AN ENERGY GAP PHENOMENON FOR THE WHITNEY SPHERE

For an immersed Lagrangian submanifold, let Ǎ be the Lagrangian trace-free second fundamental form. In this note we consider the equation ∇ ∗ T = 0 on Lagrangian surfaces immersed in C, where T = −2∇∗(Ǎyω), and we prove a gap theorem for the Whitney sphere as a solution to this equation.


Introduction
Gap phenomena are common in differential geometry hence are classical and prolific throughout the literature till now. Generally speaking, there are two categories of gap phenomena. One of them includes those rigidity theorems in submanifolds theory and we can take some examples from [1,6,8]. Another type occurs when we conduct the blow-up analysis for geometric flows, such as the well known Sacks-Uhlenbeck energy gap result in [12]for harmonic maps. The method to deal with the second type is usually evolved from PDE combined with related Bochner identity.
Let Ψ be any given differential operator that acts on immersions between two manifolds, consider a tensor field T on domain manifold. Then there exists a universal constant ǫ > 0 such that T = 0 if T L 2 ≤ ǫ and Ψ = 0.
The small energy condition here is quite natural in the sense of variation since they can be interpreted geometrically by stating that the deviation of the immersion from being simplest geometric objects such as planes and spheres is sufficiently small in an averaged sense. If we turn to the Lagrangian geometry case, we may consider the objects being the Lagrangian planes, the Clifford torus and the Whitney sphere. Luo and Wang have already considered the case when the second fundamental form is small and they obtained the following results: Theorem. Let l : Σ → C 2 be a properly immersed HW surface, then there exists ǫ 0 (n) > 0 such that if A L 2 < ǫ 0 (n), then it must be a Lagrangian plane.
Instead of consideringÅ in the Lagrangian frame, the Lagrangian trace-free second fundamental formǍ (see definition in section 2) seems to be a better candidate. As our personal interests, we introduce a (0,2)-tensor T := ∇(H ω)− 1 2 divJH ·g and consider the equation ∇ * T = 0. In the point of view of geometric, the tensor T measures the deviation of the mean curvature vector field to be a conformal vector field. We can check by a straightforward calculation that the equation ∇ * T = 0 bears the Whitney sphere as one of its solution, hence it's reasonable to consider if the Whitney sphere is its only solution givenǍ small enough. We obtain the following result: As a corollary we obtained the following gap results: We expect that there is a gap theorem for the Whitney sphere as the HW surface given the smallness ofǍ and our results can be used to provide some idea in proving it in the future. One of the difficulties here is the smallness pfǍ can't provide us any information on the mean curvature H which is much different as the standard Euclidean case. Hence instead of writing the Bochner type identity ofǍ in terms of the mean curvature itself, we write it in terms of the tensor T which also allows us utilizing the information of the equation better.
This note is organized as follows: in Section 2 we introduce some elementary notions on Lagrangian submanifolds as well as the Willmore functional. Section 3 is devoted to an curvature estimate for Lagrangian surfaces which is essential for us to get the main gap theorem.

Preliminary
In this section, we will introduce some elementary notions in the Lagrangian geometry and the Willmore functional. Let C 2 = R 4 be the 2-dimensional complex plane with standard metric ds 2 = dx 2 i +dy 2 i and the standard symplectic structure Definition 2.1. Let Σ be a surface in C 2 , with tangent and normal bundles, T Σ and N Σ, respectively. Then Σ is Lagrangian if and only if one of the following equivalent conditions holds: The simplest Lagrangian surfaces of C 2 are the totally geodesic ones, i.e. the Lagrangian subspaces or planes, another important family of such immersion are the Whitney spheres.
is a family of Lagrangian immersion. Here we embed S 2 into R 3 with center at the origin.The image of Φ in C 2 is called the Whitney sphere S W , and the constants r and − → C will be referred as the radius and the center receptively.
In view of topology, there is no embedded sphere in C 2 as a Lagrangian submanifold. Whitney spheres have the best possible behavior among them because their only non-embedding points are the poles. On the other hand their second fundamental forms also have some simple symmetric property: (1) V is called a Lagrangian vector field if the associated 1-form For immersed surfaces f : Σ → R n ,the Willmore functional is defined as whereÅ := A − 1 2 g ⊗ H denotes the trace-free part of the second fundamental form A = (D 2 f ) ⊥ and µ is the induced area measure from the target manifold R n by f . For any normal vector field φ, we recall the Laplace operator on the normal bundle as e i φ is the adjoint covariant derivative on the normal bundle. Without further notice, we will use ∇ to simplify ∇ ⊥ in the following. Then the Euler-Lagrange operator of (2.3) is by some calculations, where H = g ij A ij is the mean curvature vector field while Q(Å) given by the formula: where we use Einstein summation convention and g-orthonormal basis {e 1 , e 2 }. Now if we consider the Willmore functional for Lagrangian surfaces, we shall search the minimizer for (2.3) among all Lagrangian immersions from Σ to R n . Since we have two kinds of ways of variation by definition 2.3, there will be two kinds of minimizers which are called LW-surface and HW-surface respectively in Luo-Wang [9].
Proof. By definition ofǍ, and by Codazzi property of second fundamental form A and mean curvature H(orH ω) we have The next lemma is on the relationship of T and Willmore functional. Furthermore we have dual version in the sense of symplectic form ω: Proof. By (2.4) and (2.5), we have On the other hand, we use Ricci identity, Gauss equation and codazzi property to obtain: where K is the Gaussian curvature of the surface. Substitute with the above two formulas in (2.4) we complete the proof.

Proposition 3.2 (A Bochner type identity
). If f : Σ → C 2 is a properly immersed Lagrangian surface, then under local normal coordinates, Using the Ricci identity and the relationship between T and the mean curvature H we havě whereǍ imm,jk = 0 in the 4-th equality due to the fact thatǍ is trace-free, and we substituted h i,jk with (T ij,k + 1 2 δ ij h l,lk ) in the 6th equality to get the 7th. Then we apply the method of symmetrization to get (3.4) as desired.
3.2. Curvature estimates. Lemma 3.2. Assume f : Σ → C 2 is a properly immersed Lagrangian surface (compact or noncompact), cut-off function γ ∈ C 1 c (Σ) satisfies |∇γ| ≤ C 0 R , then we have: Proof. Multiplying (3.4) byǍ then by integrating we get For the L.H.S., just integrating by parts: now for the first term of R.H.S. we use definition of T and integrate by parts again to get: Hence we achieve by |T | ≤ c|∇Ǎ| and |∇γ| ≤ C 0 The following theorem from [10] allows us absorbing the highest order term of A above: Theorem (Michael-Simon Sobolev inequality). Let f : Σ → C 2 be an immersion and v be a non-negative C 1 c (U ) function on Σ, where U ⊆ C 2 is a domain contains f (Σ). Then where H is mean curvature vector and c is a constant independent of f .
Proof. Substitute v = |Ǎ| 2 γ in (3.7), we have: Combining the above two lemmas we get the following gradient estimate foř A: Theorem 3.1. Assume f : Σ → C 2 is a properly immersed Lagrangian surface (compact or noncompact), cut-off function γ ∈ C 1 c (Σ) satisfies |∇γ| ≤ C 0 R , then there exists ǫ 0 > 0 such that if As the first application we can deduce the Gap theorem for the equation ∇ * T = 0 with small Ǎ L 2 (Σ) for immersed Lagrangian surface. First we need a classification theorem by Castro-Urbano [4] or Ros-Urbano [11] as following: Theorem (Lagrangian umbilical surfaces, [4,11]). Let Ψ : M → C n be a Lagrangian immersion of an n-dimentional manifold M, theň Remark 3.1. If we only consider closed surfaces category, the gap phenomenon holds for Willmore equation as well. It's well known that Urbano and Castro proved in [5] that the Whitney sphere is the only Lagrangian sphere solution to W = 0 and they also classified Willmore Lagrangian tori. Now with the help of (3.1), Gauss equation and Gauss-Bonnet formula, we have: hence if Σ |Ǎ| 2 dµ is sufficiently small, the Whitney sphere is the only Willmore Lagrangian surface. Since the small energy condition implies that χ(Σ) to be non-negative, it's either a sphere or a torus. Applying the results of [7] it must be in the sphere category.