Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space forms

In this paper, we study Hamiltonian stationary Lagrangian surfaces in complex space forms. We first show that when the mean curvature is a non-zero constant, the second fundamental form is parallel. We then consider the case in which the mean curvature is a non-constant harmonic function. Under the additional assumption that the Gaussian curvature is constant, we obtain a complete classification of such Lagrangian surfaces.

💡 Research Summary

The paper investigates Hamiltonian stationary Lagrangian surfaces immersed in complex space forms ˜M²(4ε), where ε∈{−1,0,1} denotes the sign of the holomorphic sectional curvature. A submanifold M²⊂˜M²(4ε) is Lagrangian if the Kähler form vanishes on tangent vectors, and it is Hamiltonian stationary when it is a critical point of the volume functional under all Hamiltonian variations. This condition is equivalent to the divergence‑free equation div(JH)=0, where H is the mean curvature vector and J the complex structure.

The authors first treat the case where the mean curvature magnitude |H| is a non‑zero constant. By choosing a local orthonormal frame {e₁,e₂} with Je₁ parallel to H, the second fundamental form h can be written in terms of three scalar functions a, b, c (equation (3.1)). Using the Codazzi equations, the Gauss equation, and the Hamiltonian stationary condition, they show that the connection 1‑form ω₂¹ vanishes, which forces the Gaussian curvature K to be zero. Consequently a, b, c are constant and the covariant derivative of h vanishes (∇̄h=0). This reproduces the known class of Lagrangian surfaces with parallel second fundamental form, already classified in earlier works (Chen, Dillen, van der Veken, etc.).

The main contribution concerns the more delicate situation where |H| is a non‑constant harmonic function (Δ|H|=0) while the Gaussian curvature K is assumed constant. Again a local frame is chosen so that Je₁ aligns with H, leading to the same expression (3.1) for h. The harmonicity of |H| translates into the Laplace equation a_{uu}+a_{vv}=0 for a=2|H|. Combining this with the constant‑K assumption yields K<0, and after a series of manipulations the authors obtain a representation a_u=√{−K} cos θ, a_v=√{−K} sin θ, where θ is a function on M. The Codazzi equations force θ to be constant, so a is linear in the local coordinates (equation (3.19)).

At this stage the analysis splits into several algebraic cases depending on whether the functions b and c vanish. When b≡0, two sub‑cases arise: (i.1) c≡0, which forces ε=K=−1 and yields a metric g=m²e^{2y}(du²+dy²) together with a second fundamental form h(∂_u,∂_u)=J∂_u, all other components zero. This surface is congruent to the explicit immersion (1) listed in Theorem 3.1. (i.2) a v≡0, leading again to ε=K=−1, a²=u², and after a linear change of variables the metric becomes g=2(x+y)²(dx²+dy²) with h(∂_x,∂_x)=J∂_x, h(∂_y,∂_y)=J∂_y. This corresponds to immersion (2) in Theorem 3.1.

The remaining case b≠0 is handled by a massive symbolic computation. The authors write down the full Codazzi system, differentiate it, and use a computer algebra system to enforce the compatibility condition c_{uv}=c_{vu}. This produces a high‑degree polynomial relation (3.44) whose vanishing forces the quantity δ=K−ε to be zero, i.e. K=ε=−1. Substituting back shows that the only solutions reduce to the previously obtained b≡0 families.

Thus the paper proves the following classification theorem (Theorem 3.1): if a Hamiltonian stationary Lagrangian surface in a complex space form has nowhere‑vanishing, non‑constant harmonic mean curvature and constant Gaussian curvature, then the ambient space must be the complex hyperbolic plane (ε=−1), the Gaussian curvature must be K=−1, and locally the surface is the Hopf lift Π∘ϕ of one of two explicit immersions ϕ: M→H₁⁵(−1)⊂ℂ^{3,1}. The immersions are given by

1. ϕ(x,y)= ( m e^{y}+e^{−y}+2i m² x e^{y²}, m e^{i x}+y, e^{−y}+2i m² x e^{y²} ),
2. ϕ(x,y)= ( (1−i(1+m²)) m² x+y, m√{1+m²} e^{i x} m² x+y, √{1+m²} e^{i y} m² x+y ), with a positive real parameter m. When m=1 these coincide with the examples previously constructed by Chen, Garay and Zhou.

In summary, the paper completely separates Hamiltonian stationary Lagrangian surfaces into two regimes: constant mean curvature surfaces (already classified) and non‑constant harmonic mean curvature surfaces with constant Gaussian curvature, which exist only in the complex hyperbolic setting and are explicitly described by the two families above. The work combines classical differential‑geometric techniques (Gauss, Codazzi, Ricci equations) with modern symbolic computation to resolve a highly nonlinear compatibility problem, thereby extending the classification of Hamiltonian stationary Lagrangian submanifolds in complex space forms.