colding-Minicozzi [3, 5] for round spheres.The key new geometric data, derived in Section 3, are the (first) variation formulae for τ and a further auxiliary quantity P . We then prove estimates for entire graphs over thecylinders in Section 4, using Taylor expansion of τ in place of the second order analysis in[10, Section 5.2]. Note that the first order analysis of φ is still required. Some preliminariesare included in Section 2 and the Lojasiewicz inequalities of Theorems 1.1 and 1.2 are provenin Section 5. Acknowledgements.
The author would like to thank Prof. Bill Minicozzi for his encour-agement and for several insightful discussions. This work was supported in part by theNational Science Foundation under grant DMS-1802984 and the Australian Research Coun-cil under grant FL150100126. 2.
Preliminaries
We consider smooth, properly immersed submanifolds Σ n ⊂ R N .For a vector V we denote by V T the projection to the tangent bundle, and V ⊥ = Π( V )the projection to the normal bundle N Σ. Given a vector field U on Σ with k U k C smallenough, the graph Σ U is the submanifold given by the immersion X U ( p ) = X ( p ) + U ( p ). Wesay Σ U is a normal graph if U T = 0.The second fundamental form is the 2-tensor with values in the normal bundle defined by A ( Y, Z ) = ∇ ⊥ Y Z , and the mean curvature (vector) is H = − A ii . Here, and henceforth, wetake the convention that repeated lower indices are summed with the metric, for instance LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 3 A ii = g ij A ij . We denote the shrinker mean curvature by φ = x ⊥ − H and the principalnormal by N = H | H | . A submanifold is a shrinker if φ ≡ V we denote A V = h A, V i . The Hessian on the normal bundle is given by( ∇ ⊥ ∇ ⊥ V )( Y, Z ) = ∇ ⊥ Z ∇ ⊥ Y V − ∇ ⊥∇ TZ Y V .For graphs Γ U over a fixed submanifold Γ, we use subscripts to denote the values ofgeometric quantities on Γ U . We also consider these quantities as second order functionalson (normal) vector fields U . For instance, there is a smooth function ϕ such that φ U = ϕ ( p, U, ∇ U, ∇ U ). For variations of such quantities, we use the shorthand notation D ϕ ( U )to mean the variation D ϕ | ([ U, ∇ U, ∇ U ]) evaluated at 0, and so forth.The Gaussian weight is ρ = ρ n = (4 π ) − n/ e −| x | / . Here n is the dimension of the sub-manifold and will be omitted when clear from context. By L p , W k,p we denote the weightedSobolev spaces with respect to ρ . The Gaussian area functional is F (Σ) = R Σ ρ . The entropyis λ (Σ) = sup y,s> F ( s (Σ − y )). For a shrinker, λ (Σ) = F (Σ). Note that finite entropy λ (Σ) ≤ λ implies Euclidean volume growth | Σ ∩ B R | ≤ C ( λ ) R n .We will use the following elliptic operators: the drift Laplacian L = ∆ − ∇ x T ; and theJacobi operator L = L + + P k,l h· , A kl i A kl . The drift Laplacian is defined on functions andtensors, whilst L is defined on sections of the normal bundle (via ∇ ⊥ ). For such operators,unless otherwise indicated, ker will refer to the W , kernel, for instance K = ker L .We set h x i = (1 + | x | ) . On a curve Γ ⊂ R , we denote the geodesic curvature by κ anduse dots ˙ κ = ∂ σ κ to denote differentiation with respect to the arclength parameter σ .We use C to denote a constant that may change from line to line but retains the stateddependencies.2.1. Mean convex self-shrinkers.
In this article, we say that a submanifold Γ n ⊂ R N has‘codimension one’ if the minimal affine subspace containing Γ has dimension dim span(Γ) = n + 1. Note that for shrinkers, the minimal subspace necessarily contains the origin since L x = − x . Consider a codimension one shrinker Γ. Up to ambient rotation we haveΓ n ⊂ R n +1 × R N − n − . Moreover the normal bundle is trivial and is spanned by N and ∂ z α ,where z α are standard coordinates on R N − n − .An orientable codimension one submanifold is mean convex (up to change of orientation)if | H | >
0. By the work of Colding-Minicozzi [4] (see also [7]), the only mean convexself-shrinkers with finite entropy are cylinders Γ = ˚Γ k × R n − k , where ˚Γ is either a roundshrinking sphere S k √ k or an Abresch-Langer curve ˚Γ a,b (see [1]). We further decompose R N = R k +1 × R n − k × R N − m − n so that ˚Γ ⊂ R k +1 , and let ˚ x, y, z be the projection of x to eachrespective factor.Given ˚Γ k , we denote by C n (˚Γ) the set of all rotations of ˚Γ × R n − k ⊂ R N about the origin.2.2. The auxiliary quantities τ and P . For submanifolds Γ on which H never vanishes,Colding-Minicozzi [5] considered the 2-tensor τ = A | H | and showed that |∇ ⊥ τ | satisfies a JONATHAN J. ZHU certain elliptic PDE with inhomogenous term given by P = | A | | A N | − | A | + X ijlm (cid:0) h A jl , A im ih A lm , A ij i − h A ij , A ml i (cid:1) + | A | | H | (cid:0) | A N ( x T , · ) | − | A ( x T , · ) | (cid:1) . (2.1)Here A is the real-valued 2-tensor ( A ) ij = h A im , A mj i .If Γ = ˚Γ k × R n − k is either a round cylinder or an Abresch-Langer cylinder, one has τ = − k N ˚ g ij and in particular ∇ ⊥ τ = 0.As before, for graphs Γ U there are smooth functions T and P so that for sufficiently small δ > k U k C < δ we have |∇ ⊥ τ | U = T ( p, U, ∇ U, ∇ U ) and P U = P ( p, U, ∇ U, ∇ U ).The quantity P vanishes on submanifolds of codimension one.2.3. Jacobi fields.
The space of Jacobi fields on a shrinker Γ is the ( W , ) kernel K = ker L .It contains the subspace K of Jacobi fields generated by ambient rotations, and we denoteits L -orthocomplement in K by K .The following summarises the Jacobi fields on Γ = ˚Γ × R n − k , where ˚Γ is a round shrinkingsphere or an Abresch-Langer curve (see [10, Sections 2.6 and 4.1]). Proposition 2.1.
Let Γ = ˚Γ × R n − k where ˚Γ is a round shrinking sphere or an Abresch-Langer curve. Then the space K is spanned by normal vector fields of the following forms:(1) ˚ x i ∂ ⊥ ˚ x j − ˚ x j ∂ ⊥ ˚ x i ;(2) y j ∂ ⊥ ˚ x i ;(3) ˚ x i ∂ z α ; and y j ∂ z α .Moreover, the space K is spanned by the normal fields { ( y i y j − δ ij ) H } . Corollary 2.2.
Let Γ = ˚Γ × R n − k where ˚Γ is a round shrinking sphere or an Abresch-Langer curve. Let r = diam(˚Γ) + 1. There exists C so that for any J ∈ K we have | J | ≤ C h x i k J k L ( B r ) , |∇ J | + |∇ J | ≤ C h x i k J k L ( B r ) and |∇ J ( · , y ) k ≤ C h x ik J k L ( B r ) .3. Auxiliary variation analysis
In this section we compute the variation of the auxiliary quantities, insofar as to establishPropositions 3.2 and 3.3 for ∇ ⊥ τ , and Lemma 3.4 and Proposition 3.5 for P . The benefitof using these quantities is that their first variation will be sufficient to establish the formalsecond order obstruction of φ .We consider a submanifold Γ with a fixed immersion X : Γ n → R N , and a one-parameterfamily of immersions X : I × Γ n → R N with X (0 , p ) = X ( p ). We use s for the coordinateon I = ( − ǫ, ǫ ), and subscripts to denote differentiation with respect to s . For instance, X s = ∂∂s X . If p i are local coordinates on Σ, we have the tangent frame X i = X ∗ ( ∂∂p i ).All geometric quantities such as Π , g, A should be considered as functions of s, p , given bythe value of each quantity at X ( s, p ) on the submanifold defined by X ( s, · ). For instance, themetric g ij ( s, p ) is given by g ij = h X i , X j i . Recall Π is the projection to the normal bundle.Also recall that repeated lower indices are contracted via the (inverse) metric g ij . LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 5
The following first variations were calculated in [5]:
Proposition 3.1 ([5]) . Suppose X s = V = V ⊥ ; then at s = 0:Π s ( W ) = − Π( ∇ W T V ) − X j g ij h Π( ∇ X i V ) , W i , (3.1) ( g ij ) s = − A Vij , ( g ij ) s = 2 g il A Vlm g mj , (3.2) ( A ij ) s = − X l h∇ ⊥ X l V, A ij i + ( ∇ ⊥ ∇ ⊥ V )( X i , X j ) − A Vil A jl , (3.3) | H | s = −h N , ∆ ⊥ V + A Vij A ij i . (3.4)3.1. Variation of ∇ ⊥ τ . We begin with a general submanifold Γ. Assume that on Γ, wehave | H | 6 = 0 on Γ (so τ is well-defined) and ∇ ⊥ τ = 0. Then at s = 0,(3.5) ( |∇ ⊥ τ | ) s = 0 , ( |∇ ⊥ τ | ) ss = 2 | ( ∇ ⊥ τ ) s | . The first variation of ∇ ⊥ τ is given by(3.6) ( ∇ ⊥ τ ) s = Π s ( ∇ ⊥ τ ) + Π( ∇ ( τ s )) . Since ∇ ⊥ τ = 0, we have ∇ l τ ij = −h τ ij , A lm i e m . Using (3.1), this gives the followingformula for the first term on the right in (3.6),(3.7) Π s ( ∇ l τ ij ) = 1 | H | h A ij , A lm i∇ ⊥ m V. For the last term in (3.6), we calculate(3.8) Π( ∇ ( τ s )) = − ∇| H || H | Π( A s − | H | s τ ) + ∇ ⊥ A s | H | − ∇| H | s | H | τ. The first two terms in (3.8) are already known from Proposition 3.1, and we proceed tocalculate the last two: Differentiating | H | s = −h N , ∆ ⊥ V + A Vij A ij i gives ∇| H | s = − h∇ ⊥ N , ∆ ⊥ V + A Vij A ij i − h N , ∇ ⊥ ∆ ⊥ V i− A N ij ( h∇ ⊥ A ij , V i + h A ij , ∇ ⊥ V i ) − A Vij h N , ∇ ⊥ A ij i . (3.9)Now differentiating ( A ij ) s = − X l h∇ ⊥ l V, A ij i − A Vil A jl + ( ∇ ⊥ ∇ ⊥ V )( X i , X j ), we have ∇ ⊥ l ( A ij ) s = − A ml h∇ ⊥ m V, A ij i − h∇ ⊥ l A im , V i A jm − h A im , ∇ ⊥ l V i A jm − A Vim ∇ ⊥ l A jm + ( ∇ ⊥ ∇ ⊥ ∇ ⊥ V )( X i , X j , X l ) . (3.10)In the remainder of this subsection we consider the cases where ˚Γ is either: a shrinkingsphere S k √ k ; or an Abresch-Langer curve ˚Γ a,b . In both cases, Γ has codimension one andindeed satisfies ∇ ⊥ τ = 0 and A = A N N . Furthermore, ∇ ⊥ N = 0, and it follows that anynormal variation on Γ may be written U = u N + u α ∂ z α , with ∇ ⊥ i U = ( ∇ i u ) N + ( ∇ i u α ) ∂ z α ,and so forth. JONATHAN J. ZHU
Round cylinders.
Proposition 3.2.
Let ˚Γ = S k √ k and Γ ∈ C n (˚Γ). If U ∈ K , then kD T ( U, U ) k L = k k U k L . Proof.
A shrinking sphere ˚Γ = S k √ k satisfies ˚ A ij = − √ k N ˚ g ij . Consider a variation by U = u N . As in [5], it follows that at s = 0 we have( A ij ) s = ˚ g ij √ k ∇ u − ˚ g ij k u N + ( ∇ u ) ij N , | H | s = − ∆ u − u . Using the variation formulae above, we may also compute that:Π s ( ∇ l τ ij ) = 1 √ k ˚ g ij ˚ g ml ( ∇ m u ) N , ∇ ⊥ l ( A ij ) s = − k ˚ g ij ˚ g ml ( ∇ m u ) N − k ˚ g ij ( ∇ l u ) N + ( ∇ u ) ijl N , ∇ l | H | s = −∇ l ∆ u − ∇ l u. We now specialise to U ∈ K , so that u = u ( y ) = P ij c ij ( y i y j − δ ij ). In particular( L + 1) u = 0 and ∇ u = 0.Combining the above according to (3.6), (3.7) and (3.8) then gives(3.11) ( ∇ ⊥ τ ) ijl,s = r k ˚ g ij ( ∇ l u − ˚ g lm ∇ m u ) N and therefore(3.12) 12 ( |∇ ⊥ τ | ) ss = 12 D T ( U, U ) = 2 k |∇ u | . Using the identity R Γ |∇ u | ρ = − R Γ u ( L u ) ρ = − R Γ u ρ then completes the proof. (cid:3) Abresch-Langer curves.
Proposition 3.3.
Let ˚Γ = ˚Γ a,b be an Abresch-Langer curve and Γ ∈ C n (˚Γ).If U ∈ K , then kD T ( U, U ) k L ≥ B (˚Γ) k U k L , where B (˚Γ) = R ˚Γ κ ρ R ˚Γ κ ρ > Proof.
The curve ˚Γ = ˚Γ a,b satisfies ˚ A ij = − κ N ˚ g ij , where κ is the geodesic curvature. Con-sider a variation by U = u N . It follows that at s = 0 we have:Π s ( ∇ l τ ij ) = κ ˚ g ij ˚ g ml ( ∇ m u ) N , ( A ij ) s = κ ˚ g ij ∇ u − κ u ˚ g ij N + ( ∇ u ) ij N , ∇ ⊥ l ( A ij ) s = − κ ˚ g ij ˚ g ml ( ∇ m u ) N − κ ˚ g ij u ( ∇ l κ ) N − κ ˚ g ij ∇ l u N + ( ∇ u ) ijl N , LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 7 | H | s = − ∆ u − κ u, ∇ l | H | s = −∇ l ∆ u − κu ∇ l κ − κ ∇ l u. We now specialise to U ∈ K , so that u = f ( y ) κ and f ( y ) = P ij c ij ( y i y j − δ ij ); inparticular note that ( L + 1) f = 0.Take coordinates on Γ so that the i = 0 index corresponds to the arclength parameter on˚Γ and the remaining indices i > R n − . Thenthe metric on ˚Γ satisfies ˚ g ij = δ ij .Combining the above according to (3.6), (3.7) and (3.8) then gives, for i = j = 0 and l >
0, that ( ∇ ⊥ τ ) l,s = ( − κ ∇ l u − κ − ∇ l ∆ u + κ − ( ∇ u ) l ) N = ( − κ ∇ l f − κ − ∇ l (¨ κf + 2 κ ) + κ − ¨ κ ∇ l f ) N = − κ ( ∇ l f ) N . Therefore(3.13) 12 ( |∇ ⊥ τ | ) ss = 12 D T ( U, U ) = | ( ∇ ⊥ τ ) s | ≥ κ |∇ f | . Again using the identity R Γ |∇ f | ρ = − R Γ f ( L f ) ρ = − R Γ f ρ completes the proof. (cid:3) Variation of P . The first and second variations of P were studied in [5] for the caseof round cylinders. Here we consider the case where ˚Γ is an Abresch-Langer curve. Againwe write normal variations as U = u N + u α ∂ z α . Lemma 3.4. DP = 0 on Γ = ˚Γ a,b × R n − . Proof.
The variation of the first four terms of P proceeds similarly to the round cylinder casein [5, Section 5.3], so we only list the results of some key calculations. The basic ingredientsare (evaluated at s = 0):(3.14) h A ij , A ml i = κ ˚ g ij ˚ g ml , h ( A ij ) s , A ml i = − κ ( u ij − κ ˚ g ij u )˚ g ml , (3.15) ( g ij ) s = 2 A Vij − κu ˚ g ij , (3.16) − H s = κ ∇ u + (∆ u + κ u ) N + (∆ u α ) ∂ z α , N s = −∇ u − κ (∆ u α ) ∂ z α . Combining these as in [5] gives ( | A | ) s = − κ (¨ u + κ u ) , h A ij , N i s = u ij − κ ˚ g ij u, ( | A N | ) s = − κ (¨ u + κ u ) , ( A ij ) s = − κ ( u im ˚ g mj + u mj ˚ g im ) , ( | A | ) s = − κ (¨ u + κ u ) , ( h A ij , A ml i ) s = − κ (¨ u + κ u ) , ( h A jl , A im ih A lm , A ij i ) s = − κ (¨ u + κ u ) . JONATHAN J. ZHU
Since | A | = | A N | = κ , from the above one can see that the first variation of the firstfour terms of P cancels to zero. It remains to check the variation of the last term | A | | H | ( | A N ( x T , · ) | ) − | A ( x T , · ) | ) . Since A = A N N , the factor in brackets vanishes at s = 0, so it is enough to show that itsvariation is zero too.Indeed, we find that ( A N ( x T , · )) s = h ( A ( x T , · )) s , N i + h A ( x T , · ) , N s i = h ( A ( x T , · )) s , N i since h N s , N i = 0. Therefore we have( | A N ( x T , · ) | ) s = 2 h ( A N ( x T , · )) s , A N ( x T , · ) i = 2 h ( A ( x T , · )) s , A ( x T , · ) i = ( | A ( x T , · ) | ) s . This completes the proof. (cid:3)
Using Lemma 3.4, the same proof as [5, Corollary 5.36] (using that ˚Γ and its variationsby K have codimension one) now gives Proposition 3.5.
Let Γ = ˚Γ a,b × R n − . If V ∈ N Γ with LV = 0 and k V k W , < ∞ then( D P )( V, V ) = 0.
Remark 3.6.
One may also check directly that D P ( V, V ) = 0 for V ∈ K using the secondvariation formulae in [10, Section 3].4. Estimates for entire graphs
We now proceed to prove estimates for entire graphs using the auxiliary quantities. Thesetup for this subsection follows that of Section 4. In particular we consider a normalvariation fields U with compact support over a cylinder Γ = ˚Γ × R n − k with k U k C < δ .Set | V | m = P j ≤ m |∇ j V | , so that k V k qW m,q = R Γ | V | qm ρ .First, we note the following crude bounds to be used for Taylor expansion. Lemma 4.1.
Let ˚Γ be a compact shrinker with | ˚ H | >
0. If Γ = ˚Γ × R n − k , then thereexists C, δ such that for any vector field U on Γ with k U k C < δ , we have kT k C ≤ C and kPk C ≤ C h x i . Proof.
As in [5, Lemma 5.30], each quantity in the definition of ∇ ⊥ τ and P is a smoothfunction of ( p, U, ∇ U, ∇ U ). The position vector x ( p ) does not enter the definition of ∇ ⊥ τ but it enters the definition of P quadratically. (cid:3) In the remainder of this section, ˚Γ is either a round sphere or an Abresch-Langer curve.4.1.
First order decomposition.
Given a compactly supported normal field U on Γ, wehave the orthogonal decomposition U = J + h , where J ∈ K and h ∈ K ⊥ . We may furtherdecompose J = U + J ′ , where U ∈ K and J ′ ∈ K .The first order expansion of φ implies the following estimates (see [10, Proposition 5.4]): Proposition 4.2 ([10]) . Let Γ = ˚Γ × R n − k . There exists ǫ and C so that if U is a compactlysupported normal field on Γ with k U k C < ǫ , then(4.1) k h k W , ≤ C ( k φ U k L + k U k L ) , LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 9 and for any κ ∈ (0 , Z Γ h x i | U | ρ ≤ C ( κ )( k φ U k κ L + k U k L ) . Estimates by ∇ ⊥ τ and φ . We proceed with U = J + h = U + J ′ + h decomposed asin Section 4.1. Lemma 4.3.
There exists C so that for any U as above, we have the pointwise estimate(4.3) ||∇ ⊥ τ | U − D T ( J ′ , J ′ ) | ≤ C ( | U | + 2 | J | | h | + | h | + 2 | J ′ | | U | + | U | ) . Proof.
Let T ( s ) = T ( p, sU, s ∇ U, s ∇ U ). Since kT k C ≤ C by Lemma 4.1, Taylor expansionabout s = 0 gives | T (1) − T (0) − T ′ (0) − T ′′ (0) | ≤ C | U | . Note that T (1) = |∇ ⊥ τ | U , T (0) = 0, T ′ (0) = DT ( U ) = 0 and T ′′ (0) = D T ( U, U ).Expanding the bilinear form D T ( U, U ) according to the decomposition of U , and using kT k C ≤ C to estimate the remaining terms except D T ( J ′ , J ′ ) finishes the proof. (cid:3) We may now estimate the variation field U in terms of φ U and |∇ ⊥ τ | U . Proposition 4.4.
Let Γ = ˚Γ × R n − k , where ˚Γ is a round shrinking sphere or an Abresch-Langer curve. There exists ǫ > κ ∈ (0 ,
1] and U is a compactly supportednormal vector field on Γ with k U k C ≤ ǫ , then(4.4) k U k L ≤ C ( κ )( k U k L + k|∇ ⊥ τ | U k L + k φ U k κ L ) , where U = π K ( U ). Proof.
By Proposition 3.2 or 3.3 respectively we have δ k J ′ k L ≤ kD T ( J ′ , J ′ ) k L for some δ >
0. Now to estimate the right hand side, we integrate estimate (4.3), using Corollary 2.2to estimate the Jacobi terms. This gives C − kD T ( J ′ , J ′ ) k L ≤ k|∇ ⊥ τ | U k L + k U k W , + k h k W , + k U k L + k U k L Z Γ h x i | h | ρ + k U k L k U k L ≤ k|∇ ⊥ τ | U k L + k U k W , + (1 + ǫ − ) k h k W , + k U k L + 2 ǫ k U k L + ǫ − k U k L . (4.5)Here for the second line we have used Cauchy-Schwarz, so that R Γ h x i | h | ρ ≤ C k h k W , ,as well as the elementary inequality 2 ab ≤ ǫa + ǫ − b .Now for small enough ǫ , we certainly have k U k L < k φ U k W , <
1, so lower powersdominate. Using Proposition 4.2 then gives(4.6) C − δ k J ′ k L ≤ k|∇ ⊥ τ | U k L + ǫ k U k L + C ( ǫ, κ ) (cid:18) k φ U k κ L + k U k L (cid:19) . Since(4.7) k U k L ≤ k U k L + k h k L + k J ′ k L ≤ k U k L + C ( k φ U k L + k U k L ) + k J ′ k L , if we choose ǫ < Cδ then the k U k L term may be absorbed into the left hand side; thus(4.8) k U k L ≤ C ( κ )( k|∇ ⊥ τ | U k L + k φ U k κ L + k U k L ) . (cid:3) Remark 4.5.
Proposition 4.4 is analogous to [5, Proposition 4.47] and to [3, Proposition2.1]. Indeed, following the cutoff and rotation method we use in the proof of Theorem 5.1below, or in [10, Theorem 7.1], yields a corresponding estimate by |∇ ⊥ τ | and φ for graphsover a (sufficiently large) subdomain.4.3. Estimates by φ . We now observe as in [5] that the shrinker quantity φ controls theauxiliary quantity ∇ ⊥ τ , to a degree consistent with a second order obstruction.First, we need a PDE estimate proven by Colding-Minicozzi [5]: Proposition 4.6 ([5]) . Let Γ = ˚Γ × R n − k , where ˚Γ is a round shrinking sphere or anAbresch-Langer curve. There exist δ > C = C ( k U k C ) so that whenever k U k C < δ ,we have(4.9) k|∇ ⊥ τ | U k L ≤ C ( k P U k L + k φ U k W , + k φ U k W , ) . Proof.
Since U is compactly supported, we may apply [5, Theorem 2.2] to Γ U (with ψ = 1).The desired estimate follows immediately after using the identity ∇ ⊥ i H = − A ( x T , X i ) −∇ ⊥ i φ (cf. [5, Proof of Theorem 7.4]) and that A , | H | − and |∇ H | are uniformly bounded. (cid:3) Second, we have the following Taylor expansion estimate for P : Proposition 4.7.
Let Γ = ˚Γ × R n − k , where ˚Γ is a round shrinking sphere or an Abresch-Langer curve. There exist C, C κ so that for any κ ∈ (0 , k P U k L ≤ C κ ( k U k L + k φ U k κ L ) + C k U k L k φ U k L + C k φ U k W , . Proof.
The case of round cylinders was proven in [5, Proposition 6.1], and the proof forAbresch-Langer cylinders proceeds exactly the same way. For the readers’ convenience, weemphasise the corresponding ingredients:Set P ( s ) = P ( p, sU, s ∇ U, s ∇ U ); then P (0) = 0 and Lemma 3.4 and Proposition 3.5imply that P ′ (0) = 0 and | P ′′ (0) | ≤ C h x i | h | ( | J | + | h | )respectively. Lemma 4.1 then gives the Taylor expansion estimate | P U | ≤ C h x i | h | ( | J | + | h | ) + C h x i | U | . The proof now follows as in [5, Proposition 6.1], using Corollary 2.2 to estimate the J terms and Proposition 4.2 for the h, U terms. (cid:3) We may now prove the main estimate for entire graphs:
LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 11
Theorem 4.8.
Let Γ = ˚Γ × R n − k , where ˚Γ is a round shrinking sphere or an Abresch-Langercurve. There exists ǫ > κ ∈ (0 ,
1] and U is a compactly supported normalvector field on Γ with k U k C ≤ ǫ and k U k C ≤ M , then(4.10) k U k L ≤ C ( κ, M )( k U k L + k φ U k W , + k φ U k W , + k φ U k κ L ) , where U = π K ( U ). Proof.
Combining Propositions 4.4, 4.6 and 4.7 and noting that lower powers dominate, wefind that(4.11) k U k L ≤ C ( κ )( k U k L + k U k L k φ U k L + k φ U k W , + k φ U k W , + k φ U k κ L + k U k L ) . Using the elementary inequality 2 ab ≤ ǫa + ǫ − b on the second term on the right andabsorbing the resulting k U k L terms into the left hand side gives the result. (cid:3) Lojasiewicz inequalities via τ In this section, we conclude the Lojasiewicz inequalities from the main estimate Theorem4.8, using the rotation and cutoff procedure in [10, Section 7]. For convenience set δ R := R n e − R / . Theorem 5.1.
Let ˚Γ be a round shrinking sphere or an Abresch-Langer curve.There exists ǫ > ǫ , λ , C j there exist R , l such that if l ≥ l , Σ n ⊂ R N has λ (Σ) ≤ λ and:(1) For some R > R , we have that B R ∩ Σ is the graph of a normal field U over somecylinder in C n (˚Γ) with k U k C ( B R ) ≤ ǫ and k U k L ( B R ) ≤ ǫ /R ;(2) |∇ j A | ≤ C j on B R ∩ Σ for all j ≤ l ;then there is a cylinder Γ ∈ C n (˚Γ) and a compactly supported normal vector field V over Γwith k V k C ,α ≤ ǫ , such that Σ ∩ B R − is contained in the graph of V , and k V k L ≤ C (cid:16) k U k a l L + k φ k a l L ( B R ) + k φ k a l L ( B R ) + δ a l R − (cid:17) , where C = C ( n, l, C l , λ , ǫ ) and a l ր l → ∞ . Proof.
Let a l := a l, ,n be the exponent from interpolation (see Appendix A). Followingprecisely the proof of [10, Theorem 7.1], except using Theorem 4.8 in place of [10, Theorem5.8] yields a vector field V , supported on Γ ∩ B R − and such that Σ ∩ B R − is contained inthe graph of V , satisfying the estimate(5.1) k V k L ≤ C ( k π K ( V ) k L + k φ V k W , + k φ V k W , + k φ V k κ L ) , where the rotation part satisfies(5.2) k π K ( V ) k L ≤ C ( k φ k a l L ( B R ) + k U k a l L + δ a l R − + δ R − ) . Moreover, for any s ∈ [1 ,
2] we have the cutoff estimate(5.3) k φ V k sL s ≤ k φ k sL s ( B R ) + C ( s ) δ R − , Using interpolation on the φ V terms in (5.1) and then the cutoff estimate, we have k V k L ≤ C ( k φ k a l L ( B R ) + δ a l R − + k U k a l L + δ R − )+ C ( k φ k a l L ( B R ) + δ a l R − + k φ k a l L ( B R ) + δ a l R − + k φ k κ L ( B R ) + δ κ R − ) . (5.4)We take κ so that κ > a l . Then since lower powers dominate, we have(5.5) k V k L ≤ C (cid:16) k U k a l L + k φ k a l L ( B R ) + k φ k a l L ( B R ) + δ a l R − (cid:17) . (cid:3) Proof of Theorem 1.1.
Apply Theorem 5.1 and note that k U k a l L is dominated by the expo-nential error term for large enough l . (cid:3) Proof of Theorem 1.2.
Let V, Γ be as given by Theorem 1.1. Exactly as in the proof of [10,Theorem 1.4], we have(5.6) | F (Γ V ) − F (Γ) | ≤ C ( k φ V k L + k V k L ) , and | F (Σ) − F (Γ V ) | ≤ Cδ R − . Consider l large enough so that a l > . The conclusion ofTheorem 1.1, together with H¨older’s inequality imply that k V k L ≤ C ( k φ k al L + δ al R − ) . Using the cutoff estimate (5.3) and collecting dominant terms, we conclude that(5.7) | F (Σ) − F (Γ) | ≤ C ( k φ k al L + δ R − ) . (cid:3) Remark 5.2.
If in addition to the hypotheses of Theorem 1.1 one also has k φ k L ≤ e − R / ,then arguing as in the proof of [10, Theorem 7.2] we obtain(5.8) k V k L ≤ C (cid:18) e − a l R +( R − a l n e − a l ( R − (cid:19) . Since l may be chosen so that a l is arbitrarily close to 1, this gives an alternative proof of[10, Theorem 7.2], and hence of [10, Theorems 1.1 and 1.2], for the special cases where Γ isa round cylinder or an Abresch-Langer cylinder. Appendix A. Interpolation
Here we recall some interpolation inequalities; see also [10, Appendix A] and [3, AppendixB]. In this appendix, L p refers to unweighted space, with L pρ the ρ -weighted space. Lemma A.1.
There exists C = C ( m, j, n ) so that if u is a C m function on B nr , then for j ≤ m , setting a m,j,n = m − jm + n we have r j k∇ j u k L ∞ ( B r ) ≤ C (cid:16) r − n k u k L ( B r ) + r j k u k a m,j,n L ( B r ) k∇ m u k − a m,j,n L ∞ ( B r ) (cid:17) . LOJASIEWICZ INEQUALITIES FOR MEAN CONVEX SELF-SHRINKERS 13
The lemma above also holds for tensor quantities on a manifold with uniformly boundedgeometry. It follows that on a generalised cylinder Γ = ˚Γ k × R n − k , for large enough R (A.1) k u k W j,pρ ( B R − ) ≤ C ( m, j, n, p, M , M m ) k u k a m,j,n L pρ ( B R ) . References [1] U. Abresch and J. Langer. The normalized curve shortening flow and homothetic solutions.
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Mathematical Sciences Institute, Australian National University, Hanna Neumann Build-ing, Science Road, Canberra, ACT 2601, Australia and Department of Mathematics, Prince-ton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA
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