Biwave Maps into Manifolds

Abstract

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if 𝑓 is a biwave map into a Riemannian manifold under certain circumstance, then 𝑓 is a wave map. We verify that if 𝑓 is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then 𝑓 is a wave map. We finally obtain a theorem involving an unstable biwave map.

1. Introduction

Harmonic maps between Riemannian manifolds were first introduced and established by Eells and Sampson [1] in 1964. Afterwards, there were two reports on harmonic maps by Eells and Lemaire [2, 3] in 1978 and 1988. Biharmonic maps, which generalized harmonic maps, were first studied by Jiang [4, 5] in 1986. In this decade, there has been progress in biharmonic maps made by Caddeo et al. [6, 7], Loubeau and Oniciuc [8], Montaldo and Oniciuc [9], Chiang and Wolak [10], Chiang and Sun [11, 12], Chang et al. [13], Wang [14, 15], and so forth.

Wave maps are harmonic maps on Minkowski spaces, and their equations are the second-order hyperbolic systems of partial differential equations, which are related to Einstein’s equations and Yang-Mills fields. In recent years, there have been many new developments involving local well-posedness and global-well posedness of wave maps into Riemannian manifolds achieved by Klainerman and Machedon [16, 17], Shatah and Struwe [18, 19], Tao [20, 21], Tataru [22, 23], and so forth. Furthermore, Nahmod et al. [24] also studied wave maps from 𝑅×𝑅^𝑚 into (compact) Lie groups or Riemannian symmetric spaces, that is, gauged wave maps when 𝑚≥4, and established global existence and uniqueness, provided that the initial data are small. Moreover, Chiang and Yang [25] , Chiang and Wolak [26] have investigated exponential wave maps and transversal wave maps.

Bi-Yang-Mills fields, which generalize Yang-Mills fields, have been introduced by Ichiyama et al. [27] recently. The following connection between bi-Yang Mills fields and biwave equations motivates one to study biwave maps.

Let 𝑃 be a principal fiber bundle over a manifold 𝑀 with structure group 𝐺 and canonical projection 𝜋, and let 𝒢 be the Lie algebra of 𝐺. A connection 𝐴 can be considered as a 𝒢-valued 1-form 𝐴=𝐴_𝜇(𝑥)𝑑𝑥^𝜇 locally. The curvature of the connection 𝐴 is given by the 2-form 𝐹=𝐹_𝜇𝜈𝑑𝑥^𝜇𝑑𝑥^𝜈 with 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇+[𝐴_𝜇,𝐴_𝜈]. (1.1) The bi-Yang-Mills Lagrangian is defined 𝐿₂1(𝐴)=2_𝑀‖𝛿𝐹‖²𝑑𝑣_𝑀, (1.2) where 𝛿 is the adjoint operator of the exterior differentiation 𝑑 on the space of 𝐸-valued smooth forms on 𝑀 (𝐸=End(𝑃), the endormorphisms of 𝑃). Then the Euler-Lagrange equation describing the critical point of (1.2) has the form (𝛿𝑑+𝐹)𝛿𝐹=0, (1.3) which is the bi-Yang-Mills system. In particular, letting 𝑀=𝑅×𝑅² and 𝐺=SO(2), the group of orthogonal transformations on 𝑅², we have that 𝐴_𝜇(𝑥) is a 2×2 skew symmetric matrix 𝐴_𝜇^𝑖𝑗. The appropriate equivariant ansatz has the form 𝐴_𝜇^𝑖𝑗(𝑥)=(𝛿^𝑖_𝜇𝑥^𝑗−𝛿^𝑗_𝜇𝑥^𝑖)ℎ(𝑡,|𝑥|), (1.4) where ℎ∶𝑀→𝑅 is a spatially radial function. Setting 𝑢=𝑟²ℎ and 𝑟=|𝑥|, the bi-Yang-Mills system (1.3) becomes the following equation for 𝑢(𝑡,𝑟): 𝑢_{𝑡𝑡𝑡𝑡}−𝑢_{𝑟𝑟𝑟𝑟}−3𝑟𝑢_𝑟𝑟𝑟+2𝑟²𝑢_𝑟𝑟−2𝑟³𝑢_𝑟=𝑘(𝑡,𝑟), (1.5) which is a linear nonhomogeneous biwave equation, where 𝑘(𝑡,𝑟) is a function of 𝑡 and 𝑟.

Biwave maps are biharmonic maps on Minkowski spaces. It is interesting to study biwave maps since their equations are the fourth-order hyperbolic systems of partial differential equations, which generalize wave maps. This is the first attempt to study biwave maps and their relationship with wave maps. There are interesting and difficult problems involving local well posedness and global well posedness of biwave maps into Riemannian manifolds or Lie groups (or Riemannian symmetric spaces), that is, gauged biwave maps for future exploration.

In Section 2, we compute the first variation of the bi-energy functional of a biharmonic map using tensor technique, which is different but much easier than Jiang’s [4] original computation. In Section 3, we prove in Theorem 3.3 that if  𝑓∶𝑅^𝑚,1→𝑁₁ is a biwave map and  𝑓₁∶𝑁₁→𝑁₂  is a totally geodesic map, then  𝑓₁∘𝑓∶𝑅^𝑚,1→𝑁₂  is a biwave map. Then we can apply this theorem to provide many biwave maps (see Example 3.4). We also can construct biwave nonwave maps as follow: Let  ℎ∶Ω⊂𝑅^𝑚,1→𝑆^𝑛(1/√2) be a wave map on a compact domain and let  𝑖∶𝑆^𝑛(1/√2)→𝑆^𝑛+1(1)  be an inclusion map. The map  𝑓=𝑖∘ℎ∶Ω→𝑆^𝑛+1(1)  is a biwave nonwave map if and only if  ℎ  has constant energy density, compare with Theorem 3.5. Afterwards, we show that if  𝑓∶Ω→𝑁 is a biwave map on a compact domain into a Riemannian manifold satisfying −|𝜏_□𝑓|²_𝑡+^𝑚_𝑖=1|𝜏_□𝑓|²_𝑥𝑖−𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇≥0, (1.6) then 𝑓 is a wave map (cf. Theorem 3.6). This theorem is different than the theorem obtained by Jiang [4]: if  𝑓  is a biharmonic map from a compact manifold into a Riemannian manifold with nonpositive curvature, then  𝑓  is a harmonic map. In Section 4, we verify that if  𝑓  is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then  𝑓  is a wave map (cf. Theorem 4.5). We also prove that if  ℎ∶Ω→𝑆^𝑛(1/√2)  is a wave map on a compact domain with constant energy density, then  𝑓=𝑖∘ℎ∶Ω→𝑆^𝑛+1(1) is an unstable biwave map (cf. Theorem 4.7).

2. Biharmonic Maps

A biharmonic map 𝑓∶(𝑀^𝑚,𝑔_𝑖𝑗)→(𝑁^𝑛,ℎ_𝛼𝛽) from an 𝑚-dimensional Riemannian manifold 𝑀 into an 𝑛-dimensional Riemannian manifold 𝑁 is the critical point of the bi-energy functional 𝐸₂1(𝑓)=2_𝑀‖(𝑑+𝑑^∗)²𝑓‖²1𝑑𝑣=2_𝑀‖(𝑑^∗𝑑)𝑓‖²1𝑑𝑣=2_𝑀‖𝜏(𝑓)‖²𝑑𝑣, (2.1) where 𝑑𝑣 is the volume form on 𝑀.

Notations
𝑑^∗ is the adjoint of 𝑑 and 𝜏(𝑓)=trace(𝐷𝑑𝑓)=(𝐷𝑑𝑓)(𝑒_𝑖,𝑒_𝑖)=(𝐷_𝑒𝑖𝑑𝑓)(𝑒_𝑖) is the tension field. Here 𝐷 is the Riemannian connection on 𝑇^∗𝑀⊗𝑓⁻¹𝑇𝑁 induced by the Levi-Civita connections on 𝑀 and 𝑁, and {𝑒_𝑖} is the local frame at a point of 𝑀. The tension field has components 𝜏(𝑓)^𝛼=𝑔^𝑖𝑗𝑓^𝛼_𝑖|𝑗=𝑔^𝑖𝑗(𝑓^𝛼_𝑖𝑗−Γ^𝑘_𝑖𝑗𝑓^𝛼_𝑘+Γ^′𝛼_𝛽𝛾𝑓^𝛽_𝑖𝑓^𝛾_𝑗), (2.2a) where Γ^𝑘_𝑖𝑗 and Γ^′𝛾_𝛼𝛽 are the Christoffel symbols on 𝑀 and 𝑁, respectively.

In order to compute the Euler-Lagrange equation of the bi-energy functional, we consider a one-parameter family of maps {𝑓_𝑡}∈𝐶^∞(𝑀×𝐼,𝑁) from a compact manifold 𝑀 (without boundary) into a Riemannian manifold 𝑁. Here 𝑓_𝑡(𝑥) is the endpoint of a segment starting at 𝑓(𝑥)(=𝑓₀(𝑥)), determined in length and direction by the vector field 𝑓(̇𝑥) along 𝑓(𝑥). For a nonclosed manifold 𝑀, we assume that the compact support of 𝑓(̇𝑥) is contained in the interior of 𝑀 (we need this assumption when we compute 𝜏(𝑓) by applying the divergence theorem). Then we have 𝑑𝐸𝑑𝑡²(𝑓_𝑡)|_𝑡=0=𝐸̇₂(𝑓)=_𝑀(𝐷_𝑡𝜏𝑓,𝜏𝑓)_𝑡=0𝑑𝑣. (2.3)

Let 𝜉=𝜕𝑓_𝑡/𝜕𝑡. The components of 𝐷_𝑡𝜏𝑓 are 𝑓^𝛼_{𝑖|𝑗|𝑡}=(𝜕𝑓^𝛼_𝑖|𝑗/𝜕𝑡)+Γ^′𝛼_𝜇𝛾𝑓^𝜇_𝑖|𝑗𝜉^𝛾. We can use the curvature formula on 𝑀×𝐼→𝑁 and get 𝑓^𝛼_{𝑖|𝑗|𝑡}=𝑓^𝛼_{𝑖|𝑡|𝑗}+𝑅^′𝛼_𝛽𝛾𝜇𝑓^𝛽_𝑖𝑓^𝛾_𝑗𝜉^𝜇, (2.4) where 𝑅^′ is the Riemannian curvature of 𝑁. But 𝑓^𝛼_𝑖|𝑡=𝑓^𝛼_𝑡|𝑖=𝜉^𝛼_|𝑖, therefore, 𝐷_𝑡𝜏𝑓 has components 𝜉^𝛼_|𝑖|𝑗+𝑅^′𝛼_𝛽𝛾𝜇𝑓^𝛽_𝑖𝑓^𝛾_𝑗𝜉^𝜇. We can rewrite (2.3) as 𝑑𝐸𝑑𝑡²(𝑓_𝑡)|_𝑡=0=_𝑀(𝐽_𝑓(𝜏𝑓),𝜏𝑓)𝑑𝑣, (2.5) where 𝐽^𝛼_𝑓(𝜉)=𝑔^𝑖𝑗𝜉^𝛼_|𝑖|𝑗+𝑔^𝑖𝑗𝑅^′𝛼_𝛽𝛾𝜇𝑓^𝛽_𝑖𝑓^𝛾_𝑗𝜉^𝜇=Δ𝜉^𝛼+𝑅^′𝛼(𝑑𝑓,𝑑𝑓)𝜉 (2.6) is a linear equation for 𝜉(=𝜏(𝑓)), and Δ(𝜉)=𝐷^∗𝐷(𝜉) is an operator from 𝑓⁻¹𝑇𝑁 to 𝑓⁻¹𝑇𝑁. Solutions of 𝐽_𝑓(𝜉)=0 are called Jacobi fields. Hence, we obtain the following definition from (2.3), (2.5), and (2.6).

Definition 2.1. 𝑓∶𝑀→𝑁 is a biharmonic map if and only if the bitension field 𝜏₂(𝑓)^𝛼=𝐽_𝑓(𝜏𝑓)^𝛼=Δ𝜏(𝑓)^𝛼+𝑅^′𝛼(𝑑𝑓,𝑑𝑓)𝜏(𝑓)=𝑔^𝑖𝑗(𝑓^𝛼_𝑖𝑗−Γ^𝑘_𝑖𝑗𝑓^𝛼_𝑘+Γ^′𝛼_𝛽𝛾𝑓^𝛽_𝑖𝑓^𝛾_𝑗)+𝑔^𝑖𝑗𝑅^′𝛼_𝛽𝛾𝜇𝑓^𝛽_𝑖𝑓^𝛾_𝑗𝜏(𝑓)^𝜇=0, (2.7) that is, the tension field 𝜏(𝑓), is a Jacobi field.
If 𝜏(𝑓)=0, then 𝜏₂(𝑓)=0. Thus, harmonic maps are obviously biharmonic. Biharmonic maps satisfy the fourth-order elliptic systems of PDEs, which generalize harmonic maps. Our computation for the first variation of the bi-energy functional presented here using tensor technique is different but much easier than Jiang’s [4] original computation (it took him four pages).
Caddeo et al. [7] showed that a biharmonic curve on a surface of nonpositive Gaussian curvature is a geodesic (i.e., is harmonic) and gave examples of biharmonic nonharmonic curves on spheres, ellipses, unduloids, and nodoids.

Theorem 2.2 (see [4]). Let 𝑓∶𝑀^𝑚→𝑆^𝑚+1(1) be an isometric embedding of an 𝑚-dimensional compact Riemannian manifold 𝑀 into an (𝑚+1)-dimensional unit sphere 𝑆^𝑚+1(1) with nonzero constant mean curvature. The map 𝑓 is biharmonic if and only if ‖𝐵(𝑓)‖²=𝑚, where 𝐵(𝑓) is the second fundamental form of 𝑓.

Example 2.3. In 𝑆^𝑚+1(1), the compact hypersurfaces, whose Gauss maps are isometric embeddings, are the Clifford surfaces [28]: 𝑀^𝑚_𝑘(1)=𝑆^𝑘(1√2)×𝑆^𝑚−𝑘(1√2),0≤𝑘≤𝑚. (2.8) Let 𝑓∶𝑀^𝑚_𝑘(1)→𝑆^𝑚+1(1) be a standard embedding such that 𝑘≠𝑚/2. Because ‖𝐵(𝑓)‖²=𝑘+𝑚−𝑘=𝑚 and 𝜏(𝑓)=𝑘−(𝑚−𝑘)=2𝑘−𝑚≠0,𝑓 is a biharmonic nonharmonic map by Theorem 2.2.

3. Biwave Maps

Let 𝑅^𝑚,1 be an 𝑚+1 dimensional Minkowski space 𝑅×𝑅^𝑚 with the metric (𝑔_𝑖𝑗)=(−1,1,…,1) and the coordinates 𝑥⁰=𝑡,𝑥¹,𝑥²,…,𝑥^𝑚 and let (𝑁,ℎ_𝛼𝛽) be an 𝑛-dimensional Riemannian manifold. A wave map is a harmonic map on the Minkowski space 𝑅^𝑚,1 with the energy functional 1𝐸(𝑓)=2_𝑅^𝑚,1(−|𝑓_𝑡|²+|∇_𝑥𝑓|²1)𝑑𝑡𝑑𝑥=2_𝑅^𝑚,1ℎ_𝛼𝛽(−𝑓^𝛼_𝑡𝑓^𝛽_𝑡+^𝑚_𝑖=1𝑓^𝛼_𝑖𝑓^𝛽_𝑖)𝑑𝑡𝑑𝑥_𝑖. (3.1) The Euler-Lagrange equation describing the critical point of (3.1) is 𝜏^𝛼_□(𝑓)=□𝑓^𝛼+Γ^′𝛼_𝛽𝛾(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)=0, (3.2) where □=−(𝜕²/𝜕𝑡²)+Δ_𝑥 is the wave operator on 𝑅^𝑚,1 and Γ^′𝛼_𝛽𝛾 are the Christoffel symbols of 𝑁. 𝑓 is a wave map iff the wave field 𝜏^𝛼_□(𝑓) (i.e., the tension field on a Minkowski space) vanishes. The wave map equation is invariant with respect to the dimensionless scaling 𝑓(𝑡,𝑥)→𝑓(𝑐𝑡,𝑐𝑥),𝑐∈𝑅. But, the energy is scale invariant in dimension 𝑚=2.

If 𝑓∶𝑅^𝑚,1→𝑁 is a smooth map from a Minkowski space 𝑅^𝑚,1 into a Riemannian manifold 𝑁, then the bi-energy functional is, from (2.1), 𝐸₂1(𝑓)=2_𝑅^𝑚,1‖(𝑑+𝑑^∗)²𝑓‖²=1𝑑𝑡𝑑𝑥2_𝑅^𝑚,1‖𝑑^∗𝑑𝑓‖²1𝑑𝑡𝑑𝑥=2_𝑅^𝑚,1‖𝜏_□(𝑓)‖²𝑑𝑡𝑑𝑥. (3.3) The Euler-Lagrange equation describing the critical point of (3.3), from (2.5), is (𝜏₂)_□(𝑓)=𝐽_𝑓(𝜏_□𝑓)=Δ𝜏_□(𝑓)+𝑅^′(𝑑𝑓,𝑑𝑓)𝜏_□(𝑓)=0. (3.4)

Definition 3.1. 𝑓∶𝑅^𝑚,1→𝑁 from a Minkowski space into a Riemannian manifold is a biwave map if and only if the biwave field (i.e., the bitension field on a Minkowski space), (𝜏₂)_□(𝑓)^𝛼=𝐽_𝑓(𝜏_□𝑓)^𝛼=Δ𝜏_□(𝑓)^𝛼+𝑅^′𝛼(𝑑𝑓,𝑑𝑓)𝜏_□(𝑓)=□𝜏_□(𝑓)^𝛼+Γ^′𝛼_𝜇𝛾(−𝜏_□(𝑓)^𝜇_𝑡𝜏_□(𝑓)^𝛾_𝑡+^𝑚_𝑖=1𝜏_□(𝑓)^𝜇_𝑖𝜏_□(𝑓)^𝛾_𝑖)+𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇=0, (3.5) that is, the wave field 𝜏_□(𝑓), is a Jacobi field on the Minkowski space.
Biwave maps satisfy the fourth-order hyperbolic systems of PDEs, which generalize wave maps. If 𝜏_□(𝑓)=0, then (𝜏₂)_□(𝑓)=0. Waves maps are obviously biwave maps, but biwave maps are not necessarily wave maps.

Example 3.2. Let 𝑢∶𝑅^𝑚,1→𝑅 be a function defined on a Minkowski space satisfying the following conditions: □²𝑢(𝑡,𝑥)=□(□𝑢)=𝑢_{𝑡𝑡𝑡𝑡}−2𝑢_{𝑡𝑡𝑥𝑥}+𝑢_{𝑥𝑥𝑥𝑥}=0,(𝑡,𝑥)∈(0,∞)×𝑅^𝑚,𝑢=𝑢₀,𝑢_𝑡=𝑢₁,𝑢=𝑢₀,𝜕𝜕𝑡𝑢=𝜕𝑢𝜕𝑡=𝑢₁,(𝑡,𝑥)∈{𝑡=0}×𝑅^𝑚, (3.6) where the initial data 𝑢₀ and 𝑢₁ are given. Since this is a fourth-order homogeneous linear biwave equation with constant coefficients, it is well known that 𝑢(𝑡,𝑥) can be solved by [18, 29].

Let 𝑓∶𝑅^𝑚,1→𝑁₁ be a smooth map from a Minkowski space 𝑅^𝑚,1 into a Riemannian manifold 𝑁₁ and let 𝑓₁∶𝑁₁→𝑁₂ be a smooth map between two Riemannian manifolds 𝑁₁ and 𝑁₂. Then the composition 𝑓₁∘𝑓∶𝑅^𝑚,1→𝑁₂ is a smooth map. Since 𝑅^𝑚,1 is a semi-Riemannian manifold (i.e., a pseudo-Riemannian manifold), we can define a Levi-Civita connection on 𝑅^𝑚,1 by O’Neill [30]. Let 𝐷, 𝐷^′,𝐷,𝐷^′,𝐷^”,𝐷,𝐷^′,𝐷^” be the connections on 𝑅^𝑚,1,𝑇𝑁₁,𝑓⁻¹𝑁₁,𝑓₁⁻¹𝑇𝑁₂,(𝑓₁∘𝑓)⁻¹𝑇𝑁₂,𝑇^∗𝑅^𝑚,1⊗𝑓⁻¹𝑇𝑁₁,𝑇^∗𝑁₁⊗𝑓₁⁻¹𝑇𝑁₂,𝑇^∗𝑅^𝑚,1⊗(𝑓₁∘𝑓)⁻¹𝑇𝑁₂, respectively, and let 𝑅^𝑁2(,),𝑅^{𝑓1−1𝑇𝑁2}(,) be the curvatures on 𝑇𝑁₂,𝑓₁⁻¹𝑇𝑁₂, respectively. We first have the following two formulas: 𝐷^”_𝑋𝑑(𝑓₁∘𝑓)(𝑌)=(𝐷^′_{𝑑𝑓(𝑋)}𝑑𝑓₁)𝑑𝑓(𝑌)+𝑑𝑓₁∘𝐷_𝑋𝑑𝑓(𝑌), (3.7a) for 𝑋,𝑌∈𝑅^𝑚,1, and 𝑅^𝑁2(𝑑𝑓₁(𝑋^′),𝑑𝑓₁(𝑌^′))𝑑𝑓₁(𝑍^′)=𝑅^{𝑓1−1𝑇𝑁2}(𝑋^′,𝑌^′)𝑑𝑓₁(𝑍^′), (3.7b) for 𝑋^′,𝑌^′,𝑍^′∈Γ(𝑇𝑁₁).

Theorem 3.3. If 𝑓∶𝑅^𝑚,1→𝑁₁ is a biwave map and 𝑓₁∶𝑁₁→𝑁₂ is totally geodesic between two Riemannian manifolds 𝑁₁ and 𝑁₂, then the composition 𝑓₁∘𝑓∶𝑅^𝑚,1→𝑁₂ is a biwave map.

Proof. Let 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 be the coordinates of a point 𝑝 in 𝑅^𝑚,1 and let 𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),𝑒₂=(0,1,0,…,0),…,𝑒_𝑚=(0,…,0,1) be the frame at 𝑝. We know from [4] that 𝐷^”∗𝐷^”=𝐷^”_𝑒𝑘𝐷^”_𝑒𝑘−𝐷^”_{𝐷𝑒𝑘𝑒𝑘}. Since 𝑓₁ is totally geodesic, we have 𝜏_□(𝑓₁∘𝑓)=𝑑𝑓₁∘𝜏_□(𝑓) by applying the chain rule of the wave field to 𝑓₁∘𝑓 as [1]. Then we get 𝐷^”∗𝐷^”_𝜏□(𝑓₁∘𝑓)=𝐷^”∗𝐷^”(𝑑𝑓₁∘𝜏_□=(𝑓))𝐷^”_𝑒𝑘𝐷^”_𝑒𝑘(𝑑𝑓₁∘𝜏_□(𝑓))−𝐷^”_{𝐷𝑒𝑘𝑒𝑘}(𝑑𝑓₁∘𝜏_□(𝑓)). (3.8) Recalling that 𝜏_□(𝑓)=𝐷_𝑒𝑗𝑑𝑓(𝑒_𝑗), we derive from (3.7a) that 𝐷^”_𝑒𝑘(𝑑𝑓₁∘𝜏_□(𝑓))=𝐷^”_𝑒𝑘(𝑑𝑓₁∘𝐷_𝑒𝑗𝐷𝑑𝑓(𝑒𝑗))=(𝐷^′_{𝑒𝑗𝑑𝑓(𝑒𝑘)}𝑑𝑓₁)(𝐷_𝑒𝑗𝑑𝑓(𝑒_𝑗))+𝑑𝑓₁∘𝐷_𝑒𝑘(𝐷_𝑒𝑗𝑑𝑓(𝑒_𝑗))=𝑑𝑓₁∘𝐷_𝑒𝑘𝜏_□(𝑓), (3.9) since 𝑓₁ is totally geodesic. Therefore, we have 𝐷^”_𝑒𝑘𝐷^”_𝑒𝑘(𝑑𝑓₁∘𝜏_□(𝑓))=𝐷^”_𝑒𝑘(𝑑𝑓₁∘𝐷_𝑒𝑘𝜏(𝑓))=𝑑𝑓₁∘𝐷_𝑒𝑘𝐷_𝑒𝑘𝜏_□(𝑓),𝐷^”_{𝐷𝑒𝑘𝑒𝑘}(𝑑𝑓₁∘𝜏(𝑓))=𝑑𝑓₁∘𝐷_{𝐷𝑒𝑘𝑒𝑘}𝜏_□(𝑓). (3.10) Substituting (3.10) into (3.8), we arrive at 𝐷^”∗𝐷^”𝜏_□(𝑓₁∘𝑓)=𝑑𝑓₁∘𝐷^∗𝐷𝜏_□(𝑓), (3.11) where 𝐷^∗𝐷=𝐷_𝑒𝑘𝐷_𝑒𝑘−𝐷_{𝐷𝑒𝑘𝑒𝑘}.
On the other hand, we have by (3.7b) 𝑅^𝑁2(𝑑(𝑓₁∘𝑓)(𝑒_𝑖),𝜏_□(𝑓₁∘𝑓))𝑑(𝑓₁∘𝑓)(𝑒_𝑖)=𝑅^{𝑓1−1𝑇𝑁2}(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝑑𝑓₁(𝑑𝑓(𝑒_𝑖))=𝑑𝑓₁∘𝑅^𝑁1(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝑑𝑓(𝑒_𝑖). (3.12) We obtain from (3.11) and (3.12) 𝐷^”∗𝐷^”(𝑓₁∘𝑓)+𝑅^𝑁2(𝑑(𝑓₁∘𝑓)(𝑒_𝑖),𝜏_□(𝑓₁∘𝑓))𝑑(𝑓₁∘𝑓)(𝑒_𝑖)=𝑑𝑓₁∘[𝐷^∗𝐷𝜏_□(𝑓)+𝑅^𝑁1(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝑑𝑓(𝑒_𝑖)], (3.13) that is, (𝜏₂)_□(𝑓₁∘𝑓)=𝑑𝑓₁∘(𝜏₂)_□(𝑓). Hence, if 𝑓 is a biwave map and 𝑓₁ is totally geodesic, then 𝑓₁∘𝑓 is a biwave map. Note that the total geodesicity of 𝑓₁ cannot be weakened into a harmonic or biharmonic map.

Example 3.4. Let 𝑁₁ be a submanifold of 𝑁. Are the biwave maps into 𝑁₁ also biwave maps into 𝑁? The answer is affirmative iff 𝑁₁ is a totally geodesic submanifold of 𝑁, that is, 𝑁₁ geodesics are 𝑁 geodesics. 𝑁₁ is a geodesic 𝛾(𝑡)=(𝛾¹,…,𝛾^𝑛)∶𝑅→𝑁⊂𝑅^𝑛 with |𝛾̇(𝑡)|=1 iff 𝛾̇ is parallel, that is, 𝐷_{𝜕/𝜕𝑡}𝛾̇=0 iff 𝛾̈⟂𝑇_𝛾𝑁. For a map 𝑢∶𝑅^𝑚,1→𝑅, letting 𝑓=𝛾∘𝑢=(𝑓¹,…,𝑓^𝑛)∶𝑅^𝑚,1→𝑁⊂𝑅^𝑛, we have by (3.13) the following: (𝜏₂)_□(𝑓)=𝑑𝛾∘(𝜏₂)_□(𝑢)=𝑑𝛾∘□²𝑢, (3.14) since 𝛾 is a geodesic. Hence, 𝑓=𝛾∘𝑢 is a biwave map if and only if 𝑢 solves the fourth-order homogeneous linear biwave equation □²𝑢=0 as in Example 3.2. It follows from Theorem 3.3 that there are many biwave maps 𝑓∶𝑅^𝑚,1→𝑁 provided by geodesics of 𝑁.

We also can construct examples of biwave nonwave maps from some wave maps with constant energy using Theorem 3.5. Let 𝑆^𝑛(1√2)=𝑆^𝑛(1√21)×{√2}={(𝑥₁,𝑥₂,…,𝑥_𝑛+1,1√2)∣𝑥²₁+⋯+𝑥²_𝑛+1=12} (3.15) be a hypersphere of 𝑆^𝑛+1(1). Then 𝑆^𝑛(1/√2) is a biharmonic nonminimal submanifold of 𝑆^𝑛+1(1) by Theorem 2.2 and Example 2.3. Let 𝜁=(𝑥₁,…,𝑥_𝑛+1,−1/√2) be a unit section of the normal bundle of 𝑆^𝑛(1/√2) in 𝑆^𝑛+1(1). Then the second fundamental form of the inclusion 𝑖∶𝑆^𝑛(1/√2)→𝑆^𝑛+1(1) is 𝐵(𝑋,𝑌)=𝐷𝑑𝑖(𝑋,𝑌)=−(𝑋,𝑌)𝜁. By computation, the tension field of 𝑖 is 𝜏(𝑖)=−𝑛𝜁, and the bitension field is 𝜏₂(𝑖)=0.

Theorem 3.5. Let ℎ∶Ω→𝑆^𝑛(1/√2) be a nonconstant wave map on a compact space-time domain Ω⊂𝑅^𝑚,1 and let 𝑖∶𝑆^𝑛(1/√2)→𝑆^𝑛+1(1) be an inclusion. The map 𝑓=𝑖∘ℎ∶𝑅^𝑚,1→𝑆^𝑛+1(1) is a biwave nonwave map if and only if ℎ has constant energy density 𝑒(ℎ)=(1/2)|𝑑ℎ|².

Proof. Let 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 be the coordinate of a point 𝑝 in Ω⊂𝑅^𝑚,1 and let 𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),𝑒₂=(0,1,0,…,0),…,𝑒_𝑚=(0,…,0,1) be the frame at 𝑝. Recall that 𝜁 is the unit section of the normal bundle. By applying the chain rule of the wave field to 𝑓=𝑖∘ℎ, we have 𝜏_□(𝑓)=𝑑𝑖(𝜏_□(ℎ))+trace𝐷𝑑𝑖(𝑑ℎ,𝑑ℎ)=−2𝑒(ℎ)𝜁, (3.16) since ℎ is a wave map. We can derive the following at the point 𝑝 by straightforward calculation: 𝐷^∗𝐷𝜏_□(𝑓)=−𝐷^𝑓_𝑒𝑖𝐷^𝑓_𝑒𝑖𝜏_□(𝑓)=−𝐷^𝑓_𝑒𝑖𝐷^𝑓_𝑒𝑖(−2𝑒(ℎ)𝜁)=2(𝑒_𝑖𝑒_𝑖𝑒(ℎ))𝜁−2𝑒(ℎ)(𝑑ℎ(𝑒_𝑖),𝑑ℎ(𝑒_𝑖))𝜁+4𝑑𝑓[(𝑒_𝑖𝑒(ℎ))𝑒_𝑖]+2𝑒(ℎ)𝐷𝑑ℎ(𝑒_𝑖,𝑒_𝑖𝑅),_𝑆𝑛+1(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝑑𝑓(𝑒_𝑖)=−(𝑑ℎ(𝑒_𝑖),𝑑ℎ(𝑒_𝑖))𝜏(𝑓)=2(𝑑ℎ(𝑒_𝑖),𝑑ℎ(𝑒_𝑖))𝑒(ℎ)𝜁. (3.17) Therefore, we obtain 𝜏_2□(𝑓)=−2(Δ𝑒(ℎ))𝜁+4𝑑𝑓(grad𝑒(ℎ)). (3.18) Suppose that 𝑓=𝑖∘ℎ∶Ω→𝑆^𝑛(1/√2)×{1/√2}→𝑆^𝑛+1(1) is a biwave nonwave map (𝜏_□(𝑓)≠0). As the 𝜁-part of 𝜏_2□(𝑓), Δ𝑒(ℎ) vanishes, which implies that 𝑒(ℎ) is constant since Ω is compact. The converse is obvious.

Let 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 be the coordinates of a point in a compact space-time domain Ω⊂𝑅^𝑚,1 and 𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),𝑒₂=(0,1,0,…,0),…,𝑒_𝑚=(0,…,0,1) be the frame at the point. Suppose that 𝑓∶Ω→𝑁 is a biwave map from a compact domain Ω into a Riemannian manifold 𝑁 such that the compact supports of 𝜕𝑓/𝜕𝑥_𝑖 and 𝐷_𝑒𝑖𝜕𝑓/𝜕𝑥_𝑖 are contained in the interior of Ω.

Theorem 3.6. If 𝑓∶Ω→𝑁 is a biwave map from a compact domain into a Riemannian manifold such that −|𝜏_□𝑓|²_𝑡+^𝑚_𝑖=1|𝜏_□𝑓|²_𝑥𝑖−𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇≥0, (3.19) then 𝑓 is a wave map.

Proof. Since 𝑓 is a biwave map, we have by (3.4) (𝜏₂)_□(𝑓)=Δ𝜏_□(𝑓)+𝑅^′(𝑑𝑓,𝑑𝑓)𝜏_□(𝑓). (3.20) Recall that 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 are the coordinates of a point in Ω⊂𝑅^𝑚,1 and 𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),𝑒₂=(0,1,0,…,0),…,𝑒_𝑚=(0,…,0,1). We compute 12Δ‖𝜏_□(𝑓)‖²=(𝐷_𝑒𝑖𝜏_□(𝑓),𝐷_𝑒𝑖𝜏_□(𝑓))+(𝐷^∗𝐷𝜏_□(𝑓),𝜏_□=(𝑓))_𝑚𝑖=0(𝐷_𝑒𝑖𝜏_□(𝑓),𝐷_𝑒𝑖𝜏_□(𝑓))−(𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇,𝜏_□(𝑓))=−|𝜏_□𝑓|²_𝑡+^𝑚_𝑖=1|𝜏_□𝑓|²_𝑥𝑖−(𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇,𝜏_□(𝑓)). (3.21) By applying the Bochner’s technique from (3.19) and the assumption that the compact supports of 𝜕𝑓/𝜕𝑥_𝑖 and 𝐷_𝑒𝑖𝜕𝑓/𝜕𝑥_𝑖 are contained in the interior of Ω, we know that ‖𝜏_□(𝑓)‖² is constant, that is, 𝑑𝜏_□(𝑓)=0. If we use the identity _Ωdiv(𝑑𝑓,𝜏(𝑓))𝑑𝑧=_Ω(|𝜏(𝑓)|²+(𝑑𝑓,𝑑𝜏(𝑓)))𝑑𝑧,𝑧=(𝑡,𝑥) (3.22) and the fact 𝑑𝜏_□(𝑓)=0, then we can conclude that 𝜏_□(𝑓)=0 by applying the divergence theorem.

Corollary 3.7. If 𝑓∶Ω→𝑁 is a biwave map on a compact domain such that ∑^𝑚_𝑖=1|𝜏_□𝑓|²_𝑥𝑖≥|𝜏_□𝑓|²_𝑡 and 𝑅^′𝛼_𝛽𝛾𝜇(−𝑓^𝛽_𝑡𝑓^𝛾_𝑡+∑^𝑚_𝑖=1𝑓^𝛽_𝑖𝑓^𝛾_𝑖)𝜏_□(𝑓)^𝜇≤0, then 𝑓 is a wave map.

Proof. The result follows from (3.19) immediately.

4. Stability of Biwave Maps

Let 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 be the coordinates of a point in a compact space-time domain Ω⊂𝑅^𝑚,1 and let 𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),…,𝑒_𝑚=(0,0,…,1) be the frame at the point. Suppose that 𝑓∶Ω→𝑁 is a biwave map from a compact space-time domain Ω into a Riemannian manifold 𝑁 such that the compact supports of 𝜕𝑓/𝜕𝑥_𝑖 and 𝐷_𝑒𝑖𝜕𝑓/𝜕𝑥_𝑖 are contained in the interior of Ω. Let 𝑉∈Γ(𝑓⁻¹𝑇𝑁) be a vector field such that 𝜕𝑓/𝜕𝑡|_𝑡=0=𝑉. If we apply the second variation of a biharmonic map in [4] to a biwave map, we can have the following.

Lemma 4.1. If 𝑓∶Ω→𝑁 is a biwave map from a compact domain into a Riemannian manifold, then 12𝑑²𝑑𝑡²𝐸₂(𝑓)|_𝑡=0=_Ω‖Δ𝑉+𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝑉)𝑑𝑓(𝑒_𝑖)‖²+𝑑𝑧_Ω<𝑉,(𝐷^′_{𝑑𝑓(𝑒𝑖)}𝑅^𝑁)(𝑓(𝑒_𝑖),𝜏_□(𝑓))𝑉+(𝐷^′_{𝜏□(𝑓)}𝑅^𝑁)(𝑑𝑓(𝑒_𝑖),𝑉)𝑑𝑓(𝑒_𝑖)+𝑅^𝑁(𝜏_□(𝑓),𝑉)𝜏_□(𝑓)+2𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝑉)𝐷_𝑒𝑖𝜏_□(𝑓)+2𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝐷_𝑒𝑖𝑉>𝑑𝑧, (4.1) where 𝑧=(𝑡,𝑥)∈𝑅×𝑅^𝑚,𝐷^′ is the Riemannian connection on 𝑇𝑁, and 𝑉 is the vector field along 𝑓.

Definition 4.2. Let 𝑓∶𝑅^𝑚,1→𝑁 be a biwave map. If (𝑑²/𝑑𝑡²)𝐸₂(𝑓)|_𝑡=0≥0, then 𝑓 is a stable biwave map.
If we consider a wave map, that is, 𝜏_□(𝑓)=0 as a biwave map, then by (4.1) we have (𝑑²/𝑑𝑡²)𝐸₂(𝑓)|_𝑡=0≥0 and 𝑓 is automatically stable.

Definition 4.3. Let 𝑓∶𝑅^𝑚,1→(𝑁,ℎ) be a smooth map from a Minkowski space into a Riemannian manifold (𝑁,ℎ). The stress energy is defined by 𝑆(𝑓)=𝑒(𝑓)𝑔−𝑓^∗ℎ, where 𝑒(𝑓)=(1/2)|𝑑𝑓|² is the energy function and 𝑔=(⁻¹⁰_0𝐼). The map 𝑓 satisfies the conservation   law if div𝑆(𝑓)=0.

Proposition 4.4. Let 𝑓∶𝑅^𝑚,1→(𝑁,ℎ) be a smooth map from a Minkowski space into a Riemannian manifold (𝑁,ℎ). Then div𝑆(𝑓)(𝑋)=−⟨𝜏_□(𝑓),𝑑𝑓(𝑋)⟩,𝑋∈𝑅^𝑚,1. (4.2)

Proof. Let 𝑥⁰=𝑡,𝑥¹,…,𝑥^𝑚 be the coordinates of a point in 𝑅^𝑚,1,𝑒₀=𝜕/𝜕𝑡,𝑒₁=(1,0,…,0),…,𝑒_𝑚=(0,0,…,1) and 𝑔=(⁻¹⁰_0𝐼), where 𝐼 is an 𝑚×𝑚 matrix. We calculate div𝑆(𝑓)(𝑋)=𝐷_𝑒𝑖𝑆(𝑓)(𝑒_𝑖,𝑋)=𝐷_𝑒𝑖(12|𝑑𝑓|²−100𝐼−𝑓^∗ℎ)(𝑒_𝑖,𝑋)=𝐷_𝑒𝑖(12|𝑑𝑓|²−100𝐼)(𝑒_𝑖,𝑋)−(𝐷_𝑒𝑖𝑓^∗ℎ)(𝑒_𝑖,𝑋)=(−(𝐷𝜕𝑓,𝜕𝑡𝜕𝑓𝜕𝑡)(−1))(𝑒₀,𝑋)+(𝐷𝜕𝑓𝜕𝑥_𝑖,𝜕𝑓𝜕𝑥_𝑖)𝐼(𝑒_𝑖,𝑋)−𝐷_𝑒𝑖(𝑓_∗𝑒_𝑖,𝑓_∗𝑋)=(𝐷𝜕𝑓,𝜕𝑡𝜕𝑓𝜕𝑡)(𝑒₀,𝑋)+(𝐷𝜕𝑓𝜕𝑥_𝑖,𝜕𝑓𝜕𝑥_𝑖)(𝑒_𝑖,𝑋)−(𝐷_𝑒𝑖𝑓_∗𝑒_𝑖,𝑓_∗𝑋)−(𝑓_∗𝑒_𝑖,𝐷_𝑒𝑖𝑓_∗𝑋)=((𝐷_𝑋𝑑𝑓)𝑒_𝑖,𝑓_∗𝑒_𝑖)−(𝜏_□(𝑓),𝑓_∗𝑋)−(𝑓_∗𝑒_𝑖,𝐷_𝑒𝑖𝑓_∗𝑋), (4.3) where the first term and the third term are canceled out and 𝐷_𝑒𝑖𝑓_∗𝑒_𝑖=𝜏_□(𝑓).

Theorem 4.5. Let Ω⊂𝑅^𝑚,1 be a compact domain and let (𝑁,ℎ) be a Riemannian manifold with constant sectional curvature 𝐾>0. If 𝑓∶Ω→𝑁 is a stable biwave map satisfying the conservation law, then 𝑓 is a wave map.

Proof. Because 𝑁 has constant sectional curvature, the second term of (4.1) disappears and (4.1) becomes 12𝑑²𝑑𝑡²𝐸₂(𝑓_𝑡)|_𝑡=0=_Ω‖Δ𝑉+𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝑉)𝑑𝑓(𝑒_𝑖)‖²+𝑑𝑧_Ω⟨𝑉,𝑅^𝑁(𝜏_□(𝑓),𝑉)𝜏_□(𝑓)+2𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝑉)𝐷_𝑒𝑖𝜏_□(𝑓)+2𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝐷_𝑒𝑖𝑉⟩𝑑𝑧. (4.4) In particular, let 𝑉=𝜏_□(𝑓). Recalling that 𝑓 is a biwave map and 𝑁 has constant sectional curvature 𝐾>0, (4.4) can be reduced to 12𝑑²𝑑𝑡²𝐸₂(𝑓)|_𝑡=0=4_Ω⟨𝑅^𝑁(𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓))𝐷_𝑒𝑖𝜏_□(𝑓),𝜏_□(𝑓)⟩𝑑𝑧=4𝐾_Ω[⟨𝑑𝑓(𝑒_𝑖),𝐷_𝑒𝑖𝜏_□(𝑓)⟩‖𝜏_□(𝑓)‖²−⟨𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓)⟩⟨𝜏_□(𝑓),𝐷_𝑒𝑖𝜏_□(𝑓)⟩]𝑑𝑧. (4.5) Since 𝑓 satisfies the conservation law, by Definition 4.3, Proposition 4.4, and (4.2) we have ⟨𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓)⟩=0,⟨𝑑𝑓(𝑒_𝑖),𝐷_𝑒𝑖𝜏_□(𝑓)⟩=−⟨𝐷_𝑒𝑖𝑑𝑓(𝑒_𝑖),𝜏_□(𝑓)⟩=−‖𝜏_□(𝑓)‖². (4.6) Substituting (4.6) into (4.5) and applying the stability of 𝑓, we get 12𝑑²𝑑𝑡²𝐸₂(𝑓_𝑡)|_𝑡=0=−4𝐾_Ω‖𝜏_□𝑓‖⁴𝑑𝑧≥0, (4.7) which implies that 𝜏_□(𝑓)=0, that is, 𝑓∶Ω→𝑁 is a wave map.

If we apply the Hessian of the bi-energy of a biharmonic map [4] to a biwave map 𝑓∶Ω→𝑆^𝑛+1(1), then we have the following.

Lemma 4.6. Let 𝑓∶Ω→𝑆^𝑛+1(1) be a biwave map. The Hessian of the bi-energy functional 𝐸₂ of 𝑓 is 𝐻(𝐸₂)_𝑓(𝑋,𝑌)=_Ω(𝐼_𝑓(𝑋),𝑌)𝑑𝑧, (4.8) where 𝐼_𝑓(𝑋)=Δ^𝑓(Δ^𝑓𝑋)+Δ^𝑓(trace(𝑋,𝑑𝑓⋅)𝑑𝑓⋅−|𝑑𝑓|²𝑋)+2(𝑑𝜏_□(𝑓),𝑑𝑓)𝑋+|𝜏_□(𝑓)|²𝑋−2trace(𝑋,𝑑𝜏_□(𝑓)⋅)𝑑𝑓−2trace(𝜏_□(𝑓),𝑑𝑋⋅)𝑑𝑓⋅−(𝜏_□(𝑓),𝑋)𝜏_□(𝑓)+trace(𝑑𝑓⋅,Δ^𝑓𝑋)𝑑𝑓⋅+trace(𝑑𝑓,trace(𝑋,𝑑𝑓⋅)𝑑𝑓⋅)𝑑𝑓⋅−2|𝑑𝑓|²trace(𝑑𝑓⋅,𝑋)𝑑𝑓⋅+2(𝑑𝑋,𝑑𝑓)𝜏_□(𝑓)−|𝑑𝑓|²Δ^𝑓𝑋+|𝑑𝑓|⁴𝑋, (4.9) for 𝑋,𝑌∈Γ(𝑓⁻¹𝑇𝑆^𝑛+1(1)).

Theorem 4.7. Let ℎ∶Ω→𝑆^𝑛(1/√2) be a wave map on a compact domain with constant energy and let 𝑖∶𝑆^𝑛(1/√2)→𝑆^𝑛+1(1) be an inclusion map. Then 𝑓=𝑖∘ℎ∶Ω→𝑆^𝑛+1(1) is an unstable biwave map.

Proof. We have the following identities from Theorem 3.5: |𝑑𝑓|²=2𝑒(ℎ),trace(𝜁,𝑑𝑓⋅)𝑑𝑓⋅=0,(𝑑𝜏_□(𝑓),𝑑𝑓)𝜁=−4(𝑒(ℎ))²𝜁,|𝜏_□(𝑓)|²=4(𝑒(ℎ))²,trace(𝜁,𝑑𝜏_□(𝑓))𝑑𝑓⋅=0,trace(𝜏(𝑓),𝑑𝜁⋅)𝑑𝑓=0,(𝜏_□(𝑓),𝜁)𝜏_□(𝑓)=4(𝑒(ℎ))²𝜁,trace(𝑑𝑓,Δ^𝑓𝜁)𝑑𝑓⋅=(Δ^𝑓𝜁)^𝑇,(𝑑𝜁,𝑑𝑓)𝜏_□(𝑓)=−4(𝑒(ℎ))²𝜁. (4.10) Then we obtain the following formula from Lemma 4.6 and the previous identities: (𝐼_𝑓(𝜁),𝜁)=_Ω(|Δ^𝑓𝜁|²−12𝑒(ℎ)²−4𝑒(ℎ)(Δ^𝑓𝜁,𝜁))𝑑𝑧, (4.11) which is strictly negative, where Δ^𝑓𝜁=2𝑒(ℎ)𝜁. Hence, 𝑓 is an unstable biwave map.

Acknowledgment

The author would like to appreciate Professor Jie Xiao and the referees for their comments.