On VT-harmonic maps

VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.


Introduction
Let (M m , g) be a compact Riemannian manifolds with nonempty boundary ∂ M and (N n ,g) a complete Riemannian manifold without boundary. Let d : N × N → R be the distance function on N and B (1+σ )R ( p) := {q ∈ N : d( p, q) ≤ (1 + σ )R} a regular ball in N , that is, disjoint from the cut locus of its center p and of radius (1 + σ )R < π 2 √ κ , where κ = max{0, sup B (1+σ )R ( p) K N } and sup B (1+σ )R ( p) K N is an upper bound of the sectional curvature K of N on B (1+σ )R ( p), and σ > 0 is any given constant.
Let V ∈ (T M), T ∈ (⊗ 1,2 T N). We call a map u : M → N a V T -harmonic map if u satisfies τ (u) + du(V ) + Tr g T (du, du) = 0, (1.1) where τ (u) = trDdu is the tension field of the map u. This is a generalization the notion of a V -harmonic map that has been studied in recent years as a common framework including Hermitian, affine and Weyl harmonic maps into Riemannian manifolds, that is, the domain possessed a connection different from the Levi-Civita connection, but the target was a Riemannian manifold with its Levi-Civita connection. This generalized the standard harmonic map system τ (u) = 0 to a system of the form τ (u) + du(V ) = 0 with a vector field V on the domain. Here, we want to consider targets that are of the same type as the domain. That leads to the system (1.1) with an additional term arising from a 2-tensor T on the target. As this new term Tr g T (du, du), in contrast to the term du(V ), is analytically of the same weight as the elliptic operator τ (u) (which includes the Laplace-Beltrami operator of the domain), this makes the analysis more difficult and subtle. This is the problem that we are addressing in this paper. In local coordinates {x α } on M and {y i } on N , respectively, we can write (1.1) as where M is the Laplacian on (M, g), i jk stands for the Christoffel symbols of (N ,g), V := V α ∂ ∂ x α and T := T i jk ∂ ∂ y i ⊗ dy j ⊗ dy k . This is a second-order semilinear elliptic system on the manifold (M, g).
As is already the case for V -harmonic maps, in general, (1.1) is neither in divergence form, nor has a variational structure. Chen et al. [5] established a Jäger-Kaul type maximum principle for V -harmonic maps by using the method of [8], and combining this with the continuity method, the existence of V -harmonic maps into a regular ball could be proved. Therefore, it is natural to ask whether a maximum principle holds for V T -harmonic maps. However, the case of V T -harmonic maps is harder to deal with than V -harmonic maps since we now have an additional quadratic term arising from the tensor T . Due to this additional structure, the construction of the elliptic operator in [5] is no longer valid in our case. To overcome this difficulty, we use another construction as in [7] to compensate this term and obtain the following maximum principle for V T -harmonic maps: Theorem 1 Let u 1 , u 2 ∈ C 0 (M, N ) be two V T -harmonic maps into a geodesic ball B R ( p). For appropriate σ and R, there exists a constant C 0 depending only on κ, σ, R and the geometry of N , such that if max |∇T | + max |T | ≤ C 0 , (1.3) then the function : M → R defined by := q κ 4 (ρ) (q κ ((1 + σ )R) − q κ (ρ 1 )) 1 2 · (q κ ((1 + σ )R) − q κ (ρ 2 )) 1 2 (1.4) satisfies the maximum principle, namely Here the expression of q κ is given in Sect. 2, and ρ : In particular, if u 1 = u 2 on the boundary ∂ M, then u 1 ≡ u 2 on M.
Remark The explicit expression of the constant C 0 in the above and in the subsequent results can be seen in (3.5). Importantly, C 0 → ∞ for R → 0. Thus, we can also satisfy the condition on T by making the target ball sufficiently small.
For the heat flow of V T -harmonic maps, an analogous result holds. For T > 0, we set and denote the parabolic boundary of M T by We consider the heat flow of V T -harmonic maps and have In particular, if u 1 = u 2 on the boundary ∂ p M T , then u 1 ≡ u 2 on M T .
As an application of the above maximum principle, we obtain the existence of V Tharmonic maps into a geodesic ball.
Furthermore, based on Theorem 3, we shall also establish the existence of V T -harmonic maps from complete noncompact Riemannian manifolds by using a gradient estimate and the compact exhaustion method.
Theorem 4 Let (M m , g) be a complete noncompact Riemannian manifold and (N n ,g) be a complete Riemannian manifold with sectional curvature bounded above by a positive constant κ. Let B R ( p) be a geodesic ball with radius R < For appropriate σ and R, there exists a constant C 0 depending only on κ, σ, R and the geometry of N , such that if

Preliminaries
Let us first give some notations: In local coordinates {x α } on M and {y i } on N , respectively, the energy density of u is Assume the metric of N satisfies: Denote λ := min N λ and := max N ∀y 1 , y 2 ∈ B R ( p), there exists a unit speed geodesic γ : [0, ρ] → B R ( p) ⊂ N with γ (0) = y 1 , γ (ρ) = y 2 , where ρ = dist(y 1 , y 2 ). For any v j ∈ T y j N , j = 1, 2, let X be the unique Jacobi field along γ with X (0) = v 1 , X (ρ) = v 2 . Then, we define a pseudo-distance Another pseudo-distance is given by wherev 2 ∈ T y 1 N stands for the vector obtained by parallel displacement of v 2 ∈ T y 2 N along γ . Let L(T x M, T y N ) be the space of all linear maps from T x M to T y N . The pseudodistance δ on the tangent bundle can be extended to a pseudo-distance on the fibers, that is, for q 1 , q 2 ∈ ∪ y∈B R ( p) L(T x M, T y N ) (disjoint union), we define their pseudo-distance as where {e 1 , . . . , e m } is an orthonormal base for T x M.
We have the following relationship between these two pseudo-distances: There is a positive constant C depending only on B R ( p) and the geometry of N such that for any y j ∈ B R ( p) and v j ∈ T y j N , j = 1, 2, we have Remark 1 In fact, by the proof in [4] and using a well-known expression of the curvature operator (see, e.g., Lemma 4.3.3 in [12]), it is not hard to see that if the sectional curvature K on B R ( p) satisfies θ ≤ K | B R ( p) ≤ κ for a constant θ < 0, then the constant C can be expressed as 14(κ − θ).
The following estimates will also be important for us: Then, . and

The maximum principle
Proof of Theorem 1 Let We consider the operator By direct computation, we obtain For any x ∈ M, we letγ be the unique geodesic connecting u 1 (x) and u 2 (x). Choosing a parallel orthonormal frame {E i (t)} alongγ with E 1 =γ , and a local orthonormal frame {e α }| m α=1 around x, assuming that ∂ ∂ y i := a j i E j , we have δ 0 (T (du 1 (e α ), du 1 (e α )), T (du 2 (e α ), du 2 (e α ))) we then obtain where C = 14(κ − θ), and the constant θ is a lower bound of the sectional curvature of N on B π 2 √ κ ( p). The Cauchy inequality implies that By using the formula (2.13) in [7], it follows that then by Lemma 1, we have Namely, Therefore, The above inequality and (2.3) imply that Choosing It follows from (2.1) that It is easy to check that Therefore, where we have used the fact that s ν (ρ)ρ q ν (ρ) is nonincreasing in (0, 2R] and ρ 2 q ν (ρ) is increasing in (0, 2R]. Direct computation gives us It follows that Clearly, by choosing appropriate σ and R, we obtain then we have (For √ κ R → 0, we use the Taylor expansions of sin and cos to obtain positive values on the right-hand side of (3.5).) It is easy to see that there exists a constant C 0 depending only on κ, σ, R and the geometry of N , so that if

Proof of Theorem 2
We consider a parabolic operator of the form By using as in the proof of Theorem 1, we can conclude that L V ( ) ≥ 0 on M T . From the parabolic maximum principle, we have

Existence results
Using the maximum principle obtained in the last section, we shall prove the existence of solutions of the Dirichlet problem for V T −harmonic maps.

Proof of
For simplicity of notation, we write it in a concise form  As in the proof of Theorem 3 in [5], by a continuity method that rests on the maximum principle Theorem 2, we can conclude the global existence of a solution u(x, t) of the above flow (4.1). This solution satisfies for some α > 0. Consequently, by the parabolic regularity theory, we have the uniform estimate , as in the proof of Theorem 2, the function satisfies By the ordinary maximum principle for functions, it follows that (see pp.178-179 in [17]) for any positive integer k and some t 0 > 0. Letting σ 1 → 0, then we obtain |u t | → 0 as t → +∞, from which together with (4.2), we have u subconverges to a V T −harmonic map u ∞ satisfying (1.7) and u ∞ (M) ⊂ B R ( p).
With the Schauder and higher regularity estimates, we can improve Theorem 3 to the following then the Dirichlet problem

Weyl harmonic maps (c.f. [14])
Let (M, [g], W ∇) be a Weyl manifold. According to the definition, there exists a 1-form such that W g = ⊗ g for any g ∈ [g]. Equivalently, W ∇ is defined by where ∇ is the Levi-Civita connection and the vector field dual to w.r.t. g. Let γ αβ , W γ αβ be the Christoffel symbols corresponding to ∇ and W ∇, respectively. Let (N , [g], W∇ ) be also a Weyl manifold, and correspondingly, we denote by the 1form, and k i j , W˜ k i j are the Christoffel symbols for the Levi-Civita connection∇ and Weyl connection W∇ , respectively. Let u : (M, [g], W ∇) → (N , [g], W∇ ) be the usual smooth map. Let Then, we have

Affine harmonic maps (c.f. [9,10])
Let (M, g,∇), (N , h,∇ ) both be affine manifolds, where∇ is a global flat and torsion-free connection on M and∇ is a torsion-free connection on N . Then, we have where˜ k i j are the Christoffel symbols of∇ . Regarding (M, g) and (N , h) as Riemannian manifolds, let γ αβ and i jk be the Christoffel symbols of the Levi-Civita connections ∇ and ∇ of (M, g) and (N , h), respectively. We then have the usual tension field Then, we have τ (g,∇,∇ ) = τ (g, ∇, ∇ ) + du(V ) + Tr g (du, du). Therefore,

Magnetic harmonic maps
We now consider a case that, in contrast to the previous ones, does not arise from a structure different from the Riemannian, but from on additional structure on a Riemannian manifold. Let ( m , g) be an m-dimensional compact oriented Riemannian manifold with nonempty boundary, (N , g) a Riemannian manifold of dimension n. Let u : ( m , g) → (N , g) be a map and Z ∈ (Hom( m T N, T N)) ∼ = ( m T *  N ⊗ T N). Consider the following system: where {e 1 , . . . , e m } is a positively oriented local orthonormal frame of m . In string theory, it can be interpreted as the motion equation of an (m − 1)-brane under an extrinsic magnetic force (c.f. [13]). In [13], the author obtained the global existence of the heat flow in onedimensional case. Using a similar method as above, in the two-dimensional case, we can obtain the following then the function : 2 → R defined by satisfies the maximum principle, namely Here ρ := d(u 1 , u 2 ), ρ i := d( p, u i ), i = 1, 2.
In particular, if u 1 = u 2 on the boundary ∂ 2 , then u 1 ≡ u 2 on 2 .
For the heat flow of magnetic harmonic maps, an analogous result holds. For T > 0, we set then the function : 2 T → R defined by (5.2) with 2 replaced by 2 T satisfies the maximum principle: In particular, if u 1 = u 2 on the boundary ∂ p 2 T , then u 1 ≡ u 2 on 2 T .
As an application of the above maximum principle, we obtain the existence of magnetic harmonic maps into a geodesic ball.
For appropriate σ and R, there exists a constant C 0 depending only on κ, σ, R and the geometry of N , such that if then the initial boundary value problem can be written as k j in place of T i jk in the proof of the results for V T -harmonic maps, we can conclude the above theorems for magnetic harmonic maps.

VT-harmonic maps from complete manifolds into geodesic balls
In this section, we shall establish the existence of V T -harmonic maps from complete noncompact manifolds into geodesic balls in complete Riemannian manifolds with sectional curvature bounded above by a positive constant.
Before proving the existence theorem, we first give the following Bochner formula: Let {e α } be a local orthonormal normal frame of M at the considered point. Since , and ∇ e α (Tr g T (du, du)) = ∇ e α (T (du(e β ), du(e β ))) , du(e β )).
Therefore, we get which implies that (6.1) holds.
Using the above Bochner formula and the estimate of V r in [6] (here r denotes the distance function on M), we establish the gradient estimate for V T -harmonic maps.

Theorem 9 Let (M m , g) be a complete noncompact Riemannian manifold with
Then, we have where C 6 > 0 is a constant depending only on m, n, κ,R, V , T .
Proof Let r , ρ be the respective distance functions on M and N from some fixed points p ∈ M, p ∈ N . Let B a (p) be a geodesic ball of radius a aroundp. Define ϕ := cos( √ κρ).
Then, the Hessian comparison theorem implies Denote ψ := |∇u| ϕ•u . Clearly, f achieves its maximum at some interior point of B a (p), say q. WLOG, we assume that ∇u(q) = 0. Then, from , we obtain at q: It follows from the above two inequalities that By formula (2.4) in [6] (see also [16]), we have where r 0 > 0 is a sufficiently small constant and C 0 := max ∂ B r 0 (p) V r . Let {e α } be a local orthonormal frame field of M and s the rank of u at the point. We shall compute in normal coordinates at the considered point of N . By Newton's inequality, we get α,β R N (du(e α ), du(e β ), du(e α ), du(e β )) = α =β R N (du(e α ), du(e β ), du(e α ), du(e β )) where we have used the fact that s 0 := min{m, n} ≥ s in the third " ≤ ". The Cauchy-Schwarz inequality gives us The formula (3.12) in [2] (see also [3]) implies that for any > 0 Choosing = m, then we have By the V T -harmonic map equation (1.1), it is easy to see that Hence, from the Bochner formula (6.1), we obtain For simplicity in the following computation, we denote By direct calculation, we have The Cauchy-Schwarz implies that From the above two inequalities and (6.4), we get . Since and from (6.3), (6.11) Therefore, from (6.6), (6.7), (6.9)-(6.11), we obtain Since the condition (6.2) tells us there exists a constant ε 0 > 0, such that Choosing ε 2 = ∇T 2 L ∞ +1 4ε 0 , then we have Therefore, it follows from (6.12) that Note the elementary fact that if ax 2 − bx − c ≤ 0 with a, b, c all positive, then Hence, at the point q, 2 Ar 2 −2 Ar 0 r + 2 C 0 r + 2 a 2 − r 2 + 8r 2 (a 2 − r 2 ) 2 − C 2 .
From this, we can derive the upper bound of f , and it is easy to conclude that at every point of B a 2 (p), we have where C 6 > 0 is a constant depending only on m, n, κ, R, V , T . For any fixed x ∈ M, letting a → ∞ in (6.13), we obtain |∇u| ≤ C 6 ( √ A + 1).

Proof of Theorem 4
For any compact set K ⊂ M, there exists an integer i 0 > 0, such that K ⊂ i for i > i 0 . Then, by (6.8), where A is a positive constant depending only on the bounds for Ricci curvature of K and V C 1 (K ) . Since K is compact, there exist finitely many such geodesic balls {B a j (p j )} k 0 j=1 ⊂ M, such that ∪ k 0 j=1 B a j (p j ) ⊃ K . Hence, for any q ∈ K , there is a geodesic ball, say B a j 0 (p j 0 ) (1 ≤ j 0 ≤ k 0 ), containing q. Then, by Theorem 9, we can conclude that |∇u i |(q) ≤ C 7 j 0 ( A + 1). Hence, where C 8 is a positive constant independent of i. Then, by the standard elliptic theory, u i subconverges to a V T -harmonic map u ∈ C ∞ (M, N ) with u(M) ⊂ B R ( p) and u is homotopic to u 0 .
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.