On harmonic and biharmonic maps from gradient Ricci solitons

We study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two‐dimensional cigar soliton must be harmonic.


Introduction and results
One of the central aims in the geometric calculations of variations is to find interesting maps between Riemannian manifolds.This can often be achieved by extremizing a given energy functional.For a concrete energy functional under consideration one aims to obtain a classification of its critical points, i.e. one wants to know under which conditions critical points can exist or are obstructed.In this article, we will focus on harmonic and biharmonic maps between Riemannian manifolds and establish a number of classification results for these.Becoming more technical, we assume that (M, g) and (N, h) are two Riemannian manifolds.Consider a map φ : M → N , its energy is defined by It is well-known that the critical points of (1.1) are harmonic maps.They are characterized by the vanishing of the tension field which is defined as follows τ (φ) = Tr g ∇dφ, (1.2) where ∇ denotes the connection on the vector bundle φ * T N .The harmonic map equation τ (φ) = 0 is a semilinear elliptic partial differential equation of second order, for more details we refer to the book [27].Of course, the harmonic map equation admits trivial solutions, namely in the case that the map φ maps to a point q ∈ N .A higher order generalization of harmonic maps that receives growing attention are the so-called biharmonic maps.These are critical points of the bienergy functional and are characterized by the vanishing of the so-called bitension field which is explicitly given by τ 2 (φ) := ∆τ (φ) + Tr g R N (τ (φ), dφ)dφ.
(1.4) Here, ∆ denotes the connection Laplacian on the vector bundle φ * T N for which we choose the analysts sign convention, that is ∆ := Tr g ( ∇ ∇ − ∇∇ ).
In contrast to the harmonic map equation, the biharmonic map equation τ 2 (φ) = 0, is a fourth order semilinear elliptic partial differential equation which leads to additional technical difficulties.In particular, tools like the maximum principle, which are well-adapted to partial differential equations of second order, are no longer applicable in full generality in the case of biharmonic maps.
For an overview on biharmonic maps we refer to the survey article [16] and the recent book [19].Both harmonic and biharmonic maps have important applications in analysis and geometry, but are also utilized in elasticity theory and quantum field theory.For an overview on further higher order variational problems in Riemannian geometry, generalizing harmonic and biharmonic maps, we refer to the paper [4].
It can be directly seen that every harmonic map, that is a solution of τ (φ) = 0, automatically solves the equation for biharmonic maps which is τ 2 (φ) = 0. Hence, one is interested in constructing biharmonic maps which are non-harmonic, the latter are called proper biharmonic.On the other hand, under certain geometric and analytic assumptions a biharmonic map necessarily needs to be harmonic.For example, if both M, N are compact and N has negative sectional curvature then a direct application of the maximum principle shows that every biharmonic map is harmonic.More results of this kind can be found in [5,8].
In this article we will derive a number of inequalities that allow to get an understanding when harmonic maps are constant and when biharmonic maps need to be harmonic assuming that the domain manifold M is equipped with a special kind of Riemannian metric which are gradient Ricci solitons.
In order to approach these inequalities let us first recall a number of results on gradient Ricci solitons.The concept of Ricci solitons is of great importance in the study of the Ricci flow as Ricci solitons evolve just by diffeomorphisms and homotheties of the initial metric under Ricci flow.For an introduction to Ricci flow we refer to the book [26].A special class of Ricci solitons are the so-called gradient Ricci solitons.They are characterized by the following equation: Here, Ric M represents the Ricci curvature of the Riemannian manifold (M, g), ∇ 2 f is the Hessian of a function f ∈ C ∞ (M ) and λ ∈ R.
In the case of a two-dimensional manifold gradient Ricci solitons can be completely classified, see [3,22].In higher dimensions it is substantially more difficult to obtain such a classification, see for example [20], [21] for important results on this matter.A gradient Ricci soliton is called trivial if f = const.In the case of M being compact gradient Ricci solitons are most often trivial as can be seen by taking the trace of (1.5) and applying the maximum principle.For this reason, we only consider the case of a non-compact manifold M in this manuscript.Let us mention several results from the literature which are closely connected to the content of this article.Sealey showed that harmonic maps of finite energy from Euclidean and hyperbolic space of dimensions bigger than two must be constant [24].Further results of this kind based on monotonicity formulas are presented in [27,Section 2.3].Harmonic functions on gradient Ricci solitons were studied in [18] by Munteanu and Sesum.Rimoldi investigated f -harmonic maps, which are critical points of a weighted version of harmonic maps, and their relation to gradient Ricci solitons in [21].Moreover, L 2 -harmonic forms on gradient shrinking Ricci solitons were investigated by Yun in [29].
A class of solitons closely related to this article are the so-called Ricci-harmonic solitons studied in [13].These solitons arise in the context of the Ricci-harmonic flow which is a combination of both Ricci and harmonic map heat flow introduced in [17].
Let us briefly describe the strategy that we are using in order to establish our main results.
For both harmonic and biharmonic maps there exists an associated stress-energy tensor which is divergence free.Testing this conservation law with the equation for a gradient Ricci soliton, using a cutoff-function in order to be able to employ integration by parts, we are led to a number of energy inequalities from which we can deduce various vanishing results.
In the following we set m := dim M and present the main results of this article.
Theorem 1.1.Assume that (M, g, f ) is a complete non-compact gradient Ricci soliton.Let φ : M → N be a smooth harmonic map which satisfies where B R denotes the geodesic ball of radius R. Then the following inequality holds where Scal M represents the scalar curvature of the manifold M and {e i }, i = 1, . . ., m denotes an orthonormal frame field on M .
In order to obtain any information from (1.7) one of course has to make additional curvature assumptions that guarantee that the left-hand side of (1.7) is positive.One possibility to obtain a kind of Liouville theorem is given by the following Corollary 1.2.Assume that (M, g, f ) is a complete, non-compact gradient Ricci soliton with m > 2 and let φ : M → N be a smooth harmonic map which satisfies (1.6).
(1) Note that the inequality (1.7) does not contain any information if m = 2 as we have that 2 Ric M = Scal M g.
(2) The assumptions in Corollary 1.2 seem to be restrictive but there does not seem to be any kind of contradiction.(3) Note that Corollary 1.2 recovers the result of Sealey [24] in the case of (M, g) = (R m , δ), where δ represents the flat Euclidean metric, by choosing f = |x| 2 2 .This particular choice of f is called Gaussian soliton in the literature.(4) It is well-known that a harmonic map of finite energy from a complete, non-compact manifold of positive Ricci curvature to a Riemannian manifold of negative sectional curvature must be constant due to a celebrated result of Schoen and Yau [23].The previous results are in the same spirit but here we only make assumptions on the geometry of the domain and do not require the target to have negative curvature.
In addition, making use of the Bochner formula for harmonic maps, we will also establish the following statement extending a result for harmonic functions [18, Theorem 4.1]: Theorem 1.4.Let (M, g, f ) be a complete, non-compact steady gradient Ricci soliton of infinite volume and (N, h) a Riemannian manifold of non-positive sectional curvature.In addition, let φ : M → N be a harmonic map satisfying Then φ must be a constant map.
Besides the above Liouville-type results for harmonic maps from gradient Ricci solitons we will also prove the following corresponding results for biharmonic maps.
(1) It is clear that the inequality (1.10) has a similar structure as the corresponding inequality for harmonic maps (1.7).While (1.10) reflects that biharmonic maps are a fourth order equation, (1.7) clearly shows the second order character of harmonic maps.However note that in Theorem 1.5 we need to require an upper bound on the Ricci curvature which is not necessary in the case of harmonic maps (Theorem 1.1).Moreover, note that the finite energy assumption (1.9) involves the full second covariant derivative ∇dφ and not only its trace given by the tension field τ (φ).
(2) If one tries to derive corresponding energy inequalities for polyharmonic maps, then one can expect that it is necessary to also require upper bounds on the covariant derivatives of the Ricci curvature.
In addition, we can also give the following kind of Liouville-type result obtained from Theorem 1.5.Let us recall a famous example of a steady gradient Ricci soliton, which is the so-called Hamilton's cigar soliton in two dimensions.In the physics literature it is known as Witten's black hole, see [26, p. 10] for some more details.It is given by the following data and has positive scalar curvature Scal M = 1 1+x 2 +y 2 > 0. For biharmonic maps from the cigar soliton we can give the following Corollary 1.8.Assume that (M, g, f ) is the two-dimensional cigar soliton defined by (1.11).Let φ : M → N be a smooth biharmonic map with finite energy.Then φ is harmonic.
(1) In the case that (M, g, f ) is a steady gradient Ricci soliton, that is λ = 0 we do not need to require that |∇f | < ∞ as this condition is automatically satisfied, see Section 2.
However, in general, we have to make the assumption that |∇f | < ∞ although very often this condition may not be necessary but this of course depends on the concrete gradient Ricci soliton.
If we want to drop the assumption |∇f | < ∞ in Theorem 1.5, then in addition to the finite energy assumption (1.9), we have to demand where B R denotes the geodesic ball of radius R.
(2) Of course there may be geometric configurations under which Corollary 1.7 states that a biharmonic map with finite energy has to be constant rather than just being harmonic.(3) Note that Theorem 1.5 recovers a result of Baird et al. [2,Theorem 3.4] in the case of (M, g) = (R m , δ), where δ represents the flat Euclidean metric, by choosing f = |x| 2 2 , see also [6,Theorem 3.2] for a different method of proof.(4) It does not seem to be possible to prove a result of the form of Theorem 1.4 for biharmonic maps as the Ricci curvature of the domain does not enter in the Bochner formula for biharmonic maps.(5) If we rescale the metric g → r 2 g for r ∈ R in the equation for a gradient Ricci soliton (1.5) then it changes to Ric M +∇ 2 f = r 2 λg.
Hence, many of the geometric conditions that we can impose to deduce a vanishing result from Theorems 1.1, 1.5 may not be invariant under rescaling of the metric.
This article is organized as follows: In Section 2 we recall the relevant background material on gradient Ricci solitons and stress-energy tensors that is utilized in this article.Afterwards, in Section 3, we provide the proofs of the main results.Finally, Section 4 provides some remarks on harmonic and biharmonic maps in the case that the domain manifold is a gradient Yamabe soliton.
Throughout this manuscript we make use of the summation convention, that is we sum over repeated indices.We use the symbol •, • to denote various different scalar products.Moreover, the letter C denotes a generic positive constant whose value may change from line to line.We use the following sign convention for the Riemannian curvature tensor for vector fields X, Y, Z. Concerning the Laplacian we use the analysts sign convention, such that ∆ξ = ξ ′′ for ξ ∈ C ∞ (R).
Acknowledgements: The author would like to thank the reviewers for many helpful comments which helped to substantially improve the content of the article.

Some background material
In this section we recall several well-known results on gradient Ricci solitons and stress-energy tensors which are applied in the proofs of the main results.We will often make use of the following identity which follows from contracting the second Bianchi identity twice.Note that (2.1) holds on every Riemannian manifold.
2.1.Some properties of gradient Ricci solitons.First, we recall the following facts on gradient Ricci soliton, for more details we refer to [12].
Lemma 2.1.Assume that (M, g, f ) is a gradient Ricci soliton.Then the following equation holds for some constant C.
Proof.First of all, we note that by taking the trace of (1.5) we get By R ij we denote the components of the Ricci tensor Ric M .Now, taking the divergence of (1.5) we get

Using (2.1) we deduce that
Combining (2.4) and (1.5) we find Lemma 2.2.Assume that (M, g, f ) is a steady gradient Ricci soliton, then Scal M ≥ 0 and for some positive constant C.
Proof.The proof uses ideas from Ricci flow.It is well-known that every steady Ricci soliton is an ancient solution to the Ricci flow.However, by [11,Corollary 2.5] we know that any ancient smooth complete solution to the Ricci flow must have non-negative scalar curvature Scal M ≥ 0. The bound on |∇f | 2 now follows from (2.2).
2.2.Stress-energy tensors.Besides the aforementioned results on gradient Ricci solitons we also recall the stress-energy tensors for both harmonic and biharmonic maps.The stress-energy tensors arise by varying the energies (1.1) and ( 1.3) with respect to the metric on the domain.
In the case of harmonic maps the stress-energy tensor was first derived in [1] and it is given by where X, Y are vector fields on M .A direct computation shows that the stress-energy tensor (2.6) satisfies the equation div S 1 = − τ (φ), dφ . (2.7) In particular, the stress-energy tensor S 1 is conserved (has vanishing divergence) if φ : M → N is a harmonic map, that is a solution of τ (φ) = 0. Now, let us reconsider the stress-energy associated with the bienergy (1.3) which is given by where X, Y are again vector fields on M .It was first stated by Jiang in [14] and later systematically studied by Loubeau, Montaldo and Oniciuc in [15].The stress-energy tensor (2.8) satisfies the following conservation law div S 2 = − τ 2 (φ), dφ . (2.9) As in the case of harmonic maps, the stress-energy tensor S 2 is conserved (has vanishing divergence) if φ : M → N is a biharmonic map, that is a solution of τ 2 (φ) = 0. Recall that the bitension field τ 2 (φ) is defined in (1.4).
The fact that the stress-energy tensors are conserved is a direct consequence of Noether's theorem as the energies (1.1) and (1.3) are invariant under diffeomorphisms on the domain.
For more details on stress-energy tensors in the context of harmonic maps, we refer to [7] where the stress-energy tensor for polyharmonic maps was investigated in detail.

Proof of the main results
In this section we provide the proofs of the main results.First, we will establish the following Lemma 3.1.Let (M, g, f ) be a complete, non-compact gradient Ricci soliton.Suppose that φ : M → N is a smooth harmonic map and η ∈ C ∞ (M ) with compact support.Then the following identity holds where S 1 represents the stress-energy tensor associated with harmonic maps.
Proof.By assumption the map φ is harmonic hence the stress-energy tensor S 1 given by (2.6) is divergence-free.Let {e i }, i = 1, . . ., m be an orthonormal basis of T M that satisfies ∇ e i e j = 0, i, j = 1, . . ., m at a fixed point p ∈ M .As η ∈ C ∞ (M ) has compact support we can use integration by parts to calculate and making use of the definition of the stress-energy tensor S 1 completes the proof.
Proof of Theorem 1.1.First of all, by taking the trace of (1.5) we get ∆f = mλ − Scal M .Now, we choose the function η as follows: Let 0 ≤ η ≤ 1 on M be such that where B R (x 0 ) denotes the geodesic ball around the point x 0 with radius R. Using this identity, the defining equation of a gradient Ricci soliton (1.5) and (3.1) we obtain Due to the finite energy assumption (1.6) this term will vanish as we take the limit R → ∞.
The proof is now complete.
Proof of Theorem 1.4.First, recall that in the case of a steady gradient Ricci soliton, that is λ = 0, we have Ric M = − Hess f, Inserting these identities into (3.1)we find where S 1 is the stress-energy tensor associated with harmonic maps (2.6).Now, making use of the Bochner formula for harmonic maps, see for example [27,Prop. 1.3.5],we find , dφ(e i ) − R N (dφ(e i ), dφ(e j ))dφ(e j ), dφ(e i ) where we used the assumption of non-positive sectional curvature of the manifold N and the Kato inequality in the second step.Combining the previous equations yields As in the previous proof we choose η as follows: Let 0 ≤ η ≤ 1 on M be such that where B R (x 0 ) denotes the geodesic ball around the point x 0 with radius R. Now, we note that where we also employed the properties of the cut-off function η.
Recall that we have By assumption (M, g, f ) is a steady gradient Ricci soliton such that |∇f | ≤ C due to (2.5).Hence, by combining the previous identities, we find Now, letting R → ∞ and using the assumption of finite energy (1.6) we get Hence, we may conclude that |dφ| 2 = C for a positive constant C.However, we require that the manifold (M, g, f ) has infinite volume and due to the finite energy assumption we conclude that |dφ| 2 = 0.
Before we turn to the proof of Theorem 1.5 we establish a technical lemma.
Lemma 3.2.Assume that (M, g, f ) is a gradient Ricci soliton and let φ : M → N be a smooth biharmonic map.For η ∈ C ∞ (M ) compactly supported the following formula holds )∇ e j f ∇dφ(e i , e j ), τ (φ) dv g , where {e i }, i = 1, . . ., m represents an orthonormal basis of T M .
Proof.By assumption φ is a smooth biharmonic map such that the stress-energy tensor associated with biharmonic maps (2.8) is divergence free.As η has compact support we use integration by parts to deduce As a next step we manipulate the term S 2 (∇f, ∇η 2 ).To this end, let {e i }, i = 1, . . ., m be an orthonormal basis of T M that satisfies ∇ e i e j = 0, i, j = 1, . . ., m at a fixed point p ∈ M .Using the definition of the stress-energy tensor (2.8) we find where we used the equation for a gradient Ricci soliton (1.5) in the second step.The result follows from adding up the different contributions.
Proof of Theorem 1.5.Again, let 0 ≤ η ≤ 1 on M be such that η(x) = 1 for x ∈ B R (x 0 ), η(x) = 0 for x ∈ B 2R (x 0 ), |∇ q η| ≤ C R q for x ∈ M, where B R (x 0 ) denotes the geodesic ball around the point x 0 with radius R and q = 1, 2.Moreover, let {e i }, i = 1, . . ., m be an orthonormal basis of T M that satisfies ∇ e i e j = 0, i, j = 1, . . ., m at a fixed point p ∈ M .In order to prove the result we estimate the terms on the right hand side of (3.3).Making use of the finite energy assumption (1.9) it is easy to infer Proof of Theorem 1.8.Recall that the cigar soliton is a steady gradient Ricci soliton meaning that it is a solution of (1.5) with λ = 0. Since we are on a two-dimensional domain we also have 2 Ric M = Scal M g.
In the case of a steady gradient Ricci soliton we always get a pointwise bound on ∇f , see Lemma 2.2 and it is also straightforward to see that we have an upper bound on the Ricci curvature of the cigar soliton.Hence, we may apply Theorem 1.1 and inserting the above data into (1.10)yields Scal M |τ (φ)| 2 dv g ≤ 0.
As the cigar soliton has positive scalar curvature we can deduce τ (φ) = 0 yielding the claim.
4. Some remarks about harmonic and biharmonic maps from Yamabe solitons Another class of Riemannian manifolds that is closely connected to gradient Ricci solitons are manifolds that admit a concircular vector field.In this case we have where both F, ϕ are smooth functions on M .Such kinds of manifolds have been intensively studied by Tashiro [25].A special case of (4.1) is the equation for a gradient Yamabe soliton given by where Scal M represents the scalar curvature of M and ρ ∈ R.These solitons arise as self-similar solutions of the Yamabe flow and have been classified in [9] and [10].
As in the case of gradient Ricci solitons, a Yamabe soliton is called steady if ρ = 0, shrinking if ρ > 0 and expanding if ρ < 0.
We have seen that in the case of a steady gradient Ricci soliton we automatically get a pointwise bound on ∇f , see Lemma 2.2.There does not seem to be a corresponding result for gradient Yamabe solitons, see [28] for more details.However, it is straightforward to prove the following