{'fraction_non_alphanumeric': 0.06056517213462354, 'fraction_numerical': 0.024673271907166814, 'mean_word_length': 3.708658558722091, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Unique continuation properties for polyharmonic maps between Riemannian manifolds

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic case: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on a open subset, then they agree everywhere; (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.


Introduction and results
Harmonic maps are among the most studied geometric variational problems in differential geometry. The geometric setup is the following. We consider a map ϕ : M → N between two Riemannian manifolds (M m , g) and (N n , h). Then the energy of ϕ is defined by where {e j }, j = 1, . . . , m = dim M , is a local orthonormal frame field tangent to M and the second fundamental form ∇dϕ is defined by ∇ being the connection on the vector bundle ϕ −1 T N . Harmonic maps are precisely the solutions of equation (1.2). We observe that the harmonicity equation is a second order semilinear elliptic system. Since this system of equations is of second order, powerful tools such as the maximum principle help to obtain a deep understanding of both analytic and geometric properties of harmonic maps (we refer to the classical surveys of Eells and Lemaire [9,10] for an introduction and background on this topic). Another geometric variational problem which received growing attention in the recent years is that of the so-called biharmonic maps. These maps are characterized as the critical points of the bienergy for maps between two Riemannian manifolds, which is given by Here, the Euler-Lagrange equation is a fourth order semilinear elliptic system and is expressed by means of the vanishing of the bitension field τ 2 (ϕ), that is 0 = τ 2 (ϕ) :=∆τ (ϕ) + m j=1 R N (dϕ(e j ), τ (ϕ))dϕ(e j ) , (1.4) where∆ is the so-called rough Laplacian, i.e., the connection Laplacian on ϕ −1 T N . For background and research on biharmonic maps we refer to [26] and the recent book [27].
In contrast to the harmonic map equation, (1.4) is of fourth order, a fact which entails significant additional technical difficulties. For instance, classical tools such as the maximum principle are no longer applicable (for instance, see [21,22]). There exist different systematic approaches which generalize the notions of harmonic and biharmonic maps to energy functionals that contain derivatives of higher order.
In this paper we shall focus on the following k-order versions of the energy functional: if k = 2s, s ≥ 1, (1. 5) In the case that k = 2s + 1: A polyharmonic map of order k (briefly, a k-harmonic map) is a critical point of the k-energy functional E k (ϕ). The functionals (1.5), (1.6) are probably the simplest higher order version of the classical energy functional in a Riemannian geometric setting. These functionals were first studied systematically in an interesting series of papers by Maeta. Particularly, he established their main variational equations and properties, and proved some basic characterizations of proper (i.e. non-harmonic) triharmonic submanifolds into a sphere (see [14,15,16]). More recently, these k-energy functionals have been intensively studied. For instance, the stress-energy tensor for polyharmonic maps was recently calculated in [3]. Vanishing results for polyharmonic maps into Euclidean spaces have been obtained in [18,25] and for arbitrary targets in [4]. Moreover, proper k-harmonic immersions into spheres, ellipsoids and rotation hypersurfaces were constructed in [20,23,24]. Some other results on triharmonic maps were achieved in [17,19]. The Euler-Lagrange equations of (1.5), (1.6) were calculated in [14,16] and can be described as follows (note that we set∆ −1 = 0): (1) The critical points of (1.5) are those which satisfy 0 = τ 2s (ϕ) :=∆ 2s−1 τ (ϕ) − R N (∆ 2s−2 τ (ϕ), dϕ(e j ))dϕ(e j ) − s−1 ℓ=1 R N (∇ e j∆ s+ℓ−2 τ (ϕ),∆ s−ℓ−1 τ (ϕ))dϕ(e j ) − R N (∆ s+ℓ−2 τ (ϕ),∇ e j∆ s−ℓ−1 τ (ϕ))dϕ(e j ) . (1.7) (2) The critical points of (1. (1. 8) In this article we shall focus on one specific analytic aspect of solutions of the polyharmonic map equation, namely, the so-called unique continuation principle. We refer to the work of Kazdan [12] for an introduction to this topic in a geometric setting. In particular, Kazdan exhibited an artificial counterexample, but also stated that it is reasonable to expect that the unique continuation property should hold in all geometrically meaningful situations. Indeed, for harmonic maps unique continuation properties were proven by Sampson in [30] and recently generalized to biharmonic maps in [7]. In this paper, all manifolds are assumed to be connected and we shall work with smooth objects only. Our first result is the following:  The Euler-Lagrange system of equations which defines a k-harmonic map is elliptic of order 2k and has a rather complicated expression which depends on the Riemannian curvature tensor field of the target. Therefore, although all our proofs are based just on the application of the classical Aronszajn's unique continuation principle for second order elliptic operators, the technical steps that we shall have to carry out are quite demanding and require a delicate use of suitable new variables. For the sake of completeness, we point out that another interesting generalization of both harmonic and biharmonic maps can be obtained by studying the critical points of the following higher order energies: The study of these functionals was proposed by Eells and Sampson in 1965 (see [11]) and, later, by Eells and Lemaire in 1983 [9, p.77, Problem (8.7)]. A rigorous mathematical investigation of (1.9) has recently been initiated by the authors in [6] and was further developed in [5,20]. We point out that, in general, the functional (1.9) coincides with E k (ϕ) introduced above only when k = 1, 2, 3. In Section 3 we will show that Theorems 1.1 and 1.2 also hold for the critical points of (1.9) in the case k = 4, which is the only case where the Euler-Lagrange equations are explicitly computed (see [6]). However, since the technical difficulties are huge, we have preferred not to investigate other possible extensions of the unique continuation principle. Throughout this article we shall use the following sign conventions and notations. The Riemannian curvature tensor field on a manifold N is Y ] Z and, when the context is clear, we shall simply write R instead of R N . As for the rough Laplacian on ϕ −1 T N , we shall use∆ := − Tr g (∇∇ −∇ ∇ ). Similarly, the sign of the Laplace operator ∆ on functions is such that ∆f = −f ′′ on R.
In general, we will use the same symbol ·, · to indicate the Riemannian metrics on various vector bundles. We also note that on 0-forms, that is on sections, d = ∇. When the range is not explicitly specified we will use Latin indices i, j, k for indices on the domain ranging from 1 to m and Greek indices α, β, γ for indices on the target which take values between 1 and n. When the range of the indices is from 1 to q, for some positive integer q, we will often denote them by a, b, c. We will use the Einstein summation convention, i.e., we will sum over repeated indices in the diagonal position. Most of our computations will be carried out in local charts and our convention concerning the indices of the sectional curvature tensor field is (1.10)

Proof of the main results
We recall the following classical result due to Aronszajn [1, p.248]: Theorem 2.1. Let A be a second-order linear elliptic differential operator of class C ∞ defined on an open subset D of R m . Let u = (u 1 , . . . , u q ) be a function on D satisfying the inequality for some C > 0. If u = 0 in an open subset of D, then u = 0 throughout D.
Remark 2.2. In the literature, some unique continuation results for higher order elliptic equations are available (for instance, see [29]). However, the great generality of the elliptic operator A in (2.1) persuaded us that Theorem 2.1 is the most effective available tool to achieve unique continuation in the context of polyharmonic maps. Moreover, as pointed out in [30], also the strong version of the unique continuation principle holds, i.e., the conclusion of Theorem 2.1 is still true if u = 0 to infinite order at some point. Therefore, as in [30], both Theorems 1.1 and 1.2 admit a strong formulation and the proof is the same.
In order to prove our results we need to write down a suitable local expression for the equations for polyharmonic maps (1.7), (1.8).
To this end, let us choose a local chart (U, x i ) on M and a local chart (V, y α ) on N such that ϕ(U ) ⊂ V . To simplify the notation, we shall denote by ϕ the expression of ϕ in the two local charts.
It is well-known that, in local coordinates, the tension field is given by where Γ α ϑβ represent the Christoffel symbols of the manifold N . We also recall that the Laplace-Beltrami operator ∆ acts locally on a function f : U → R as follows:  and, for simplicity, ϕ β i = ∂ϕ β /∂x i .
Proof of Theorem 1.1. Since the proof will involve several rather technical arguments, we have preferred to start giving all the geometrical details in the special case that k = 3. We believe that this should help the reader to follow the various steps which will be necessary to handle the general case. Case k = 3. Using (1.8), we write (2.5) We fix the notation as follows: It is important to point out that A α is linear with respect to η ϑ and ξ ϑ i . All of this can be iterated once more and yields where now, of course, ∂u ϑ 1 ∂x i . Now, let us assume that τ 3 (ϕ) = 0. Using (2.5) and computing we find that this is locally equivalent to the following system of equations: where where {e α } is the canonical basis of R n and u 0 is thought of as an R n -valued function defined on U . The components of the 1-form v 0 give rise to an R mn -valued function defined on U : For simplicity, we keep the same notation for the 1-form and the R mn -valued function. Note that here the index i in v α 0i does not mean the derivative with respect to the variable x i (v α 0 does not even exist!). With this notation, (2.8) can be rewritten as follows: Moreover, for future use, we deduce from (2.6) and inspection of (2.9) that there exists C > 0 such that on D we have where D is an open subset of M such that its closure is compact and contained in U . Indeed, in (2.6) and (2.9) it is possible to bound from above by means of a constant any of the functions which appear as a multiplicative coefficient of u 0 , v 0 , u 1 and {∂u α 1 /∂x i } i,α . From this (2.10) can be obtained easily. We shall also need to estimate the Laplacian of the R mn -valued function v 0 . To this purpose, we perform a computation which gives the following output: Now, we call F 2 = F 2 α i the right hand side of (2.11). Thus, F 2 depends on u 0 , v 0 , {∂v α 0i /∂x j } i,j,α and {∂u α 1 /∂x i } i,α and it is linear in each of them. Then, similarly to (2.10), it is easy to deduce from (2.11) the following estimate on D: for a suitably large constant C > 0. Finally, we also define the function F 1 = F 1 α as follows: We note that F 1 depends on u 0 , v 0 and u 1 and it is linear in each of them. It follows that there exists C > 0 such that The next step is to consider u 0 , v 0 , {∂v α 0i /∂x j } i,j,α , u 1 and {∂u α 1 /∂x i } i,α not as vector-valued functions defined on U , but as a set of independent variables. More specifically, we define where r 1 = n + nm + nm 2 + n + nm and r 2 = n + nm + n. Thus, formally Remark 2.3. In the previous proof, we have worked with the partial derivatives {∂v α 0i /∂x j } i,j,α . This choice is the most suitable for the purposes of this paper and, particularly, for the extensions to the case k ≥ 4 which will be illustrated below. However, we point out that it is also possible to work with covariant derivatives Consequently, the basic estimates (2.10), (2.12) and (2.13) still hold, conceptually with the same proofs. In particular, the choice of working with covariant derivatives was adopted in [7].
Remark 2.4. We point out that in [7] the definition of F 2 = F 2 α i should have been given as Note that a different sign convention was used in [7]. Since the additional term is linear in [7] can be completed with minor changes.
Case k ≥ 4. First, we provide a short illustration of the case k = 4 because this step may help to understand the idea behind the introduction of a suitable set of recursively defined new variables, a fact which is a key point. We recall from (1.7) that, when k = 4, the 4-tension field is Now, let us assume that τ 4 (ϕ) = 0. First, we set again Now, using (2.16), we find that in our notation the assumption τ 4 (ϕ) = 0 is equivalent to ∂y α , where we have set: We use the following set of variables: Now, ∆u = F + terms linear in v 0 and its first derivatives for the second component of F and + terms linear in v 1 and its first derivatives for the fourth component as in (2.15). Using the same technique that we employed for the first 2 vector components of F in the case k = 3, now the first 2 vector components of F and, analogously, the vector components 3, 4, can be estimated to ensure the validity of (2.1). As for F 4 , the explicit analysis of (2.17) yields from which (2.1) follows easily. This ends the case k = 4. The general case can be handled similarly. Indeed, for any fixed value k ≥ 4, we have recursively defined functions for 0 ≤ i ≤ k − 2 and also A α k−1 is defined. Then we introduce the vector-valued function as follows: . ] is the right-hand side of (1.7) or (1.8) without the first term. Again, by construction and it is easy to see that the first 2(k − 2) vector components of F can be estimated in such a way that (2.1) holds. Finally, direct inspection of (1.7) or (1.8) (compare with (2.18)) shows that also F k can be estimated so that (2.1) is verified. So the proof ends by application of Theorem 2.1.
Proof of Theorem 1.2. Let ϕ,φ be two k-harmonic maps which coincide on an open subset. To simplify the notation, we shall also denote by ϕ,φ the vector-valued functions which represent ϕ,φ in local charts: We define a vector-valued function u for the map ϕ as follows where u 0 , . . . , u k−2 and v 0 , . . . , v k−3 are defined as in the proof of Theorem 1.1. We also need to introduce the analogous vector-valued function associated withφ, i.e., with u,ũ defined in (2.21) and (2.22) respectively. As in the proof of Theorem 1.1, the following functions are also defined: Next, we define G as follows: Now, by construction, we have Note that there are three different blocks in the definition of G. The first two rows only contain the Laplacian applied to ϕ and its first partial derivatives, and we will explain in more detail below how one should think of dϕ in the the definition of G. After that, we always have pairs of u j − A j and its first derivatives, with 1 ≤ j ≤ k − 2. In the last row we have the right hand side of the polyharmonic map equation (1.7), (1.8), denoted by F k . Note that this is the only place in (2.23) where the Euler-Lagrange equation for polyharmonic maps enters. Our aim is now to apply the theorem of Aronszajn, that is Theorem 2.1, to z. Hence, we have to estimate ∆z (equivalently, G) in terms of z and its first partial derivatives. In the following C will always represent a positive constant whose value may change from line to line. To estimate the first row in (2.23) we use the following Lemma 2.5. Let ϕ,φ be two maps with corresponding variables (2.21) and (2.22). Then the following estimate holds (2.24) Proof. Recall that where, here and below, we have shortened the notation denoting . The first three terms on the right hand side can be estimated directly. To estimate the difference of the Christoffel symbols we make use of the mean-value inequality, for more details we refer to the discussion before Lemma 2.6 in [7]. The proof is now complete.
In the following we will often apply the mean-value inequality without explicitly mentioning it. As a second step, we estimate the second line of (2.23).
Lemma 2.6. Let ϕ,φ be two maps with corresponding variables (2.21) and (2.22). Then the following estimate holds Proof. Recall that we use dϕ, dφ in (2.23) to represent the partial derivatives of ϕ andφ. Hence, when we apply the Laplacian to the second line in (2.23) we will get some correction terms as already computed in (2.11), see also Remark 2.3. Now, let us consider our two maps ϕ,φ: by combining (2.11) and using that ∆ϕ where in the last line, for clarity, we have also added the linear terms. In order to estimate the second term on the right hand side we rewrite We deduce the estimate The claim now follows by combining the equations.
In the following two lemmata we will estimate the pairs u j+1 − A j+1 and their derivatives which are in the middle block of (2.23).
Lemma 2.7. Let ϕ,φ be two maps with corresponding variables (2.21) and (2.22). Assume that 0 ≤ j ≤ k − 3. Then the following estimate holds Proof. We know that (this follows from (2.6) and using ∆ϕ α = −u α 0 + dϕ β , dϕ γ Γ α βγ ) (2.4). Hence, for two maps ϕ,φ we get . This gives the estimate . Hence, we find the estimate |u ν j dϕ ϑ ,dϕ σ C α ϑσν (ϕ) −ũ ν j dφ ϑ , dφ σ C α ϑσν (φ)| ≤ C(|ϕ −φ| + |dϕ − dφ| + |u j −ũ j |). This completes the proof. Lemma 2.8. Let ϕ,φ be two maps with corresponding variables (2.21) and (2.22). Assume that 0 ≤ j ≤ k − 3. Then the following estimate holds Proof. By a direct calculation we find Again, we have to be careful when applying the Laplacian to v j = ∇u j as the Laplacian does not commute with partial derivatives and we get several extra terms as demonstrated in (2.11). However, all these terms on the right hand side can be easily estimated in terms of v r , ∇v r . The statement of the lemma can now be derived as in the previous lemmata.
Finally, we estimate the contribution in (2.23) originating from the polyharmonic map equation. We only consider the case of polyharmonic maps of even order with 2s = k ≥ 4 as the odd case follows by exactly the same arguments.
Lemma 2.9. Let ϕ : M → N be a polyharmonic map of even order with corresponding variables (2.21). Then where E α βδϑη := R α βγδ Γ γ ϑη +R α βγϑ Γ γ δη . Proof. This follows directly from the Euler-Lagrange equation (1.7) using Lemma 2.10. Suppose ϕ,φ are two polyharmonic maps with corresponding variables (2.21) and (2.22). Then the following estimate holds Proof. Suppose we have two polyharmonic maps ϕ,φ of order k = 2s. Then, from (2.28), we get In order to estimate the first term on the right hand side we use (2.26). It is straightforward to estimate , which controls the second term on the right hand side. Concerning the first term inside the sum we rewrite Hence, we deduce the estimate . Regarding the second term in the sum we get and obtain the estimate The last term in the sum may be rewritten as which leads us to the estimate The claim now follows by also using (2.26), where now we have to write |∇u k−2 −∇ũ k−2 | instead of |v k−2 −ṽ k−2 | since these last functions do not appear in the definition of z.
Combining the inequalities (2.24), (2.25), (2.26), (2.27) and (2.29) we arrive at where we use a "˜" to indicate objects on S n−1 . The equator S n−1 is given by In the special case that N = S n we know that: Next, using (2.32), we express R α δβγ with respect to our local coordinates (ỹ, s). For our purposes we only need to compute explicitly R n δβγ . Using (1.10) and (2.32), we find that the only non-zero terms of this type are: Now, let (U, x i ) be a local chart on M and denote the domain of the above local coordinates on S n \ {N, S} by V . In addition, we assume that ϕ(U ) ⊂ V . For simplicity, we again denote the expression of ϕ in local coordinates also by ϕ, i.e. ϕ = (ϕ 1 , . . . , ϕ n ).

Assume that W is an open subset of
Clearly, the function f vanishes when restricted to W . Let D be an open subset of U such that its closure in M is compact and included in U , and W ⊂ D ⊂ U . We define a vector-valued function y for the n-th component of the map ϕ as follows where u n 0 , . . . , u n k−2 and v n 0 , . . . , v n k−3 are the n-components of the variables u 0 , . . . , u k−2 and v 0 , . . . , v k−3 defined as in the proof of Theorem 1.1. In addition, we define (2.35) Again, by construction ∆y = F + terms linear in v n j , ∇v n j , j = 0, . . . , k − 3.
In the following we will give the proof of Theorem 1.3. We will only consider the case of a polyharmonic map of even order as the odd case can be treated by exactly the same methods.
This proves the third estimate of the lemma. The estimate on ∆(df ) can be achieved by exactly the same methods.
Lemma 2.12. Let ϕ : M → S n be a map with corresponding variables (2.34). Assume that 0 ≤ j ≤ k − 3. Then the following estimate holds |dA n j+1 | ≤ C(|f | + |df | + |∇df |) . (2.38) Proof. By a direct calculation we find All the terms in this expression can be estimated by expanding the Christoffel symbols and using the same strategy as in the proof of Lemma 2.11 except the last term which requires a more careful inspection. Again, a direct calculation yields We realize that all terms in u ν j dϕ ϑ , dϕ σ ∂C n ϑσν ∂y δ ϕ δ i can be estimated by the same reasoning used before, only the contribution that is proportional to the second derivative of the Christoffel symbols needs to be treated in more detail. Hence, let us have a closer look at Using (2.31) we obtain ab ∂y c ∂y d and we can conclude that all terms can be estimated in such a way that the statement of the lemma holds true.
Lemma 2.13. Let ϕ : M → S n be a polyharmonic map of even order with corresponding variables (2.34). Then the following estimate holds where E n βδϑη :=R n βγδ Γ γ ϑη + R n βγϑ Γ γ δη .
First, we use the third inequality in (2.36) to estimate A n k−1 . It remains to estimate the terms involving the curvature tensor. Inserting the non-zero components of the curvature tensor given in (2.33) we find u δ k−2 dϕ γ , dϕ β R n βγδ = u δ k−2 dϕ n , dϕ β R n βnδ + u n k−2 dϕ γ , dϕ β R n βγn . Hence, we may estimate |u δ k−2 dϕ γ , dϕ β R n βγδ | ≤ C |df | + |u n k−2 | . By the same reasoning we find  (2.40) To prove this claim, we observe that f = ϕ n − π/2, and consequently df , vanish because ϕ maps W into the equator. Next, using ∆f = 0 on W and the explicit expression (2.31) of the Christoffel symbols we deduce that u n 0 = 0. The functions A n j , j = 1, . . . , k − 2 also vanish identically on W . This follows easily from the definition (2.37), using again the explicit expression (2.31) of the Christoffel symbols together with dϕ n = 0. Finally, from these facts it is easy to deduce that all the components of y vanish on W and so the claim (2.40) holds. Next, using the inequalities (2.36), (2.38) and (2.39) we find Now, because of (2.40) and the estimate (2.41), the assumptions of Theorem 2.1 are satisfied and we can conclude that y = 0 on D, which in particular implies that ϕ n = π 2 , i.e. ϕ maps the whole of D into S n−1 . We finish the proof of Theorem 1.3 by setting A := {p ∈ M : ϕ(p) ∈ S n−1 } and using the same globalization argument as above.
Remark 2.14. Theorems 1.1, 1.2 and 1.3 have some important applications. For instance, these results can be used to simplify proofs and obtain uniqueness and reduction results for k-harmonic submanifolds in spheres. For simplicity, we shall illustrate these facts in the case that k = 2: (a) A result of J.H. Chen (see [8]) says that a compact proper biharmonic hypersurface M m in S m+1 with |A| 2 ≤ m is CMC (and thus |A| 2 = m). The proof (in Chinese) is long and skillful, but it can be simplified using Theorem 1.1. This simplified proof was given in [2,26]. (b) All CMC proper biharmonic immersions from R 2 in S m are given in [13].

Unique continuation theorems for ES-4-harmonic maps
In this section we study unique continuation properties for critical points of the ES-4-energy (1.9). We shall prove two unique continuation results for k = 4, which is the only case for which we know the explicit form of the Euler-Lagrange equations (see [6]). The energy functional for ES-4-harmonic maps (corresponding to (1.9) with k = 4) is given by Note that here and in the sequel we shall omit to write the symbol when it is clear from the context. The first variation of (3.1) was calculated in [6, Section 3] and is characterized by the vanishing of the ES-4-tension field τ ES 4 (ϕ) given by the following expression τ ES 4 (ϕ) = τ 4 (ϕ) +τ 4 (ϕ).
At the end all addends in (3.5) have a tensorial character and replacing the geodesic frame field {e i } by the local coordinates frame field {∂/∂x i } they can be estimated by u α 0 , ∂u α 0 /∂x i and u α 1 . At this point we have realized that all terms on the right hand side of (3.3) can be expressed in terms of the required variables.
Proof of Theorem 3.1 and Theorem 3.2. In order to prove Theorem 3.1 we define the vectorvalued function and by the same analysis as in the proof of Theorem 1.1 we show using Aronszajn's Theorem that u = 0. In order to prove Theorem 3.2 we define the vector-valued function andũ will be defined accordingly. Employing the same strategy as in the proof of Theorem 1.2, now it is easy to complete the proof of Theorem 3.2.
string theory". The second and the last author were supported by Fondazione di Sardegna (project STAGE) and Regione Autonoma della Sardegna (Project KASBA).