Some remarks on energy inequalities for harmonic maps with potential

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions.


Introduction and results.
One of the most studied partial differential equations for a scalar function u : R n → R is the Poisson equation, that is, where f : R n → R is some given function. If one allows the function f to also depend on u, that is, one calls the equation nonlinear Poisson equation. Following the terminology from the literature, we call a solution u ∈ C 3 (R n ) of the nonlinear Poisson equation entire. For entire solutions of the nonlinear Poisson equation, the following results characterizing its qualitative behavior have been obtained: (1) Suppose that F ∈ C 2 (R) is a nonnegative function that is a potential for f , that is, F (u) = f (u). Let u be an entire, bounded solution of the nonlinear Poisson equation. Then the following energy inequality holds [1] |∇u| 2 ≤ 2F (u). (1.1) (2) Making use of the Modica-type estimate (1.1), the following Liouville theorem was given [1,Theorem 1]: suppose F ∈ C 2 (R) is a nonnegative function that is a potential for f and u an entire, bounded solution of the nonlinear Poisson equation. If F (u(x 0 )) = 0 for some x 0 ∈ R n , then u must be constant. (3) Again, making use of the Modica-type estimate, the following monotonicity formula has been established in [2]. Let u : R n → R be an entire, bounded solution of the nonlinear Poisson equation. Then the following monotonicity formula holds d dr where B r (x) denotes the ball around the point x ∈ R n with radius r. (4) Another kind of Liouville theorem was achieved in [3]: suppose that u is an entire solution of Δu = f (u, Du).
If ∂f ∂u ≥ 0 and both u and Du are bounded, then u must be constant. (5) Recently, a maximum principle has been established for solutions of Δu = ∇F (u) in the vector valued-case [4,5], that is, u : A ⊂ R n → R m , where A ⊂ R n is some domain. Here it is assumed that the potential F vanishes at the boundary of a closed convex set. In this note we focus on the study of a geometric generalization of the nonlinear Poisson equation, which leads to the notion of harmonic maps with potential. To this end let (M, h) and (N, g) be two Riemannian manifolds, where we set n = dim M . For a map φ : M → N we consider the Dirichlet energy of the map, that is, M |dφ| 2 dM . In addition, let V : N → R be a smooth scalar function. We consider the following energy functional The Euler-Lagrange equation of the functional (1.3) is given by where τ (φ) ∈ Γ(φ * T N) denotes the tension field of the map φ. Note that in contrast to the Laplacian acting on functions, the tension field of a map between Riemannian manifolds is a nonlinear operator. Solutions of (1.4) are called harmonic maps with potential. Since (1.4) is a nonlinear partial differential equation, its solution are not necessarily smooth. However, throughout this note we will mostly assume that φ is a smooth solution of (1.4).
We want to point out that motivated from the physics literature one defines (1.3) with a minus sign in front of the potential.
Harmonic maps with potential have been introduced in [6]. It is shown that due to the presence of the potential, harmonic maps with potential can have a qualitative behavior that differs from the one of harmonic maps. Existence results for harmonic maps with potential have been obtained by the heat flow Vol. 109 (2017) Energy inequalities for harmonic maps with potential 153 method [7,8] under the assumption that the target has negative curvature. In addition, an existence result for harmonic maps with potential from compact Riemannian manifolds with boundary was obtained in [9], where it is assumed that the image of the map lies inside a convex ball. Besides the aforementioned existence results, there also exist several Liouville theorems for harmonic maps with potential. For a compact domain manifold M , these were derived by the maximum principle under curvature assumptions in [6,Proposition 4]. A Liouville theorem for harmonic maps with potential from a complete noncompact Riemannian manifold and the assumption that the image of the map φ lies inside a geodesic ball is given in [10]. A monotonicity formula for harmonic maps with potential together with several Liouville theorems was derived in [11].
For functions on Riemannian manifolds, several generalizations of the Modica-type estimate (1.1) have been established, see [12,13]. These results hold under the assumption that the manifold has positive Ricci curvature.
However, it was also noted that estimates of the form (1.1) do not hold if we consider vector-valued functions [14,15].
It is the aim of this article to discuss if the results obtained for the nonlinear Poisson equation stated in the introduction still hold when considering harmonic maps with potential.
This article is organized as follows: In Sect. 2 we discuss in which sense the Modica-type estimate (1.1) for solutions of the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. In the last section we will give a Liouville theorem for harmonic maps with potential under curvature and boundedness assumptions.

Energy inequalities for harmonic maps with potential.
Before we turn to deriving energy inequalities let us make the following observation: If we want to model the trajectory of a point particle in a curved space, we can make use of harmonic maps with potential from a onedimensional domain, which are just geodesics coupled to a potential. To this end we fix some interval I and consider a curve γ : I → N that is a solution of (1.4), which in this case reads Here represents the derivative with respect to the curve parameter, which we will denote by s. For a curve γ satisfying this equation the total energy is conserved, that is, This can easily be seen by calculating d ds where we used the equation for harmonic maps with potential in the last step. This fact is well known in classical mechanics, that is, the mechanics of point particles governed by Newton's law. The total energy consists of the sum of the kinetic and the potential energy and it is conserved when the equations of motion are satisfied.
However, if the dimension of the domain M is greater then one, we cannot expect that a statement about the conservation of the total energy will hold in full generality.
We will make use of the following Bochner formula for a map φ : M → N , that is, Here e i , i = 1, . . . , n is an orthonormal basis of T M. Throughout this article we will make use of the Einstein summation convention, that is, we sum over repeated indices. In addition, by the chain rule for composite maps, we find where we used that φ is a solution of (1.4) in the second step.
In order to obtain the Modica-type estimate (1.1) for solutions of the scalar nonlinear Poisson equation one makes use of the so-called P-function technique, which heavily makes use of the maximum principle. The generalization of the P -function to harmonic maps with potential is given by Unfortunately, it turns out that the P -function does not satisfy a "nice" inequality in the case of harmonic maps with potential.

Lemma 2.2.
Let φ : M → N be a smooth harmonic map with potential. Then the P -function (2.3) satisfies the following inequality Proof. Using the Bochner-formulas (2.1), (2.2) a direct calculation yields In addition, we apply the Kato-inequality and find Since we cannot derive energy inequalities by making use of the techniques that were developed for solutions of the scalar nonlinear Poisson equation, we will apply ideas that were used to derive gradient estimates and Liouville theorems for harmonic maps between complete Riemannian manifolds [17].
Here, one assumes that the image of the map φ lies inside a geodesic ball in the target.

Gradient estimates for harmonic maps with potential.
In the following we will make use of the following Then the following inequality holds Proof. This follows from the Bochner formula (2.1) and the identity d|dφ| 2 2 ≤ 4|dφ| 2 |∇dφ| 2 .
Now we fix a point x 0 in M , and by r we denote the Riemannian distance from the point x 0 . Let η : N → R be a positive function. On the geodesic ball B r (x 0 ) in M , we define the function Clearly, the function F vanishes on the boundary of B a (x 0 ), hence F attains its maximum at an interior point x max . We can assume that the Riemannian distance function r is smooth near the point x max , see [18,Sect. 2]. In the following we will apply the Laplacian comparison theorem, see [19, p. 20], that is, with some positive constant C L . Moreover, we make use of the Gauss Lemma, that is, |dr| 2 = 1. Ric M − Hess V ≥ −A V and that the sectional curvature K N of N satisfies K N ≤ B. Then the following inequality holds (2.7) Proof. At the maximum x max the first derivative of (2.6) vanishes, yielding Applying the Laplacian to (2.6) at x max gives Squaring (2.8) we find Inserting (2.5) and (2.10) into (2.9) and using the Gauss Lemma we get the claim.
To obtain a gradient estimate from (2.7) for noncompact manifolds M and N , we have to specify the function η.
First, we choose a function η that is adapted to the geometry of the target manifold motivated by a similar calculation for harmonic maps between complete manifolds [17]. Let ρ be the Riemannian distance function from the point y 0 in the target manifold N . We define the positive number d will be specified in the assumptions of Theorems 2.7 and 2.20. By B R (y 0 ) we will denote the geodesic ball of radius R around the point y 0 in N . We will assume that R < π/(2 √ d), thus 0 < ξ(R) < √ d on the ball B R (y 0 ).

Lemma 2.6.
On the geodesic ball B R (y 0 ) we have the following estimate We will also make use of the following fact: if c 1 x 2 − c 2 x − c 3 ≤ 0 for c i > 0, i = 1, 2, 3, then the following inequality holds (2.14) where the positive constant C 2 depends on the geometry of N .
Proof. We choose the function ξ defined in (2.11) and insert it for η in (2.7). By the Hessian comparison theorem (2.12), we find where we also used that φ is a harmonic map with potential. In addition, making use of the assumption on the image of φ(M ), there exists a positive constant C 2 such that −2B + 2d holds. Inserting this into (2.7) we find The claim then follows from (2.13).

Corollary 2.8.
Under the assumptions of Theorem 2.7, we can take the limit a → ∞ while keeping the point x 0 in M fixed and obtain the estimate If M has positive Ricci curvature and if the potential V (φ) is concave, then the following inequality holds which can be interpreted as a Modica-type estimate for harmonic maps with potential. There is another way how we can obtain a gradient estimate from (2.7), by assuming that the potential V (φ) has a special structure. More precisely, we have the following Theorem 2.9. Let φ : M → N be a smooth harmonic map with potential. Suppose that the Ricci curvature of M satisfies Ric M ≥ −A and that the sectional curvature K N of N satisfies K N ≤ B. Moreover, assume that the potential V satisfies Then the following energy estimate holds (2.15) The constant C 3 depends on the geometry of N .
Making use of the assumptions on the potential V (φ), we note that In addition, again by the assumptions on the potential V (φ), we get for some positive constant C 3 . Inserting into (2.7) then yields The statement follows from applying (2.13) again.

Corollary 2.10.
Under the assumptions of Theorem 2.9, we can take the limit a → ∞ while keeping the point x 0 in M fixed and obtain the estimate If M has nonnegative Ricci curvature, then φ is trivial. Remark 2.11. In Theorem 2.7 we assume that φ(M ) is contained in a geodesic ball. On the other hand, in Theorem 2.9, we require that the potential V (φ) has a special structure such that we can drop the assumption of φ(M ) being contained in a geodesic ball.

Generalized monotonicity formulas.
In the following we will make use of the stress-energy-tensor for harmonic maps with potential, which is locally given by Vol. 109 (2017) Energy inequalities for harmonic maps with potential 159 The stress-energy-tensor is divergence-free when φ is a smooth harmonic map with potential [11], that is, Let us recall the following facts: a vector field X is called conformal if where L denotes the Lie-derivative of the metric h with respect to X and f : M → R is a smooth function.
Lemma 2.12. Let T be a symmetric 2-tensor. For a conformal vector field X, the following formula holds By integrating over a compact region U and making use of Stokes theorem, we obtain: Lemma 2.13. Let (M, h) be a Riemannian manifold and U ⊂ M be a compact region with smooth boundary. Then, for any symmetric 2-tensor and a conformal vector field X, the following formula holds where ν denotes the normal to U .
We now derive a type of monotonicity formula for smooth solutions of (1.4) for the domain being R n . Lemma 2.14. Let φ : R n → N be a smooth harmonic map with potential. Let B r (x) be a ball with radius r in R n . Then the following formula holds r ∂Br(x) Proof. For M = R n we choose the conformal vector field X = r ∂ ∂r with r = |x|. Note that div X = n. The statement then follows from Lemma 2.13 applied to (2.16).
Making use of the coarea formula, we obtain the following Theorem 2.15. Let φ : R n → N be a smooth harmonic map with potential. Let B r (x) be a ball with radius r in R n . Then the following formula holds Note that this monotonicity formula is different from the one for solutions of the nonlinear Poisson equation (1.2) since we do not have a Modica-type estimate for harmonic maps with potential.
Monotonicity formulas for harmonic maps with potential with the domain being a Riemannian manifold have been established in [11].
The results presented above also hold for harmonic maps with potential that have lower regularity. To this end we need the notion of stationary harmonic maps with potential.

Definition 2.17.
A weak harmonic map with potential is called stationary harmonic map with potential if it is also a critical point of the energy functional with respect to variations of the metric on the domain M , that is, Here k ij is a smooth symmetric 2-tensor.
Every smooth harmonic map with potential is stationary, which is due to the fact that the associated stress-energy-tensor is conserved. However, a stationary harmonic map with potential can have lower regularity.
For stationary harmonic maps with potential, we have the following result generalizing In particular, this implies that φ is constant when V (φ) ≤ 0.
Proof. We will prove the result for the case that M = R n . Let η ∈ C ∞ 0 (R) be a smooth cut-off function satisfying η = 1 for r ≤ R, η = 0 for r ≥ 2R, and |η (r)| ≤ C R . In addition, we choose Y (x) := xη(r) ∈ C ∞ 0 (R n , R n ) with r = |x|. Hence, we find We can bound the right-hand side as follows Making use of the properties of the cut-off function η, we obtain Taking the limit R → ∞ and making use of the assumptions, we find which finishes the proof for the case that M = R n . Making use of the Theorem of Cartan-Hadamard, the proof carries over to hyperbolic space.
Remark 2.19. The last theorem can be interpreted as an integral version of (1.1) for bounded harmonic maps with potential.
Now we derive a generalized monotonicity formula for harmonic maps with potential, where we take into account the pointwise gradient estimate (2.15). Theorem 2.20. Let φ : R n → N be a smooth harmonic map with potential. Suppose that the Hessian of the potential V satisfies − Hess V ≥ −A V and that the sectional curvature Then the following monotonicity-type formula holds

20)
where the positive constant C depends on B.
Proof. Throughout the proof we set from which we get the claim.
Let us make several comments on Theorem 2.20: (1) The monotonicity type-formula (2.20) can be interpreted as the generalization of (1.2) to harmonic maps with potential. (2) Making use of the monotonicity formula for harmonic maps with potential from [11] it is possible to generalize Theorem 2.20 to the case of the domain being a Riemannian manifold.

A Liouville theorem.
In this section we derive a Liouville theorem for harmonic maps with potential from complete noncompact manifolds with positive Ricci curvature. Our result is motivated from a similar result for harmonic maps, see [20,Theorem 1]. In addition, this result also generalizes the Liouville theorem for solutions of the nonlinear Poisson equation [3], which is stated in detail in the introduction. Proof. We follow the presentation in [21, pp. 26]. For a solution φ of (1.4) and by the standard Bochner formula (2.1), we find  In addition, by the Cauchy-Schwarz inequality we find |de(φ)| 2 ≤ 2e(φ)|∇dφ| 2 . Now let x 0 be a point in M and let B R , B 2R be geodesic balls centered at x 0 with radii R and 2R. We choose a cutoff function η satisfying In addition, we choose η such that 0 ≤ η ≤ 1, |∇η| ≤ C R for a positive constant C. Then we find We therefore obtain