Unique continuation theorems for biharmonic maps

Abstract We prove several unique continuation results for biharmonic maps between Riemannian manifolds.


Introduction and results
Finding interesting maps between Riemannian manifolds is one of the most challenging problems in modern Riemannian geometry. Suppose that (M, g) and (N, h) are two Riemannian manifolds. Moreover, let φ : M → N be a smooth map. One option of finding such maps is to find extrema of their energy which are called harmonic maps and are characterized by the vanishing of the so-called tension field, that is 0 = τ (φ) := Tr g ∇dφ, (1.2) where τ (φ) ∈ Γ(φ * T N). Many results on harmonic maps have been obtained in the past decades, we refer to [5] for an overview. Currently, there is a growing interest in a geometric variational problem that generalizes harmonic maps, the so-called biharmonic maps, which were first studied in [6]. In this case, one looks for critical points of the bienergy of a map φ : M → N , which is defined as The critical points of (1.3) are characterized by the vanishing of the so-called bitension field, that is R N (dφ(e i ), τ(φ))dφ(e i ), (1.4) where {e i }, i = 1, . . . , m = dim M is a local orthonormal frame field tangent to M and Δ represents the Laplacian on φ * T N. For recent results on biharmonic submanifolds we refer to [12], see also the older survey [9]. The harmonic map equation (1.2) is a second-order elliptic partial differential equation, whereas the biharmonic map equation (1.4) is of fourth order, which makes it substantially harder to characterize the qualitative behaviour of its solutions. In this article, we want to focus on one particular aspect regarding the qualitative behaviour of harmonic and biharmonic maps, namely the unique continuation property. For harmonic maps, the question of unique continuation was settled by Sampson [15,Theorem 1], see also [17,Section 1.4.2]. More precisely, the following result was proved: Theorem 1.1 (Sampson). Let φ 1 , φ 2 : M → N be two harmonic maps. If they agree on an open subset, then they are identical; and indeed the conclusion holds if φ 1 and φ 2 agree to infinitely high order at some point. In particular, a harmonic map which is constant on an open subset is a constant map.
In addition, Sampson established the following geometric unique continuation property of harmonic maps (see [15,Theorem 6]): There is a strong belief that the unique continuation property holds for all solutions of secondorder elliptic partial differential equations arising in geometry [7]. On the other hand, one can construct explicit counterexamples of solutions of fourth-order elliptic partial differential equations where the unique continuation property does not hold [7,Example 1.11].
In this article, we will prove the following results for biharmonic maps:

biharmonic map. If φ is harmonic on an open subset, then it is harmonic everywhere.
Since there is a big interest in biharmonic maps to spheres we will first establish a unique continuation result for spherical targets as also the proof is considerably simpler here. Afterwards, we will prove a unique continuation theorem for biharmonic maps to an arbitrary target manifold. Finally, we will also provide a geometric unique continuation property for biharmonic maps generalizing Sampson's result for harmonic maps [15,Theorem 6]. First, we will give the following version for a spherical target: In addition, we are also able to prove a corresponding version of the above theorem for an arbitrary target.  [8,Theorem 4,Proposition 3] for the particular case of CMC biharmonic immersions into spheres. In the given reference, the following idea was employed: A CMC biharmonic immersion into S n composed with the canonical inclusion of the sphere in the ambient Euclidean space R n+1 gives an immersion that can be written as a sum of two R n+1 -valued eigenmaps of the Laplace operator on (M m , g). These maps induce harmonic maps into S n of appropriate radius. Then, in contrast to the article at hand, the results follow directly from the classical results of Sampson [15,Theorems 1,6] for harmonic maps. Here, our results are more general and the technique we are using is completely different.
(2) Theorem 1.3 was first proved in [4] by simply applying [16,Proposition 1.2.3], but here, the proof is more clear and based on the classical result from Aronszajn [2].
Throughout this article, we will use the following sign conventions. For the Riemannian curvature tensor field, we use In general, we will use the same symbol ·, · to indicate the Riemannian metrics on various vector bundles, and the same symbol ∇ for the corresponding Riemannian connections.
All manifolds are assumed to be connected and we will work only with smooth objects. Whenever we will make use of indices, 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, that is, repeated indices that are in the diagonal position indicate the sum.

Proofs of the theorems
In this section, we will prove the results obtained in this article.
Our strategy of proof is to cleverly rewrite the biharmonic map equation such that we are effectively dealing with a second-order problem to which we can apply the classical result from Aronszajn [2]   Let φ : M → N be a smooth map and let σ be a section in the pull-back bundle φ * T N. We consider a local chart (U, x i ) on M and a local chart (V, y α ) on N such that φ(U ) ⊂ V . The section σ can be written as The Laplacian Δσ is given by We find where u α i = ∂u α ∂x i and (φ β ) β is the corresponding expression for φ in local coordinates. Then, by a straightforward computation, we get (see also [14 We can go a bit further and notice that Thus, we obtain the following Proof of Theorem 1.3. Let us denote Clearly, A is closed in M and its topological interior, Int A, is non-empty. If the boundary of Int A, ∂(Int A), is empty, then, as M is connected, Assume that there exists p 0 ∈ ∂(Int A). Furthermore, let U be an arbitrary open subset containing p 0 . Clearly p 0 does not belong to Int A and U ∩ Int A = ∅.
On the other hand, we have Denote 3), we have that on U , and so on D, From the above equality, we get the following inequality on D as all functions g ij , φ α and their derivatives, Γ θ βα and their derivatives, are bounded on D. Applying Theorem 2.1 we can deduce that u = 0, which implies that τ (φ) vanishes everywhere in D. This contradiction implies ∂(Int A) = ∅ and we end the proof.

The case of a spherical target
In this subsection, we will prove Theorem 1.4. First, we will exploit the fact that we are considering a spherical target to bring (1.4) into a simpler form.
To this end, we consider the inclusion map ι : S n → R n+1 and form the composite map

4)
where Ric M denotes the Ricci tensor field on the domain manifold M and {e i }, i = 1, . . . , m is an orthonormal frame field.
Proof. It is well known that for a spherical target of constant curvature 1 the following formula holds true Note that if σ ∈ Γ(φ * T N), then we can also think of σ as a section in the pull-back bundle ϕ * T R n+1 and the connections along φ and ϕ are related via . By a direct calculation one finds that As a next step, we prove that To this end, we fix p ∈ M arbitrary and let {X i }, i = 1, . . . , m be a geodesic frame field around p. We calculate at the point p To manipulate Δ|dϕ| 2 we again compute at p Recall that Tr g ∇ 2 dϕ = −Δdϕ + dϕ(Ric M ), where Δ is the Hodge-Laplacian acting on R n+1valued one-forms. Due to our sign convention, we have −Δdϕ = dΔϕ = ∇Δϕ such that we find Finally, we have at the point p and the claim follows by combining the previous equations. Now, we define new variables v = ∇ϕ and w = Δϕ. In terms of a local chart (U, x i ) on M and {e a } denoting the canonical basis of R n+1 , a = 1, . . . , n + 1, we can write v = ϕ a i dx i ⊗ e a , where ϕ a i = ∂ϕ a ∂x i . Then any solution of (2.4) satisfies the second-order elliptic equation At this point we consider two biharmonic maps ϕ 1 , ϕ 2 and their corresponding new variables v 1 , v 2 , w 1 , w 2 . We set u := y 1 − y 2 . Note that the function u takes values in R (n+1)(m+2) , which can be seen from Then, we find where we think of u as being defined on the image of U and u . As a next step, we prove that Δ(w 1 − w 2 ) can be expressed as a sum of the components of u and their first order derivatives multiplied by terms that do not contain the components of u or their derivatives. This allows us to obtain the estimate for |Δ(w 1 − w 2 )|.
where the positive constant C depends on D and the derivatives of ϕ 1 and ϕ 2 up to third order.
Proof. Using (2.5), we find In the following, we will rewrite all the terms on the right-hand side by adding suitable zeros, The next contribution can be manipulated as follows: Moreover, we perform the following manipulation The next term can be rewritten as In addition, we have Finally, we obtain The result then follows directly since the closure of D is assumed to be compact and both ϕ 1 , ϕ 2 are smooth by assumption.

The case of a general target
In this section, we prove Theorem 1.5.
First, we consider a biharmonic map φ : M → N and a local chart (U, x i ) on M and (V, y α ) on N such that φ(U ) ⊂ V . In order to avoid any notational confusion the corresponding expression for φ in local coordinates will be denoted byφ, that iŝ Lemma 2.5. Let φ : M → N be a biharmonic map. In terms of local coordinates, it satisfies Proof. Using the local form of the tension field u θ = τ θ (φ) = Δφ θ + dφ α , dφ β Γ θ αβ in (2.3) and the fact that By a direct calculation we obtain The claim follows by combining the equations. Now, as in the spherical case, we define new variables v = ∇φ = dφ = (dφ α ) and w = Δφ = (Δφ α ).
We can write v = φ α i dx i ⊗ e α , where φ α i = ∂φ α ∂x i and {e α } denotes the canonical basis of R n . Then, any solution of (2.7) satisfies the second-order elliptic equation Here, F 3 is given by At this point, we consider two biharmonic maps φ 1 , φ 2 : M → N and let (U, x i ) be a local chart on M and (V, y α ) a local chart on N such that φ 1 (U ) ⊂ V and also φ 2 (U ) ⊂ V .
We set u := y 1 − y 2 . Note that the function u takes values in R n(m+2) , which can be seen from Then, we find In order to obtain the estimate for |Δ(w 1 − w 2 )|, we need the image of V in R n to be convex and with compact closure such that we can apply the mean value theorem for functions defined on the image of V . Then, when applying the mean value inequality, the standard norm of the differential of the function will be bounded on the image of V . So, first, we consider the image of V to be an open ball of radius ε in the Euclidean space R n and then we shrink it to a ball of radius ε 2 . Accordingly, we consider a smaller domain U . Thus, we will make use of the mean value theorem for functions defined on the open ball of radius ε 2 but which are also defined on the closed ball of radius ε 2 . More precisely, for f : B n ( ε 2 ) → R we will apply the following inequality where y * belongs to the standard segment that joins y 1 with y 2 . Then, as a next step, we prove that Δ(w 1 − w 2 ) can be expressed as a sum of the components of u and their first-order derivatives multiplied by terms that do not contain the components of u or their derivatives.
We have Lemma 2.6. Let D ⊂ U be an open subset such that its closure in M is compact and included in U . Then, the following estimate holds on D where the positive constant C depends on D and the derivatives ofφ 1 andφ 2 up to third order.
Proof. Using (2.7), we find Now, we have to start estimating all the terms on the right-hand side by adding suitable zeros. The first term can be controlled as follows For the second term we rewrite where we applied the mean value inequality to the last term. In the following we will frequently apply the mean value inequality without mentioning it explicitly. The third contribution can be manipulated as follows Regarding the fourth term we find For the fifth term we obtain The sixth contribution can be estimated as The seventh term can be estimated as The claim follows by adding up the different contributions.
Proof of Theorem 1.4. Making use of (2.9) we get the following inequality where a, b = 1, . . . , n(m + 2). Applying Theorem 2.1 we can deduce that u = 0 on D. We finish the proof by denoting A := {p ∈ M : φ 1 (p) = φ 2 (p)} and using the same arguments as in the proof of Theorem 1.3.

Proof of Theorem 1.6
In this section, we prove Theorem 1.6. The proof is again based on Theorem 2.1 and the concrete expressions of the Christoffel symbols on S n . Let S n be the Euclidean unit sphere and denote by N and S the north and south pole, respectively. It is well known that Let (y a ) be local coordinates on S n−1 , a = 1, . . . , n − 1. Then, (y 1 , . . . , y n−1 , y n = s) are local coordinates on S n \ {N, S}.
In this geometric setup the Christoffel symbols on S n are given by Γ n bn (y α ) =0, where we use a˜to indicate objects on S n−1 .
The equator S n−1 is given by Now, let (U, x i ) be a local chart on M and we denote the domain of the above local coordinates on S n \ {N, S} by V . In addition, we assume that φ(U ) ⊂ V .
We denote the corresponding expression for φ in local coordinates byφ, that is, Assume that W is an open subset of U and φ(W ) ⊂ S n−1 , that is, φ(W ) ⊂ S n−1 ∩ V . Hence, in W we have φ n = π 2 . Now, define f : U → R, f := φ n − π 2 . Clearly, 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 will prove the following estimate: Lemma 2.7. The function f : U → R defined above satisfies the following estimate on D Proof. We have Δ 2 f = Δ 2 φ n and we will use (2.7) with θ = n in order to obtain the estimates.
We expand the first term as it remains to estimate the term | ∇Δφ b , dφ c Γ n bc |. We immediately obtain Therefore, for the first term on the right-hand side of (2.7), we have the estimate The second, third and last term can be estimated in the same manner as the first one. We write the fourth term in the same way as the first one and, the only term we must estimate is We directly find such that |A n abc | C|f | and The fifth and sixth terms will be estimated in the same way as the fourth term, which completes the proof.
Proof of Theorem 1.6. As in the proofs of Theorems 1.4 and 1.5 we define suitable functions u and F . Applying Theorem 2.1 we can deduce that u = 0 on D, that is, φ maps the whole of D into S n−1 .
We finish the proof by denoting A := {p ∈ M : φ(p) ∈ S n−1 } and using the same arguments as in the proof of Theorem 1.2. Theorem 1.7 In this section, we prove Theorem 1.7. Let (V, y α ) be a local chart on N such that V ∩ P : y r+1 = · · · = y n = 0, where r denotes the dimension of the submanifold P . Moreover, we assume that the image of V in R n is convex. For example, we can assume that it is the interior of an n-cube with side lengths ε/2 that lies inside an n-cube with side lengths ε.

Proof of
In addition, let (U, x i ) be a local chart on M and assume that φ(U ) ⊂ V . We denote the corresponding expression for φ in local coordinates byφ, that is, As P is totally geodesic in N , we have on the intersection V ∩ P Γ θ αβ = P Γ θ αβ for all 1 α, β r, 1 θ r, Γ θ αβ = 0 for all 1 α, β r, r + 1 θ n, (2.11) where P Γ θ αβ denote the Christoffel symbols of the submanifold P for 1 θ, α, β r. On V ∩ P we also have We will prove the following estimate: Lemma 2.8. Restricted to D, the function f : U → R n−r defined above satisfies the following estimate where the constant C depends on the geometry of N .
Proof. We make use of (2.7) and estimate each term on the right-hand side. We expand the first term of (2.7) in the following way where b, c = 1, . . . , r. Since we have it remains to estimate the term ∇Δφ b , dφ c Γ r+ã bc . To this end we define a new functionφ : U → R n byφ = (φ 1 , . . . , φ r , 0, . . . , 0) and consequently φ : U → N takes values in P . We note that, according to (2.11), we have Γ r+ã bc = Γ r+ã bc (φ) = 0 on W , but at the points in D \ W the image of the map φ may not be in P such that Γ r+ã bc = 0. However, on U , and thus also on D, we have Consequently, for the first term on the right-hand side of (2.7), we get the estimate The second, third and the last term of (2.7) can be estimated in the same manner as the first one. In order to estimate the fourth term we rewrite it in the same way as the first one, and, the only contribution we have to estimate is Remember that which establishes the desired estimate.
Proof of Theorem 1.7. As in the proofs of Theorems 1.4 and 1.5, we define suitable functions u and F . Applying Theorem 2.1 we can deduce that u = 0 on D, that is, φ maps the whole of D into P .
We finish the proof using the same arguments as in the proof of Theorem 1.6.
In the particular case when ψ defines a regular, totally geodesic submanifold, we can give an alternative proof of the above fact, similar to the proof of Theorem 1.7.

A remark on Sampson's maximum principle
Another geometric application of the unique continuation property of harmonic maps was provided by Sampson [15,Theorem 2]. More precisely, he gave the following result: Theorem 2.10 (Sampson). Assume that φ : M → N is a harmonic map, with q = φ(p). Let S be a C 2 hypersurface in N passing through q, at which point we assume that the second fundamental form is definite. If φ is not a constant mapping, then no neighbourhood of p is mapped entirely on the concave side of S.
Besides the unique continuation property, the proof of this theorem makes use of the maximum principle. This powerful tool only exists for second-order elliptic partial differential equations, not for fourth order ones, such that we cannot expect to find a generalization of Sampson's maximum principle for proper biharmonic maps.
The map φ α is harmonic if and only if where c 1 and c 2 are real constants. We note that if α(t) admits a local extremum point at t 0 such that α(t 0 ) > 0, then t 0 is a minimum point. Indeed, consider t 0 ∈ I, I being an open interval of R, such that α(t) > 0 on I. If t 0 is an extremum point, harmonicity implies α (t 0 ) = mα(t 0 ) > 0 and it follows that t 0 is a minimum point.
The (2) We also have solutions α(t) that admit a local minimum point at t 0 such that α(t 0 ) > 0. For example, we have m = 1, α(t) = e t − te −t and t 0 = 0.