Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

Let $C$ be an $m$-dimensional cone immersed in $\mathbb{R}^{n+m}$. In this paper, we show that if $F:M^m \rightarrow \mathbb{R}^{n+m}$ is a properly immersed mean curvature flow self-shrinker which is smoothly asymptotic to $C$, then it is unique and converges to $C$ with unit multiplicity. Furthermore, if $F_1$ and $F_2$ are self-expanders that both converge to $C$ smoothly asymptotically and their separation decreases faster than $\rho^{-m-1}e^{-\rho^2/4}$ in the Hausdorff metric, then the images of $F_1$ and $F_2$ coincide.


Introduction
A proper m-dimensional immersion F : M m → R n+m is called a self-shrinker of the mean curvature flow if it satisfies the following non-linear elliptic equation for every p ∈ M : where H is the mean curvature vector of the immersion and F (p) ⊥ is the component of the position vector perpendicular to the tangent plane T F (p) F (M ) ∼ = F * (T p M ).
If F satisfies this equation, then the family of rescalings p) is a solution to the backwards mean curvature flow equation, where H(p, t) denotes the mean curvature of the immersion F t at the point p ∈ M .In this paper, we will define the rescaled immersion λF : M m → R n+m by (λF )(p) = λF (p) ∈ R n+m for each p ∈ M m and for any λ > 0.
Let γ : Γ m−1 → S n+m−1 ⊂ R n+m be a closed (m − 1)-dimensional properly immersed submanifold of S n+m−1 .The cone over γ is the following immersion into R n+m : C : Γ × (0, ∞) → R n+m C(q, ℓ) = ℓγ(q) In this paper, we extend the results of Lu Wang in [23] and prove the uniqueness of higher codimension self-shrinkers which are properly immersed and locally smoothly asymptotic to a given immersed cone.More precisely, we prove the following theorem: Theorem 1.1.Let C : Γ m−1 × (0, ∞) → R n+m be a regular cone of dimension m and R 0 a positive constant.Suppose that F : M m → R n+m and F : M m → R n+m are smooth, connected and proper self-shrinking immersions into R n+m \ B R0 with boundary contained in ∂B R0 .If F and F are smoothly asymptotic to the same cone C, then F and F can be reparametrized to converge to C with unit multiplicity and are the same shrinker up to reparametrization.
The method of proof runs along similar lines to the proof for embedded hypersurfaces in [4] and [23], but numerous complications arise when considering immersions in high codimension.In particular, the major difficulties in the case of immersed shrinkers are the possibility of convergence with multiplicity and the existence of self-intersection points.These considerations motivate a new definition, Definition 2.3, of local smooth convergence for immersed shrinkers converging with multiplicity to a cone.One can check that in the case of embedded hypersurfaces (which must converge with multiplicity one), Definition 2.3 reduces to the definition of local smooth convergence given on page 4 of [23].
The first property of Definition 2.3 tells us that for every compact annular region Bρ2 \ B ρ1 ⊂ R n+m , the rescaled immersions λF must eventually be caught in some ǫ-band around the truncated cone C ρ1,ρ2 .The second and third properties assume a choice of pullback G λ of λF to the normal bundle N C ρ1,ρ2 of C ρ1,ρ2 and describe the convergence of λF in N C ρ1,ρ2 as a multi-section with constant multiplicity of convergence.Notice that this definition is equivalent to local Euclidean graphical convergence with arbitrary multiplicity over small embedded neighborhoods of the cone C.This is a very modest and natural generalization of the notion of convergence used in [23].Additionally, Definition 2.3 is consistent with the definition of asymptotic conicality given in [5] if we consider transversal sections S of the Grassmannian and multi-graphs defined over S.
After establishing definitions, we obtain estimates analogous to those found in Lemma 2.1 and 2.2 in [23] by considering graphs of vector-valued multifunctions and utilizing the mean curvature flow system in lieu of the mean curvature flow equation.We then attempt to write one self-shrinker as a normal section of the other.There is not an obvious way to do this-for example, the two shrinkers may converge to C with different multiplicities.However, outside some large radius R 0 , the annular subsets F −1 (R n+m \ B R ) ⊂ M are homotopic for all R > R 0 and evenly cover the cone.As a result, we may appeal to the theory of covering spaces and utilize the smooth lifting property to allow us to "unwrap" the covering and construct a normal projection to other shrinkers realizing the same covering space.One particularly significant consequence of this argument is Corollary 3.9, which implies that a shrinker with multiplicity may be written as a normal graph over itself.
When one shrinker can be represented as a section V of the normal bundle of the other, we generalize the approach of Jacob Bernstein in [4] to show that they coincide, instead of the parabolic backwards uniqueness used in [23].We show that (3.35) holds-that is, this normal section V is an "almost" eigensection of the drift Laplacian ∆ ⊥ − 1 2 ∇ ⊥ F T on the normal bundle of a shrinker F .In Section 4, we modify the results of [4] so that they hold for sections of vector bundles with metric connections.In Section 5, we obtain that the normal section V representing the separation between the shrinkers is actually the zero section.In particular, Corollary 3.9 and the fact that V ≡ 0 imply that every shrinker asymptotic to a cone may be reparametrized to converge with multiplicity one, significantly simplifying the immersed picture.This implies that any two shrinkers asymptotic to a cone C in the sense of Defintion 2.3 must cover C equivalently, may be written as normal graphs over one another, and thus coincide.In particular, increasing the multiplicity of the cone in the sense of geometric measure theory does not give rise to new shrinkers.
A proper m-dimensional immersion F : M m → R n+m is called a self-expander of the mean curvature flow if it satisfies the following non-linear elliptic equation for every p ∈ M : By a small modification of our arguments for shrinkers, we can also prove a similar theorem for self-expanders with a certain decay rate as in [4].
Theorem 1.2.Let C : Γ m−1 × (0, ∞) → R n+m be a regular cone of dimension m and R 0 a positive constant.Suppose that F : M m → R n+m and F : M m → R n+m are smooth, connected, proper self-expanding immersions into R n+m \ B R0 with boundary contained in ∂B R0 that are smoothly asymptotic to C. If F and F satisfy then F and F are the same expander up to reparametrization.Here dist H is Hausdorff distance.
Remark 1.4.Note that we allow the convergence to be of multiplicity greater than or equal to one.
Roughly, one may observe that estimates given in Sections 2 and 3 depend only on the relations |H| ≃ |F ⊥ |, and |(∂ t F ) ⊥ | = |H|, which hold both for shrinkers flowing backwards in unit time and expanders flowing forwards in unit time.Using these estimates, we obtain a differential inequality for the linearization of the expander equation in Corollary 3.41, which allows us to apply the theory of Bernstein.
Our results have a number of interesting consequences.A major class of examples of high codimension self-similar mean curvature flow solutions are minimal cones.One application of Theorem 1.1 is the following corollary: Corollary 1.5.In any dimension and codimension, the only smooth, complete, properly immersed self-shrinkers asymptotic to a minimal cone are linear subspaces.
We similarly obtain a weaker statement for self-expanders from Theorem 1.2.
Corollary 1.6.In any dimension and codimension, if C is a minimal cone, any non-trivial self-expander asymptotic to C may, outside some ball B R , be written as a normal multi-section over C whose magnitude is asymptotically bounded below by a constant multiple of r −m−1 e −r 2 /4 .Another application of our results is to Lagrangian mean curvature flow, which has seen many recent articles on the properties of self-shrinkers and self-expanders.Self-expanding Lagrangians, and especially those asymptotic to cones, have been a recent object of interest due to the proposed program of Joyce in [12] to resolve the singularities of the almost-calibrated Lagrangian mean curvature flow by gluing in Lagrangian expanders.Indeed, Neves proves in [20] that a singularity of an almost-calibrated Lagrangian mean curvature flow must be asymptotic to a union of special Lagrangian cones, so Corollary 1.6 provides non-trivial information about expanders that may be used to resolve these singularities.
Additionally, an example from Lagrangian mean curvature flow gives a bound on how much the decay condition in Theorem 1.2 can be weakened.Anciaux [1] and Joyce-Lee-Tsui [13] discovered a family of Lagrangian self-expanders asymptotic to transversally intersecting Lagrangian planes, and the uniqueness of these expanders was later proved by Lotay-Neves in [18] and by Imagi-Joyce-dos Santos in [11].In particular, it was found that in the immersed case, the only two expanders asymptotic to the two transversal Lagrangian planes were the Joyce-Lee-Tsui expander and the planes themselves.This is consistent with our findings, as the decay of the Joyce-Lee-Tsui expander is O(e −r 2 /4 ).
While they do not occur in the almost-calibrated case, Lagrangian self-shrinkers are an important class of singularity models for the general Lagrangian mean curvature flow.A number of recent articles have been written about compact shrinkers in this setting (see [6], [7], [19]).Moreover, in [16] and [17], Lee and Wang construct explicit examples of Hamilton stationary self-shrinkers asymptotic to Hamilton stationary Lagrangian cones.By Theorem 1.1, these are unique.
Corollary 1.7.Assume that λ j > 0 for 1 ≤ j ≤ k < n and λ j < 0 for k < j ≤ n are integers satisfying n j=1 λ j > 0. Let be the family of Hamilton stationary Lagrangians in C n constructed by Lee and Wang.The embedded shrinker V −2 n j=1 λj is the only self-shrinker asymptotic to the cone V 0 .
This article also adds to the existing literature on general mean curvature flow in arbitrary codimension.The properties of higher codimension self-shrinkers have been studied from the perspective of the F -functional (applied in [8] to hypersurfaces) in the papers [2], [3], and [15].Higher codimension solitons with the property that the principal normal is parallel have also been studied by Smoczyk [22] for self-shrinkers and by Kunikawa [14] for translating solitons.

Preliminaries and Basic Estimates
When working with immersed sumbanifolds, it will often be convenient to consider Langer charts, which can be roughly thought of as disk-like neighborhoods on the source manifold M of a proper immersion F : M m → R n+m .If p ∈ M , let A p be an arbitrary affine isometry of R n+m that takes F (p) to the origin and takes the tangent plane is called an (r, α)-immersion if for every q in M , there exists a function f q : D m r → R n with Df q (0) = 0 and |Df q | ≤ α so that the image (A q • F )(U q,r ) is equal to the graph of f q over D m r .In particular, the restriction F | Uq,r is an embedding.The next proposition gives a quantitative bound on the maximum radius of graphical Langer charts with derivative bounded by α.This bound is dependent only on the given α and the magnitude of the second fundamental form A of the immersion F .Proposition 2.2.Let α > 0. Then for any C 2 -immersed submanifold F : M m → R n+m and any r satisfying r ≤ α F is an (r, α)-immersion.
We wish to study the self-shrinkers that are smoothly asymptotic to a given cone C, so it is necessary to define an appropriate notion of local smooth convergence to make this idea rigorous.Definition 2.3.We say that an immersion F : M m → R n+m is smoothly asymptotic to an immersed cone C : Γ × (0, ∞) → R m+n with multiplicity k (which we will henceforth denote by kC) if the following properties hold.
(1) For any compact subset and let N (Γ × [ρ 1 , ρ 2 ]) be the normal bundle with fiber metric equal to the pullback metric ).Consider the intersection of λF (M ) with T ǫ (C ρ1,ρ2 ), an ǫ-tubular neighborhood of C ρ1,ρ2 in R n+m .For sufficiently small λ, this intersection can be pulled back to a smooth family of immersions G λ from the set Σ := (λF (3) For every point p ∈ Γ × [ρ 1 , ρ 2 ], there exists some r > 0 and a Langer chart U p,r of the cone C such that the intersection G λ (Σ) ∩ D ⊥ U p,r can be parametrized as the image of k sections {σ 1 λ , . . ., σ k λ } of the unit disk bundle, such that each σ i λ : U p,r → D ⊥ U p,r converges to the zero section smoothly with respect to λ as λ → 0.
Remarks 2.4.Notice that by condition (2) and the homogeneity of the cone, there exists an R > 0 and a compact set K, such that Also note that in property (3), it is not necessary to insist that each neighborhood is covered by exactly k sections if the link Γ of C is connected.If U p1,r and U p2,r have nonempty intersection and are covered by k and k ′ local sections respectively, then k = k ′ .Furthermore, the finiteness of the cover, i.e. the condition that k < ∞, follows from the properness of the immersion.
If the link Γ is disconnected, then an immersion may in principle converge with distinct multiplicity on each end.In this case, we consider the cone C to be the union of the cones over each connected component of Γ.Then, we consider separately each end of M asymptotic to each of these individual cones and can assume without loss of generality that Definition 2.3 is satisfied on a cone with connected link.
Notation 2.5.Given an immersion F : M m → R n+m and a compact set K containing the origin, we often consider the annular regions For ease of reading, we introduce the notation M K := F −1 (R n+m \K).For immersions indexed by i or by t, we denote this set by We will often consider K = B R , the ball of radius R containing the origin.For notational simplicity, we set We will also denote the sets In the case of a cone C with disconnected link (discussed in Remarks 2.4) we will consider individual connected components (i.e.individual conical ends) of M i,K separately along with their corresponding asymptotic cones with connected link.In the sequel, we conflate the notation for these connected components with that for the entire M i,K and C in the interest of simplicity.This notational convention does not affect the proofs.Lemma 2.6.Let F : M m → R n+m be a shrinker smoothly asymptotic to the cone C : Γ m−1 × (0, ∞) → R n+m , and F t the corresponding solution to the backwards mean curvature flow.There exist C 1 > 0 and R 1 > 0 such that for p ∈ M t,R1 , t ∈ (0, 1] and 0 ≤ i ≤ 2, where A Ft is the second fundamental form of the immersion F t .
Proof.The second fundamental form and its covariant derivatives are bounded, since the link Γ is a closed, properly immersed submanifold.For any ǫ > 0, for sufficiently small λ > 0, the intersection of λF (M ) with the tubular neighborhood T ǫ (C 1/2,2 ) can be pulled back to a submanifold S λ of the normal unit disk subbundle D ⊥ (Γ × [1/2, 2]) with boundary contained in D ⊥ (Γ × {1/2}) ∪ D ⊥ (Γ × {2}).By Definition 2.3, for sufficiently small λ, the submanifold S λ can be covered by finitely many images of local sections of D ⊥ (Γ × [1/2, 2]) converging smoothly to the zero section as λ → 0. Hence, the second fundamental form A S λ of S λ converges locally smoothly to the second fundamental form A C(Γ) on C 1/2,2 with a uniform rate of convergence and there exists a constant λ 1 > 0 so that there is a δ 1 > 0 such that if λ ∈ (0, δ 1 ), then By scaling, we have which completes the proof of the statement.
We collect some useful facts and notation in the following remark.
Remark 2.7.Given a point p = (p, 1) in the Riemannian manifold Γ × (0, ∞) with metric C * g R n+m , we briefly describe local graphical convergence of the family of rescaled shrinkers {F t } to a neighborhood of p.By Definition 2.3 there exists a time t 0 > 0, an r 0 > 0, and a Langer neighborhood U p,r0 of p in the cone such that for t < t 0 , the intersection of the rescaled shrinker F t with the unit disk bundle DU p,r0 is represented by k sections {σ 1 λ , . . ., σ k λ }, where λ = √ t.To simplify notation, we denote these sections by σ j t throughout the rest of the paper.As t approaches 0, the sections σ j t approach the zero section smoothly uniformly.In particular, the second fundamental forms A σ j t (q, σ j t (q)) of graph(σ j t | Up,r 0 ) will converge uniformly to A Up,r 0 (q).Thus, by Proposition 2.2, given α > 0 we can choose r ∈ (0, r 0 ) so that C 1/2,2 is an (r, α)-immersion and each U p,r is the graph of a function f : D m r → R n .For t < t 0 , we may write the images σ j t (U p,r ) ∩ (π ) as the graphs of a smoothly varying family of time dependent functions g j t : In particular, we may assume that each of the g j t 's has spatial gradient bounded by |Dg j t | ≤ 2α for all t < t 0 .The next lemma describes locally-defined functions whose graphs represent the shrinker near a point (p, ρ) ∈ Γ × [R 2 , ∞) (for R 2 > 0 large) and proves some bounds on their spatial and temporal derivatives.
Lemma 2.8.Let α > 0 and p ∈ Γ be given and let the k functions g i t be defined for (p, 1) ∈ Γ × (0, ∞) as in Remark 2.7.There exists a large radius R 2 > R 1 , and numbers 0 < ǫ 0 < 1 and C 2 > 0 depending on α, so that if p = (p, ρ) ∈ Γ × [R 2 , ∞), then the associated functions are well defined for t ∈ [0, 1] and satisfy that,  ∞).Observe that ρ −2 < t 0 /16 < t 0 , so the functions g i t : D m ǫ0 → R n associated to U (p,1),ǫ0 are defined for times t ≤ ρ −2 .By scaling and the self-similarity of we take a rescaling of the immersion G λ into the normal unit disk sub-bundle from Definition 2.3, We obtain this immersion into the ρ-disc subbundle of the normal bundle by simply composing G ρ −1 with a rescaling of the unit disc bundle by ρ.Then, we can see that the intersection of the image of ρG ρ −1 with the cylinder over the disk D m ρǫ0 ((p, ρ)) ⊂ T (p,ρ) C in the tangent plane to the cone C at p = (p, ρ), can be represented as the k graphs of the functions Note that the shrinker F can be covered by the images of such graphs by (3) in Definition 2.3 and Remark 2.7.We now establish the derivative bounds on the u i 's.We first take the spatial gradient for fixed t ∈ [0, 1].
However, we know that for t < t 0 , |Dg i t | ≤ 2α and thus |Du i | ≤ 2α.By Lemma 2.6, Lemma A.1, and Lemma A.3 we know that where C 2 is a constant depending on C 1 , α, the dimension m, and codimension n.
Next, we establish the bounds on ∂ t u i and its derivatives.For clarity of presentation, we do our calculations for a generic function u i which we simply denote by u.We have the following system for the backwards mean curvature flow of the graph of u(•, t) over D m ǫ 0 ρ 2 .Consider the graph of u given by the embedding X(w) = (w, u(w, t)), where w ∈ D m ǫ 0 ρ 2 . Then, the embedding satisfies backwards mean curvature flow if (2.9) where a k ∂ k X is a vector field generating the appropriate tangential diffeomorphisms.The equation (2.9) yields the following system: By substituting the first system into the second, we obtain the backwards mean curvature flow system for graphs: The metric g ij can be expressed as follows: Since |Du| is uniformly bounded, g ij and g ij are uniformly bounded.Since The derivatives D k g ij can be expressed as polynomials in D k g ij and lower order derivatives of g ij .From this fact and our previous arguments, we deduce that that and thus |∂ k ∂ t u| ≤ Cρ −2 .A similar calculation yields |∂ 2 lk ∂ t u| ≤ Cρ −3 .This completes the proof of the lemma.
In particular, the distance between F (p) and the image of the cone Proof.We know that for any y ∈ D m ǫ0ρ , the vector u i (y, 0) is associated to a point on the image of the cone C, and u i (y, 1) is associated to a nearby point on the shrinker F , and the displacement between them is given by the vector u i (y, 1) − u i (y, 0).Using Lemma 2.8, we estimate the magnitude of this displacement.
The bounds for the partial derivatives of the u i are proved in exactly the same way.
By assumption, there is some y ∈ D m ǫ0ρ associated to F (p), which completes the proof.

Self-Shrinking Ends as Normal Exponential Graphs
In this section, we will prove that if two shrinkers F 1 : M 1 → R n+m and F 2 : M 2 → R n+m are asymptotic to the same cone C and satisfy a topological condition, then one can be written as a normal graph over the other.More precisely, there exists a radius R 4 > R 2 so that there is a compact K ⊂ R n+m such that the end M 2,K ⊂ M 2 can be pulled back isometrically to a section of the normal bundle of the end M 1,R4 (see Notation 2.5 for the definition of these submanifolds).We will use a covering space argument to obtain this isometry.First, we will recall some well-known results about smooth covering spaces.Proposition 3.1 (Smooth Lifting Criterion).Let X, X, and Y be path-connected smooth manifolds.Suppose p : ( X, x0 ) → (X, x 0 ) is a smooth covering space and An immediate consequence of this the uniqueness of smooth covering spaces.Proposition 3.2.Let X, X1 , X2 be path-connected smooth manifolds.If p 1 : X1 → X and p 2 : X2 → X are two smooth covering maps, then there exists a diffeomor- )), the map f : X1 → X2 is the smooth lift p1 and its inverse is the lift p2 .We will use the convergence of F 1 and F 2 to C to find appropriate covering maps )), the lift p1 : M 1.R3 → M 2,K will yield a section of the pullback bundle.We then show using our estimates that this section is close by to a normal section.
We first need to show that given a shrinker F : M → R m+n asymptotic to the cone C, there is a consistent notion of a "topology at infinity."More precisely, Lemma 3.3.Let F : M → R m+n be asymptotic to the cone C. Then there exists a large radius R 3 > R 2 , such that for any compact sets K 1 , K 2 ⊃ B R3 which are radial with respect to the origin, Proof.Let r : R m+n → R be the distance function r(x) = |x| on R n+m .If we consider the restriction of this function to the cone C, we see that the tangential gradient ∇ C r = (Dr) T = Dr (where Dr indicates the Euclidean derivative), and in particular |∇ C r| = |Dr| = 1.Since the tangent planes of the shrinker F approach those of the cone C, there is a radius In particular |∇ F r| is non-vanishing and uniformly bounded away from zero.Every point in M R3 is contained in a unique flow line of the negative gradient flow ẋ = −∇ F r(x) which has velocity bounded above and below.If we consider a compact set K ⊃ B R3 that is radial with respect to the origin, we can move along the flow lines to homotope M K to M R3 .Observing that the fundamental group is a homotopy invariant concludes the proof of the lemma.Lemma 3.4.For R 3 > 0 as in Lemma 3.3, and compact, radial K ⊃ B R3 , the manifold M K is a k-fold covering space of the cone C K where K is a compact, radial subset of R n+m containing the origin.Furthermore, this covering map is realized in R n+m as projection along the normal fibers of the cone.
Proof.This follows immediately from property (3) in Definition 2.3.
be the maximum of the radii R 3 > 0 given by Lemma 3.3 applied to F 1 and F 2 .Let p 1 and p 2 be the projections of M 1,R3 and M 2,R3 in the normal bundles of C K1 and C K2 , respectively, where K 1 , K 2 are compact, radial sets containing the origin as in Lemma 3.4.If , where K is some compact, radial set containing B R3 .Furthermore, this diffeomorphism can be realized as a section of the pullback vector bundle p −1 1 N C over M 1,2R3 .Proof.The first claim is an immediate consequence of the fact that M 1,R3 and M 2,R3 are covering spaces and the uniqueness of smooth covering spaces, Proposition 3.2.To see that this diffeomorphism is a section of the pullback bundle π −1 1 N C, with base M 1,2R3 and fibers N p1(x) C for x ∈ M 1,2R3 , we realize the lifted map p1 : M 1,2R3 → M 2, K locally in Euclidean space.Around each point, there is a coordinate patch of the shrinker F 1 which by definition can be written as a local section of the normal bundle of the cone.Over this patch of the cone lie k sections representing the intersection of the shrinker F 2 with the normal fibers above that patch of the cone.The lift of p 1 chooses one of these sections, and thus the map p1 can be represented at a point x ∈ M 1 as p1 (x) = x + v, where v is a vector in Note that when restricted to the section determined by the lift, p 2 is injective.
Remark 3.6.We will see in Section 5 that Corollary 3.9 ensures that the lifting criterion is always met in the case of self-shrinkers, possibly after a reparametrization.In the case of self-expanders, the lifting condition in Lemma 3.5 may not be satisfied, i.e. p 1 * (π 1 (M 1,R3 , x 1 )) = p 2 * (π 1 (M 2,R3 , x 2 )).In this case we consider the subgroup Now, we may instead consider the locally isometric covering manifolds M1,R3 and M2,R3 of M 1,R3 and M 2,R3 respectively corresponding to subgroup G. Let Mi,R3 be an ℓ-fold cover of M i,R3 with projection P i : Mi,R3 → M i,R3 , a local isometry.We represent M i,R3 locally as a collection of k local sections {σ j } k 1 of the normal bundle over topological disks U contained C Ki , as in the proof of Lemma 3.5.Given a section σ j , the inverse image P −1 i (graph(σ j )) consists of ℓ disjoint isometric copies of graph(σ j ) in Mi,R3 .Thus, the local geometry on all relevant scales is unchanged when lifting to the covers M1,R3 and M2,R3 , so in the rest of the paper, we may replace our expanders by these covers and the isometric immersions Fi = F i • P i whenever appropriate.Note that Mi,R3 is a kℓ-fold cover of C Ki , where ℓ is equal to the index [p 1 * (π 1 (M 1,R3 , x 1 )) : G], for i = 1, 2. After this "reparametrization" by a locally isometric cover, all arguments in the paper may be applied without complication to the self-expanders F1 and F2 .
Remark 3.7.It may also be noted, continuing the thread of Remark 3.6, that if M 1,R3 is non-orientable, then we may consider the locally isometric orientable double cover π : Just as in Remark 3.6, the local geometry is unchanged and we may consider the isometric immersions Fi = F i • π without loss of generality.This removes the need to consider integration with densities on a non-orientable manifold.Proposition 3.8.Let F 1 and F 2 satisfy the hypotheses of Lemma 3.5 and let ǫ ≃ R −1 3 , where the implicit constant depends only on n, m, and , the maximum of the constants given by Lemma 2.8.Consider the ǫ-tubular neighborhood T ǫ (M 1,4R3 ) of the zero section inside the total space p −1 1 N C with the Euclidean pullback metric inherited from R n+m .There exists a compact set K containing B R3 such that the section σ : ), and normal projection from M 2,K to M 1,2R3 with respect to pullback metric on p −1 1 N C is well-defined and injective.Proof.By Corollary 2.10, for a sufficiently large R 3 and some K, the section σ representing M 2,K is ǫ-close to the zero section of the normal bundle N M 1,R3 .Thus, around a point p ∈ M 1,4R3 and for some ǫ 1 < ǫ 0 , we can realize the ǫ 1 |F 1 (p)|neighborhood of p in p −1 1 N C as a subset of an ǫ 1 |F 1 (p)|-tubular neighborhood which is itself realized in Euclidean space R n+m under the exponential map.One may choose ǫ is sufficiently small that the ǫ 1 F 1 (p)-tubular neighborhood contains a connected, embedded neighborhood U 1 of F 1 (p) and a connected, embedded neighborhood U 2 of F 2 (σ(p)).Additionally, by Lemma 2.8 we may represent these embedded pieces as graphs u 1 and u 2 with bounded gradient over the same m-disk of radius ǫ 1 |F 1 (p)|/2.Lemma 2.8 further tells us that these ).By integrating along paths out of q, for sufficiently large R 3 , we see that the the condition ensures that M 1,4R3 only intersects Bn+m dist(σ(p),q) (σ(p)) at q.This proves that nearest point projection is well-defined.Now we prove the injectivity of the nearest point projection from M 2,K to M 1,4R3 in the total space of the fiber bundle p −1 1 N C. Suppose that there exist two points q 1 and q 2 in the disk D m ǫ 1 2 |F1(p)| such that F 1 (p) is the nearest point in U 1 to both q1 = (q 1 , u 2 (q 1 )) and q2 = (q 2 , u 2 (q 2 )).This implies that the vector q2 − q1 lies in the normal space N F1(p) U 1 .The gradient bound |Du 1 | ≤ 2α implies that the inner product of the unit n-blades in Λ n (R n+m ) representing N F1(p) U 1 and the normal space {0} × R n is bounded below.
Consequently, the projection of the vector q2 − q1 to the normal space {0} × R n , which we denote (q 2 − q1 ) ⊥ , has magnitude bounded below Similarly, we can bound the magnitude of the tangential component, (q 2 − q1 ) T = q 2 − q 1 above.
Because (q 2 − q1 ) ⊥ = u 2 (q 2 ) − u 2 (q 1 ), we can estimate the difference quotient Let µ = (q 2 − q 1 )/|q 2 − q 1 | be the unit vector pointing in the same direction as q 2 − q 1 .The mean value inequality for vector valued functions tells us that there is a t 0 ∈ (0, 1) such that at q 3 = (1 , for α sufficiently small.This is a contradictiontherefore, normal projection from M 2,K to M 1,2R3 with respect to pullback metric on p −1 1 N C is well-defined and injective inside the tubular neighborhood This implies that, perhaps with a slightly modified compact set K, M 2,K can be written as a section V of the normal bundle of M 1,R4 , where R 4 > 2R 3 .Corollary 3.9.Let F : M → R n+m be a self-shrinker and let the projection map p : M R3 → C K be given by orthogonal projection in the normal bundle of C K .Let γ ∈ π 1 (C K , x 0 ) be a non-trivial deck transformation of M R3 , considered as a covering space of C K .Let F 1 = F , and define the shrinker Let p 1 and p 2 be the standard normal projections to C K restricted to the images of F 1 and F 2 respectively.The previous two lemmas allow us to write M R4 as a non-trivial section of its own normal bundle.
Proof.The deck transformation permutes the sheets of the k-fold covering space without fixed points, so the lift in Lemma 3.5 of the projection map is realized as projection along the normal fibers of C from a sheet U to the sheet γU .The statement is immediate from this remark and the previous lemmas.
In the following remark, we introduce several conventions we will use to calculate derivatives on the various bundles associated to the shrinker.Remark 3.10.In the remainder of this section, it often will be useful to move between an intrinsically defined vector field on T M 1,R or N M 1,R and its realization in R n+m .Thus, we recall here the conventions given in [21, §2] for the induced connections on the tangent and normal bundles, T M 1,R and N M 1,R .The Levi-Civita connection ∇ F1 on T M 1,R induced by the immersion where X, Y ∈ Γ(T M 1,R ), (F 1 ) * (Y ) is an arbitrary extension of (F 1 ) * (Y ) to an open neighborhood in R n+m , the operator D represents the standard differentiation of vector fields in R n+m , and the superscript T denotes projection to (F 1 ) * (T M ).
Similarly, the induced connection ∇ ⊥ on the normal bundle N M 1,R is given by ) is identified with its realization in the tangent space of R n+m and arbitrarily extended to an open set by ν, and ⊥ is projection to the orthogonal complement of (F 1 ) * (T M ).
Let V be a section of the normal bundle N M 1,R representing the shrinker F 2 in the sense of Lemma 3.8.We now obtain estimates on V and its derivatives.Lemma 3.11.For R 4 > 8R 3 , there exists C 3 > 0 such that, for the section where ∇ ⊥ is the induced connection on the normal bundle N M 1,R4 .
Proof.The strategy of the proof is along the same lines as [23, Lemma 2.3], but written in more general language and modified to accommodate the presence of a non-trivial normal bundle.Note that the case i = 0 is immediate since |V (x)| represents the distance from x to M 2,K in the normal bundle N M 1,R4 and Corollary 2.10 implies that this distance is bounded proportional to |F 1 (x)| −1 .Thus, we begin with the i = 1 case.
Let x 0 ∈ M 1,R4 and choose a local frame {n β (x)} β∈{1,...,n} of the normal bundle defined in a neighborhood of x 0 which will be made explicit at a later point in the proof.
We consider three points associated to x 0 .
• z 0 := p 1 (x 0 ), where p 1 is the standard projection from is the local inverse of projection to the cone from M 2,K described in the proof of Lemma 3.5.
For sufficiently large R 4 > 0, we may apply Lemma 2.8 and represent a neighborhood of F 1 (x 0 ) as the graph of a function u 1 defined on a domain Similarly, a neighborhood of F 2 (ζ 0 ) can be represented as the graph of a function u 2 defined on the same domain D ⊂ T z0 C. By the i = 0 case and Corollary 2.10, the distance between y 0 and ζ 0 is bounded of order O(|F 1 (x 0 )| −1 ), so F 2 (y 0 ) is contained in the graph of u 2 : that is, there exists q 0 ∈ T z0 C such that (q 0 , u 2 (q 0 )) = F 2 (y 0 ).
We will explicitly parametrize this correspondence between points in the graphs of u 1 and u 2 so that we can differentiate the nearest point projection from the image of F 1 to the image of F 2 with respect to the coordinates on the domain D ⊂ T z0 C. Assume without loss of generality that T z0 C ⊂ R n+m coincides with R m × {0} and is parametrized by coordinates (p 1 , . . ., p m ) where z 0 coincides with (0, . . ., 0).Let {e 1 , . . ., e n+m } be an orthonormal frame for R n+m such that if p ∈ R m × {0}, then p = p 1 e 1 + • • • p m e m .Given p ∈ D and p := (p, u 1 (p)) in the image of F 1 , let q be defined such that (q, u 2 (q)) = p + V β (p)n β (p).That is to say, Another way to see this relation is that q − p is the projection of V (p) to T z0 C and u 2 (q) − u 1 (p) is the projection of V (p) to the orthogonal complement (T z0 C) ⊥ = {0} × R n ⊂ R n+m .This relation between p and q is a system of equations which we restate for clarity.
).Thus, given the vector field V (p) defined on the image of F 1 in a neighborhood of F 1 (x 0 ), the following derivatives are equivalent by the chain rule: where p is considered as an independent variable and We now differentiate (3.13) and (3.14) with respect to p. (3.15) Then substitute the equations (3.15) into the equations (3.16) to obtain the following relation.
).We aim to estimate the magnitude of the derivative ∂ i V (p) at p = (0, u 1 (0)) = x 0 .Remark 3.18.We have heretofore suppressed the time dependence of the functions u j (p, t) for j = 1, 2, defined in Lemma 2.8 and only considered the t = 1 timeslice denoted by u j (p) = u j (p, 1).In the following calculation, we will use the time dependence of these functions in conjunction with Lemma 2.8 to obtain some necessary estimates.
We begin by estimating the difference | By integration and Lemma 2.8, the first summand is bounded by Note that at time t = 0, u 1 (p, 0) = u 2 (p, 0) are equal to the graph of the cone C over T z0 C, and apply Corollary 2.10 to find that Noting again in particular that u 1 (p, 0) = u 2 (p, 0) and applying the same argument yields All in all, this tells us that at p = (0, ). Intuitively, the first summand is all we care about: we wish to estimate ∇ ⊥ V (x 0 ), and for sufficiently large R 4 , the normal frame {e α } is very close to the normal frame {n β }.Thus, the normal part of the total derivative should be captured almost entirely in the first term, and the remainder in the second term should be negligible.
We show this rigorously.First, define the normal frame {n β (p)} n β=1 by Note that each n β (p) is perpendicular to the vectors ∂ i F (p), which span the tangent space T pM 1,R4 .Furthermore, the collection is linearly independent and thus constitutes a frame of the normal space.The metric h on the fibers of N M 1,R4 can be expressed with respect to this frame by (3.22) By the linear independence of the frame, h αβ is invertible and its inverse is denoted h αβ .
Next, we expand the left-hand side of (3.20) via the product rule and estimate the summands.
..m are the tangent vectors to the embedded image F (D), and that the induced metric is given by Note that at the origin, the first derivatives ∂ i u 1 (0, 0) = 0, since T z0 C is tangent to the graph of the cone at z 0 .By integration (c.f.Corollary 2.10) ) at the origin.Similarly, the normal bundle metric h αβ = n α • n β can be written as The terms involving ∂ i n β can be estimated using Lemma 2.8.We calculate Since for any γ = 1, . . ., n, we can estimate Recall that at the point x 0 , the norm . Thus, the terms in (3.23) and (3.24) involving products of Thus, we can further simplify 3.20 to the following form.
αβ is invertible with close to unit operator norm.Thus, we obtain that Which implies that when p = (0, u 1 (0)), This concludes the proof of the i = 1 case.To prove the Lemma for i = 2, we begin by differentiating the system given in (3.13) and (3.14) twice.This yields the following system: . ., n. Substituting (3.27) into (3.28) and rearranging terms, we obtain Next, use (3.15) to expand ∂ j q l (p)∂ i q k (p).
where the last line comes from the proof of the i = 1 case.Since Evaluating at x 0 the right hand side of (3.29) becomes The difference in the above expression can be estimated using precisely the same application of Lemma 2.8 and Corollary 2.10 used to estimate (3.19) in the proof of the i = 1 case.Thus, the equation (3.29) becomes (3.31) where we estimate ) by the proof of the i = 1 case, and use the bounds on |∂ j n β (x 0 )| derived in the proof of the i = 1 case.Next, we bound ).Thus, equation (3.31) can be further simplified to As in the i = 1 case, the left-hand side can be inverted at x 0 to obtain This completes the proof of the lemma.
Lemma 3.34.Let σ be a section of the normal bundle N M 1,R4 .Define a linear operator acting on such sections σ at a point x ∈ M 1,R4 where ∆ ⊥ and ∇ ⊥ are the Laplacian and covariant derivative on the normal bundle respectively.Let V be the section of N M 1,R4 given in Lemma 3.8.There exists C 4 > 0 such that at any x 0 ∈ M 1,R4 , the following equation is satisfied.
where F 1 (•) T is a vector field of T M 1,R4 and the function Q satisfies the inequality Proof.Fix a point x 0 ∈ M 1,R4 .We recall from Lemma 3.11 the parametrization ) is its image in R n+m (up to affine transformations).
If we use the exponential map to obtain a normal coordinate system at x 0 , we can pull back these coordinates to D and relabel (p 1 , . . ., p n ) so that We also recall the frame {n β } n β=1 of the normal bundle N M 1,R4 given by (3.21).In addition to this frame, we will at times use for convenience an orthonormal geodesic frame of the normal bundle {N β } n β=1 in a neighborhood of x 0 such that ∇ ⊥ N β (x 0 ) = 0 for all β.
Let V = V β n β be the section of N M 1,R4 given in Lemma 3.8.Since the shrinker F 2 can be written as a graph over F 1 outside of B R4 ⊂ R n+m , we have a parametrization of F 2 near the point y 0 = x 0 + V β (x 0 )n β (x 0 ) given by We differentiate in the coordinates (x k ) m k=1 on Ω ⊂ R m to find the tangent vectors to the shrinker F 2 . (3.37) where β = 1, . . ., n.The second fundamental form of F at 0 is the normal component of ∂ 2 ij F .To this end, we determine the values of In the last line, we used the asymptotics from Lemmas 3.11 and 2.6 to conclude that We now calculate the metric tensor with respect to the coordinate chart F .Here, we use the geodesic normal frame {N α } to achieve a simpler and more geometrically meaningful expression.Note that ∂ j N α = A α jl ∂ l F at x 0 by the vanishing of the Christoffel symbols.
We now calculate the inverse (g ij ) up to quadratic error.This reduces to inverting is a symmetric matrix by the symmetry of the second fundamental form.Let be the characteristic polynomial of S. The inverse (I + S) −1 has the following form: DV ) Throughout the above computation, we are moving terms into the quadratic error term using the decay estimates from Lemma 3.11.Note that the last line comes from observing the following bound on the normal part of Putting everything together, we find the normal component of Next we calculate the normal component of the position vector F .We use the normal frame {N α } for these computations.First find the tangential component Note that in the fourth equality, the fact that Since F is a self-shrinker, we can substitute (3.38) and (3.39) into the self-shrinker equation.
), so we can trace with respect to δ ij and place the remainder terms in Q(x 0 , V, DV ).
By the calculation (3.33) in Lemma 3.11, make the substitution Tracing and taking the normal part of our equation, we obtain ) and this term can thus be moved into Q(x 0 , V, DV ).Note also that Removing all tangential parts, we obtain We can rewrite this expression as where we are evaluating at x 0 .This concludes the proof of the lemma.
We now prove that the analogous equation holds for self-expanders.
Corollary 3.41.Suppose that F 1 and F 2 are self-expanders asymptotic to the same asymptotic cone C. All the previous conclusions of Sections 2 and 3 for selfshrinkers up to but not including Lemma 3.34 apply to expanders F 1 and F 2 .Let σ be a section of the normal bundle N M 1,R4 .Define a linear operator acting on such sections σ at a point x ∈ M 1,R4 where ∆ ⊥ and ∇ ⊥ are the Laplacian and covariant derivative on the normal bundle respectively.Let V be the section of N M 1,R4 given in Lemma 3.8.There exists C 4 > 0 such that at any x 0 ∈ M 1,R4 , the following equation is satisfied.
where F 1 (•) T is a vector field of T M 1,R4 and the function Q satisfies the inequality Proof.To see that the properties for shrinkers proved in Sections 2 and 3 apply also to expanders, notice that all the estimates in Section 2 depend only on the relations |H| ≃ |F ⊥ | and |(∂ t F ) ⊥ | = |H|, which apply equally to shrinkers and expanders.
To prove the rest of the corollary, we follow the proof of Lemma 3.34 until (3.40) which is the first time the self-shrinker equation is explicitly used.At this step, we instead plug in the self-expander equation: Arguing as in Lemma 3.34, obtain which can be rewritten as evaluated at x 0 .This completes the proof of the corollary.

Unique Continuation on Weakly Conical Ends
In the second half of the paper, we prove a unique continuation result for higher codimension self-shrinking ends asymptotic to a cone.We do this by extending to the vector-valued case a recent result [4] of Jacob Bernstein on the asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends.
A weakly conical end is a triple (Σ, g, r) consisting of a smooth m-dimensional manifold, Σ, with m ≥ 2, a C 1 -Riemannian metric, g and a proper unbounded C 2 function r : Σ → (R Σ , ∞) where R Σ ≥ 1, and such that there is a constant Λ ≥ 0 with the property that (4.1) We consider the tuple (B, p, Σ, h, ∇), which represents a vector bundle B over Σ with metric h, compatible metric connection ∇, and projection p.We furthermore assume that the connection ∇ satisfies the following condition.
In particular, Condition 4.2 is satisfied by the curvature R ⊥ of the connection ∇ ⊥ of the normal bundle on M 1,R (see Lemma 5.2).
We can define the drift Laplacian with respect to ∇ by Our generalization of [4,Theorem 1.2] to almost eigensections of vector bundles over weakly conical ends is the following: Theorem 4.4.If (Σ m , g, r) is a weakly conical end, (B, p, Σ, h, ∇) is a vector bundle over Σ with a metric connection ∇ satisfying Condition 4.2, and V ∈ C 2 (Σ; B) is a section that satisfies then there are constants R 0 and K 0 , depending on V , so that for any Moreover, V is asymptotically homogeneous of degree 2λ and tr 2λ ∞ V = a for some section a ∈ L 2 (L(Σ); B| L(Σ) ) that satisfies Here A ∈ L 2 loc (Σ; B) is the leading term of V and L(Σ) is the link of the asymptotic cone.
The proof of Theorem 4.4 is almost identical to the proof of Theorem 1.2 in [4].However, it is necessary to prove several preliminary estimates and lemmas in the vector-valued case before Bernstein's arguments can be directly applied.We introduce some of the basic notation and terminology from [4], prove the missing results, and indicate their scalar-valued analogues in the original paper.After substituting these modified results, one can follow Bernstein's proofs directly.Note that if a proposition or proof is not included in the sequel, we implicitly assume that it can be extended to the vector-valued case essentially without modification.4.1.Basic Definitions and Estimates.We now introduce some notation and facts about weakly conical ends from [4].The first condition in (4.1) ensures that r has no critical points for sufficiently large ρ ∈ (R Σ , ∞) and r is proper.Thus the level sets S ρ = r −1 (ρ) are compact C 2 -regular hypersurfaces foliating Σ.Let L(Σ) = S RL = S RΣ+1 be the link of Σ.Notice that the bound on ∇ g r guarantees that L(Σ) has the same topological type as any other S ρ .
For any Define three important C 1 vector fields on Σ: We state the following identities without proof.Derivations can be found in Section 2 of [4].The divergences of these three vector fields are The asymptotics of the Hessian are If A Sρ and H Sρ are the second fundamental form and mean curvature of S ρ , then (4.9) The restriction Π τ to the link L(Σ) is a diffeomorphism π τ : L(Σ) → S τ RL .Let g(τ ) be the C 1 -Riemannian metric induced by g on S τ .Take the limit C 0 -metric on L(Σ), ).This limit metric on the link extends to a C 0 cone metric on Σ.Let The C 0 -Riemannian cone (Σ, g C ) with link (L(Σ), g L ) is the asymptotic cone of (Σ, g, r).Let dµ C , dµ gτ , dµ g be the densities associated to the metrics g C , g τ , g respectively.For any compact set K ⊂ Σ there is a τ 0 = τ 0 (K) so that for any W ∈ C 1 (K; B) and any τ ≥ τ 0 , (4.10) We define the pullback of a section of a vector bundle as follows.Let G ∈ L 2 loc (Σ; B; dµ g ) be a section, and also denote by G a choice of pointwise a.e.representative of the equivalence class [G].Then, we define the pullback of G at the point p by the flow Π τ as follows: , where P c,t is defined as the parallel transport from the fiber of B above c(t) to the fiber above c(0) along the path c : (a, b) → Σ.Since Π t (p) is a regular path for fixed p, this operation is well defined.
A section of the bundle B restricted to the link L(Σ).We call this restriction the degree d trace of F .
where F is a homogeneous section of degree d.The section F on the limit cone is called the leading term of G.The trace at infinity of G is defined to be the degree d trace of the leading term F .
4.2.Extension to the Case of Almost L µ -harmonic Sections.Let (Σ m , g, r) be a weakly conical end of dimension m, and let (B, p, Σ, h, ∇) be a vector bundle of rank n over Σ with projection map p, bundle metric h, and metric connection ∇ satisfying Condition 4.2.Define the weight function As in [4], we will write Φ µ (p) instead of Φ µ (r(p)).The associated drift Laplacian on sections V ∈ Γ(B) will be where ∆ is the trace with respect to g of the double covariant derivative ∇ 2 on the vector bundle B. Almost L µ -harmonic sections are sections that satisfy where V and ∇V are contracted with respect to the tensors h αβ and h αβ g ij , respectively.As is standard, coordinates on the base Σ will be given by Latin indices, and coordinates in the fibers B x will be given by Greek indices.In the rest of the paper, we suppress the mention of specific metric tensors unless ambiguity arises.
As a preliminary, we must define appropriate function spaces for our analysis.For any R > R Σ , let C l ( ĒR ; B) be the space of l-times differentiable sections of B defined on E R .Consider the weighted L 2 norms on sections in C l ( ĒR ; B): Let C l µ ( ĒR ; B) and C l µ,1 ( ĒR ; B) be the space of l-times differentiable sections such that ||V || µ < ∞ and ||V || µ,1 < ∞ respectively.
Fix R > R Σ and a section V ∈ C 2 ( ĒR ; B).For each ρ > R, define the boundary and define the flux The following quotient is the associated frequency function.
The corresponding frequency function is harmonic, we can apply integration by parts for tensor fields and obtain This yields the useful identities (4.12) Fµ (ρ) = Dµ (ρ) + Lµ (ρ) and We prove a Poincaré inequality analogous to [4, Proposition 3.1].

Proposition 4.13.
There is an Proof.Define a tangential vector field W(p) = r(p)|V (p)| 2 Φ µ (p)N(p).Take its divergence: By identity (4.5) the first summand can be rewritten as Expand the second summand Combining the two summands, we obtain We use Cauchy-Schwartz for h and the absorbing inequality to estimate Combining terms, we obtain For R > R P sufficiently large, the r 2 term will dominate, and the following inequality holds As V ∈ C 2 µ,1 ( ĒR ), the proposition follows immediately from the divergence theorem.
Next, we derive the formula for the derivative of the boundary L 2 norm B(ρ), which should be compared to [4, Lemma 3.2].Lemma 4.14.We have and Proof.Calculate the first variation with respect to the vector field r We use (4.9) in the fourth equality and (4.
Proof.By the coarea formula, Differentiating once and applying (4.5) yields We analyze the last summand.By metric compatibility with covariant differentiation, We derive a weighted Rellich-Nečas identity for vector bundles to apply the divergence theorem and extract an L µ term from the Hessian term.We take the divergence of the 1-form We identify the L µ V term: To evaluate the final term, we calculate at a point p and assume that (x i ) n i=1 are normal coordinates centered at p, and we have a local geodesic frame {e α } k α=1 of the fiber space, such that ∇ ∂i e β (p) = 0, for all β = 1, . . ., n and i = 1, . . .m.Then, at the point p, we find that where the second to last equality comes from equation (4.8).The Rellich-Nečas identity is Before we can plus this identity into our original expression, we must commute X to the first position in the Hessian ∇ 2 V .We obtain We assume that R ∇ satisfies Condition 4.2 and obtain Plug this into our original expression and apply the divergence theorem.
Estimate the second integral in (4.16) using the Cauchy-Schwarz inequality and Proposition 4.13.
We plug this bound into (4.16) and obtain µ (ρ).Exactly as in [4], one establishes using the coarea formula and Fubini's theorem that Corollary 4.17.
Proof.The proof proceeds exactly as in [4], except that we obtain the last term by noticing that ρ Bµ (ρ) The modified statements in Proposition 4.15 and Corollary 4.17 are used in four places in the arguments on pp.9-15 [4] to prove the central frequency decay estimates: the proofs of [4,Proposition 4.4], [4,Proposition 4.5], [4,Lemma 5.3], and [4,Proposition 5.4].The extra terms affect the proofs only very slightly, but we will recall some of these results and pinpoint exactly how to absorb these terms.Proof.When Corollary 4.17 (resp.[4,Corollary 4.3]) is applied to bound N ′ m (ρ) near the end of the proof, notice that by (4.19) . Thus, it may be absorbed into the term 2  10 Γρ −2+2γ .The proof for [4,Proposition 5.4] can be continued without further modifications to prove the statement of the Proposition.
With these results in hand, one can directly apply the arguments on pp.14-15 [4] to prove the crucial frequency decay estimates in [4, Theorem 4.1] and [4, Theorem 5.1], which we summarize here: µ+2,1 ( ĒR ; B) satisfies (4.9) and is non-trivial, then In particular, there is a ρ −1 ≥ R so that for ρ ≥ ρ −1 and ξ In order to prove the analogue of [4, Theorem 6.1] for vector bundles, it remains to prove a vector-valued version of [4, Proof.Let G τ = Π * τ G, and take the covariant derivative at p with respect to the velocity of the flow line Π τ (p), parametrized by τ .Since the flow Π τ is the log τ flow of the vector field X, we have the following equality.
We show that covariant differentiation by 1 τ X commutes with pullback via the flow diffeomorphism Π τ .
The following application of Cauchy-Schwartz and Fubini's theorem establish that the sequence of pullbacks G R is Cauchy in Π * s |∇ X G| 2 dµ C ds Note that in the last inequality, we used the fact that P Πt(p),τ is an isometry from B Πτ (p) to B p .By (4.8), there exists τ 0 (K) such that for s ≥ R ≥ τ 0 (K), dµ C ≤ 2Π * s (r −m dµ g ).Thus, Using the bounds on |∇ g r| and the hypotheses of the proposition, for large ), and has a unique limit F K ∈ L 2 (K; B; dµ C ).Since we can find this limit for the annuli A ρ2,ρ1 which form a compact exhaustion of Σ, this shows that there is a limit F ∈ L 2 loc (Σ; B; dµ C ) such that lim We show that F is homogeneous of degree zero.For any τ ≥ 1, G is asymptotically homogeneous of degree 0. Now that we have established the preceding results, the proof of [4, Theorem 6.1] with the appropriate modifications suffices to prove the following theorem.Theorem 4.24.If V ∈ C 2 µ+2,1 (Σ; B) is a section that satisfies (4.11) then there are constants R 0 and K 0 , depending on V , so that for any Moreover, V is asymptotically homogeneous of degree 0 and tr 0 ∞ V = a for some section a ∈ L 2 (L(Σ); B| L(Σ) ) that satisfies Here A ∈ L 2 loc (Σ) is the leading term of V and L(Σ) is the link of the asymptotic cone.
In order to finish the proof of Theorem 4.4, we prove that almost eigensections of L µ corresponding to an eigenvalue λ can be transformed into almost eigensections corresponding to a different eigenvalue λ + ν of another operator in the family, L µ−4ν .In particular, almost eigensections of L 0 with negative eigenvalues −λ < 0 can be transformed into almost harmonic sections of the operator L 4λ .
Before stating the proposition, we note that when we apply the operator L µ to sections of different bundles, we will assume that we are taking all covariant derivatives with respect to the particular section to which we are applying it.In particular, when we apply L µ to scalar functions, we will assume that we are applying it to sections of the trivial bundle Σ × R endowed with the standard product metric.

Proposition 4.25. There is a constant M
is an asymptotically conical end with associated constant Λ, (B, p, Σ, h, ∇) is a vector bundle of rank n over Σ with metric h, and V ∈ C 2 (Σ; B) satisfies Proof.We calculate directly, for arbitrary µ ′ ∈ Z Note that a simple way to evaluate the trace tr(dr ⊗ ∇V ) is to calculate in a coordinate system (r, x) around p ∈ Σ, where x is a local coordinate system on S r(p) .Finally, we observe that Plug this into the previous calculation to obtain Finally, observe that ∇ V = r 2ν ∇V + 2νr 2ν−1 dr ⊗ V , and so This concludes the proof of the proposition.
Proof of Theorem 4.4.Theorem 4.4 is an immediate consequence of Theorem 4.24 and Proposition 4.25.

Unique Continuation on Self-Similar Ends
We begin by proving that the ends of asymptotically conical high codimension self-shrinkers are weakly conical ends.We follow the method of proof in [4, Lemma 8.1] almost exactly, with a few modifications to extend the proof to the case of high codimension.
Lemma 5.1.Let F : M m → R n+m be an asymptotically conical self-shrinker or self-expander.There is a radius R F so that if g is the metric pulled back from R n+m by F to M RF and r(p The self-shrinker/self-expander equation implies that there is C > C 1 such that Note that we use the self-shrinker/self-expander equation in the second to last inequality.Next, we confirm the estimate for the Hessian ∇ 2 g r 2 .Observe that ∇r 2 = 2r∇r = 2F T .Let X, Y be two tangential vector fields with respect to the shrinker/expander F . We conclude that that (M RF , g, r) satisfies all the properties of a weakly conical end.
Proof.Consider a point x 0 on M 1,R4 and let z 0 be the nearest point to x 0 on the cone C. By Lemma 2.8, a neighborhood of x 0 in M 1,R4 may be represented as the the graph of a function u(x, 1) defined on a disc D m ρ centered at z 0 in the tangent space T z0 C. Using the notation of Lemma 2.8, the function u(x, 0) represents a neighborhood of of z 0 in C on the same disc D m ρ .By Corollary 2.10, the differences |D i u(z 0 , 1) − D i u(z 0 , 0)| = O(|F 1 (x 0 )| −i−1 ).In particular, the equivalence of the Hessian of u and the second fundamental form (c.f.Lemma A.1) implies that Lemma 5.6.If a self-shrinking end M 2,K can be represented as a normal section V over a self-shrinking end M 1,R4 which satisfies both the decay estimates (3.12) and then V is identically 0 and F 1 and F 2 coincide.
Proof.By Lemma 5.1, F 1 is a weakly conical end and V is an almost eigensection of L 0 with eigenvalue 1/2.We may apply Theorem 4.4 to the almost eigensection V and obtain that V is asymptotically homogeneous of degree 2λ = 1, that tr 1 ∞ V = a ∈ L 2 (L(Σ); B| L(Σ) ), and that (5.7) We claim that the leading term A of V is equal to 0. We test this by taking the The first inequality comes from the bound on V in Lemma 3.11.Thus, the leading term A is equal to 0. This implies that the trace at infinity of V vanishes as well.
a = tr 1 ∞ (V ) = tr 1 (A) = 0. Plugging into inequality (5.7) implies that V is uniformly equal to 0 in the annular region ĒR0 .That F 1 and F 2 coincide in their region of definition follows from the analyticity of self-shrinkers.
Corollary 5.8.If a self-expanding end M 2,K can be represented as a normal section V over a self-expanding end M 1,R4 which satisfies the estimates (3.12), (1.3), and then V is identically 0 and F 1 and F 2 coincide.
By (1.3) and the weakly conical end property, we calculate This implies that ||â|| L 2 = 0, and thus that V = V = 0 on E R ′ 0 .The analyticity of self-expanders allows us to say that F 1 and F 2 coincide where defined.
Proof of Theorem 1.1.By Corollary 3.9, a connected AC shrinker end with multiplicity may be written as a section V of its own normal bundle satisfying the decay estimates (3.12) and the hypotheses of Theorem 4.4.By Lemma 5.6, the section V is identically 0 and the k sheets of the k-fold covering space coincide.Applying this argument to each connected AC end implies that the image of a shrinker F asymptotic to a cone C may be written as a single valued normal section over the cone C outside of a ball.This gives a reparametrization of F (M ) \ B R4 with multiplicity 1.
Let F 1 and F 2 be two shrinkers asymptotic to C which have been reparametrized to have multiplicity 1 over C outside some ball B R .They are homeomorphic to C R via the covering projections p 1 and p 2 .By Proposition 3.8, Lemma 3.11, and Lemma 3.34 outside some ball F 2 can be written as a section V over F 1 satisfying the decay estimates (3.12) and the hypotheses of Theorem 4.4.By Lemma 5.6, V ≡ 0 so F 1 and F 2 coincide outside the ball B R .The theorem follows from analytic unique continuation inside the ball.
Proof of Theorem 1.2.Corollary 3.9 cannot be used to reduce the multiplicity of self-expanders, because the separation between the sheets of a single expander does not necessarily satisfy condition (1.3).
For the moment, let F 1 and F 2 refer to the restrictions of these expanding immersions to one of their connected AC ends.We consider the covers F1 and F2 corresponding to the subgroup G = p 1 * (π 1 (M 1,R3 , x 1 )) ∩ p 2 * (π 1 (M 2,R3 , x 2 )), as in Remark 3.6.Proposition 3.8, Lemma 3.11, and Corollary 3.41 suffice to establish that in some annular region, M2,K can be written as a normal section V over M1,R4 which satisfies the hypotheses of Corollary 5.8 for expanders satisfying (1.3).Therefore, V vanishes and the images of F1 and F2 coincide.Recall that the induced immersion Fi is F i • P i the composition of the original immersion F i : M i,K → R n+m and the covering projection P i : Mi,K → M i,K .Hence, the images of F 1 and F 2 coincide.Then apply this argument to each connected end to obtain the theorem.
Remark 5.10.If F 1 and F 2 are self-expanders satisfying the hypotheses of Theorem 1.2, then the coincidence of their images means they must converge to C with the same "true" multiplicity after a reparametrization.In fact, the conclusion of Theorem 1.2 implies that one of the ends M i,K must evenly cover the other-thus, the desired reparametrization can be found by taking quotients with respect to deck transformations.In contrast to case of self-shrinkers, there may be many selfexpanders which converge non-trivially to the cone C with different multiplicities.

7 )
in the final equality.This completes the proof.The second identity follows immediately.Now we compute the rate of change of the weighted Dirichlet energy Dµ (ρ), analogous to [4, Proposition 4.2].The need to commute second covariant derivatives creates an error term that does not appear in [4], but appears in the adaptation to Ricci expanders by Deruelle and Schulze [10, Proposition 5.1].Proposition 4.15.
1, 2.Here, D and D 2 are the Euclidean gradient and Hessian on R m respectively, and ∂ t denotes the partial derivative with respect to t fixing points in R m .