A note on the evolution of the Whitney sphere along mean curvature flow

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.


Introduction
The Whitney sphere is the immersion F : S n → R 2n given by This immersion is Lagrangian, i.e., F * ω = 0, where ω is the standard symplectic form on R 2n . From the point of view of topology, the Whitney sphere is interesting since it has the best topological behavior, namely it fails to be embedded only at the north and south pole where it has a transversal double point. An well known result of Gromov asserts that there are no embedded Lagrangian spheres in C n . On the geometry side, this immersion can be characterized by many geometric rigidity properties, see [3,12]. In this sense, the Whitney sphere plays the role of totally umbilical hypersurfaces in R n in the class of Lagrangian submanifolds.
Another interesting aspect of the Whitney sphere is that it appears as a limit surface under Lagrangian mean curvature flow of some well-behaved Lagrangian submanifolds in R 4 . Recall that the mean curvature flow (MCF) of an immersion F 0 : M k → R m is a map F : M → [0, T ] → R m such that F (x, 0) = F 0 and satisfies the equation where H is the mean curvature vector of M n . It was shown by K.
Smoczyk that the Lagrangian condition is preserved by MCF when the ambient space is a Kähler-Einstein manifold. The Lagrangian mean curvature flow gained a lot of interest recently as a potential tool to find minimal Lagrangian (special Lagrangian) in a given homology class or Hamiltonian isotopy class of a Calabi-Yau manifold. Special Lagrangian submanifolds have the remarkable property of being area minimizing by means of calibration arguments. The classical approach of minimizing area in a given class, however, does not seem very effective to find smooth special Lagrangian as shown by Schoen and Wolfson in [13]. Ideally, one could hope that the evolution of well behaved Lagrangian submanifolds along mean curvature flow to converge to special Lagrangians. In a series of works, A. Neves showed that finite time singularities are unavoidable in the Lagrangian mean curvature flow in general, see [8,10]. It is constructed in [8] a non-compact zero Maslov class Lagrangian in R 4 with bounded Lagrangian angle and in the same Hamiltonian isotopy class of a Lagrangian plane that nevertheless develops a singularity in finite time. At the singular time the limit surface pictures like a connect sum of a smooth Lagrangian (diffeomorphic to a Lagrangian plane) with a Whitney Sphere. Such construction were later generalized to 4-dimensional Calabi-Yau manifolds, see [10].
There are very few results regarding the evolution of compact Lagrangian submanifolds in C n . Motivate by this, we investigate the evolution of the Whitney sphere along mean curvature flow. Despite its many geometric properties, it is not a self-similar solution of the flow. By exploiting its rotationally symmetries, one can reduce its mean curvature flow to a flow about curves in the plane. As a particular case of our main result we prove Let F : S n × [0, T ) → C n be the maximal existence mean curvature flow of the Whitney sphere. Then F T (x) = {0} for every x ∈ S n . The tangent flow at the origin is a Lagrangian plane with multiplicity two.
A Lagrangian submanifold L ⊂ C n is called equivariant if there exists a antipodal invariant curve γ : I → C such that L can be written as where G is a the standard embedding of S n−1 in R n . Using spherical coordinates on S n , (cos(u) G(x), sin(u)), we check that the Whitney sphere is equivariant with associated curve γ 0 : (0, 2π) → R 2 given by: , sin(u) cos(u) 1 + cos 2 (u) .
The equivariant property is preserved by the mean curvature flow and the corresponding evolution equation for γ t is Here − → k denotes the curvature vector of γ, it is defined by |γ | , and γ ⊥ denotes the normal projection of the position vector γ. This flow is known as the equivariant flow.   Remark 1.4. The assumptions in Theorem 1.3 are sharp. In Section 3 we construct for every α > π 2 a curve γ ∈ C and {γ} ⊂ Ω α that develops a non-trivial singularity along the flow (1.1) at the origin when n = 2.
The proof of Theorem 1.3 follows closely the ideas in [8,9] where it is shown that singularities for the mean curvature flow of monotone Lagrangian submanifolds in R 4 are modeled on area minimizing cones.

Preliminaries
Let L be a Lagrangian submanifold in C n . This implies that ω| L = 0, where ω = n i=1 √ −1 2 dz i ∧ dz i is the standard symplectic form on C n . Let Ω be the complex valued n-form given by A standard computation implies that The multivalued function θ is called the Lagrangian angle of L. If θ is a single valued function, then L is called zero-Maslov class. If θ = θ 0 , then L is calibrated by Re(e −iθ 0 Ω) and hence area-minimizing. In this case, L is called special Lagrangian. More generally, the Lagrangian angle and the geometry of L are related through − → H = J(∇θ). Recall also the Liouville one form given by for some c ∈ R, then L is said to be a monotone Lagrangian.
Let L be a equivariant Lagrangian submanifold in R 2n . Hence, there exists a regular curve γ in R 2 such that After choosing a parametrization of γ we have where z · w denotes the standard multiplication of complex numbers; here we consider γ as complex valued function. The Lagrangian angle relates to the geometry of L. If L t is the mean curvature flow starting at L, then L t shares the same rotational symmetries of L, i.e., L t = {γ t cos(α), γ t sin(α)), α ∈ R/2πZ}. Moreover, Although the term γ ⊥ |γ| 2 is not well defined at the origin the quantity has its meaning even when a curve goes through the origin as we can see below.
Proof. Let us write the left hand side as Using that lim s→0 γ(s) s = γ (0) and applying the L'Hopital's rule twice, we obtain (2) The curve γ i,t , i = 1, 2, solves the equation Then for all Proof. It suffices to restrict to what happens near the origin since the proposition follows from the standard maximum principle applied to the first time of tangential intersection. First notice that γ i,t can be written as a graph on [−δ, δ] for some Since f (0) = 0 and f (0) = 0 (item (1)), we can apply L'Hopital's rule to show that α and α in (2.5) have a limit when s → 0. Hence, α is twice differentiable.
Finally we consider the function u t (s) = α 1,t − α 2,t . Notice that u 0 > 0 by assumption (3) and u t (s) = u t (−s). Recall that in the case of a graph γ(s) = (s, f (s)) we have Besides, Now we proceed to find the equation for dut dt . Using that α i,t s is also smooth, one can checked that where each C k is a smooth and bounded function. By item (3), the function u t=0 is strictly positive since γ 1 and γ 2 have a non-tangential intersection at the origin. Suppose T 1 is the first time where u t has a zero say at s 0 . Hence, s 0 is a minimum point as u T 1 ≥ 0. We consider the function v t = u t e −Ct + ε(t − T 1 ) where C is very large and ε is a very small positive number. So at (s 0 , T 1 ) we have We used in the equality part that u T 1 (s ) = 0 and that u T 1 (s 0 ) = 0 and u T 1 (s ) ≤ 0 since s 0 is a minimum point for u T 1 . If s 0 = 0 then the second term in the right hand side is zero and we get a contradiction. If s 0 = 0 then that term is just u t (0)e −CT 1 by the L'Hopital's rule, hence, non-negative and we obtain a contradiction again. Proof. The symmetries of the curve γ are preserved by the equivariant flow, hence γ t is also antipodal invariant. Proposition 2.2 guarantees that the only self intersection of γ t is at the origin. Moreover, Proposition 2.2 also implies that γ t (s) non trivially intersect the line s − → v s∈R only in at most one pair of antipodal points for any t ∈ [0, T ). On the other hand, by Proposition 1.2 in [1], this intersection is never tangential unless it is trivial, i.e., is at the origin. Therefore, if γ ∈ Ω π 2 , then so is γ t . Finally, by Theorem 1.3 in [1], the number of intersections between γ and S 1 (R) is non-increasing along flow.
Lemma 2.4. If γ ∈ Ω π n , then for every t > 0 there exists δ t > 0 such that {γ t } ⊂ Ω π n −δt . Proof. Since γ ∈ C is antipodal invariant and passes through the origin, one can check that lim s→0 This implies that z 0 and −z 0 are critical points of the Lagrangian angle θ L . It can be check easily that they correspond to local minimum and local maximum critical points. The strong maximum principle applied to d dt θ = ∆θ implies that θ t (z 0 ) < θ(z 0 ) and θ t (−z 0 ) > θ(−z 0 ).
Let us use Area(γ) to denote the area enclosed by γ ∈ C. By the Stokes' theorem we have that Area(γ t ) = − 1 2 γt γ t , ν d γt , where ν is the unit outward normal vector of γ.
Proof. Let γ t (u) be a parametrization of γ t . Using that ν = i γ t |γ t | , we have that Area(γ t ) = − 1 2 γt γ t , i γ t du. Hence, The last equality follows from the Fundamental Theorem of Calculus. Hence, The last equality follows from the Divergence Theorem applied to vector field X = z |z| 2 and the fact that z = 0 is not in the interior of the region enclosed by γ t . Combining the Gauss-Bonnet theorem and the fact that the exterior angle α t of γ t at the origin is in [−π, π] we obtain The Lemma now follows if we integrate this quantity from t to T .

Proof of the Theorem
Let L t be a solution of the mean curvature flow starting on a kdimensional submanifold L in R m . Consider the backward heat kernel The following formula is known as the Huisken's monotonicity formula: where dH k denotes the k-dimensional Hausdorff measure.
Recall that if {L t } t∈[0,T ) is the Lagrangian mean curvature flow starting at L, then L σ s = σ (L T + s σ 2 − x 0 ), for s ∈ [−T λ 2 , 0), also satisfies the Lagrangian mean curvature flow and is referred as the tangent flow at x 0 . The following is a restatement of Theorem 1.3: where A( 1 η , η, 0) is an annulus centered at z = 0 with inner and outer radius η and 1 η , respectively. Proof. Let L i s be the immersed Lagrangian sphere in C 2 obtained via where H is the mean curvature vector of L i s . For the convenience of the reader let us recall the proof of this fact. It is a standard computation to check that the Lagrangian angle θ obeys the following evolution The last inequality follows from the scale invariance and monotonicity of Lt θ 2 ΦdH n . Similarly, we obtain It follows from the triangular inequality that This completes the proof of (3.2). As From previous lemma it follows that for almost every s ∈ (a, b) that This implies that γ i s converges to a union of lines in C To handle other connects components of γ i s in B 4R (0) we study the Lagrangian angle θ i s . Let β be a primitive of λ L . It is proved in [9] that ∇β = J(x ⊥ ) and d dt β = ∆β − 2θ. This implies that the function Integrating this formula from −1 to s 0 and using (3.2), we obtain Let γ i be a connected component of γ i s in B 4R (0) that intersects B R (0) and does not passes through the origin. Since |∇f (β i s )| is bounded, there exists a constant b s 0 such that lim i→∞ f (β i Note that (2.3) implies that θ i s converge to a constant in each connected component of γ i s ∩ (B R (0) − B r (0)). We claim that θ 1 = θ 2 . Otherwise, by choosing f with support near b s 0 and equal to 1 near b s 0 , we obtain contradiction.
Let us assume that γ satisfies item ii). In this case, γ i s ∩ B 4R (0) has a connected component γ i intersecting B 2R which converges in to the lines γ A and γ B with multiplicity one. Moreover, θ i s converge to a constant θ 0 on each connected component of (0)). This implies that γ A = γ B with the same orientation or the angle between γ A and γ B is π n . The first case cannot happen since I 2 (β i s , S 1 (0, r)) = 0, where I 2 (·, ·) is the intersection number mod 2. The second case cannot happen since {γ t } ⊂ Ω π n −δt by Lemma 2.4. Hence, the origin is not a singularity if we assume that γ T = {0}.
On the other hand, no singularities away from the origin occur. Indeed, in [11] J. Oaks complement the work of S. Angenent on singularities of equations of type d showing that near the singularity the curve γ t must lose a self intersection. Since Proposition 2.2 asserts the only self intersection of γ t is at the origin we are done. Now let us prove that the tangent flow at the singular point is a line through the origin with multiplicity 2. For this we choose a sequence of scale factors λ i → +∞ and we set γ i s = λ i γ T + s As discussed before γ i s converges in C 1, 1 2 loc (R 2 − {0}) to a union of two lines through the origin for almost every s fixed. Let us denote them by l A and l B . As Area(γ t ) is going to zero there exist a unique t i ∈ [0, T ) for which Area(γ t i ) = 1 In particular, if s * = − 1 3π , then lim sup i→∞ Area(γ i s * ) ≤ 1. Therefore, γ i s * must converge to 2γ A + 2γ B or γ A = γ B since γ i s * is becoming non-compact enclosing bounded area. The first case does not happen as it violates the assumptions i) and ii) as discussed above.
Let us construct equivariant Lagrangian spheres in R 4 that do not collapse to a point along the mean curvature flow. Example 3.3. Let γ 0 be the curve γ α (u) = sin( πu α ) − α π (cos(u), sin(u)) with u ∈ R. The existence of a solution of the equivariant flow starting at γ α is given in [8], let us denote it by {γ t } t∈[0,Tα) . It is shown in [8] that when α > π 2 , then T α < ∞ and γ t develops a singularity at the origin. When α ∈ (0, π), then γ α is contained in Ω α and it is asymptotic to its boundary. Consider the region U α in Ω α that is bounded by {γ α } ∪ {−γ α }. One can check that U α has infinite area. Choose β ∈ C contained in U α whose area enclosed, Area(β), is greater than 3π T α . See Figure 2 for the case α = π. Let {β t } t∈[0,T ) be the solution of the equivariant flow starting at β. By the avoidance principle, β t and γ t do not intersect. Hence, T < T α . On the other hand, by Lemma 2.5 we have that Area(β T ) ≥ Area(β) − 3πT ≥ 3π(T α − T ) > 0. Therefore, a non trivial singularity must occur at the origin. Let us show that any Type II dilation of γ t near the singularity converges to an eternal solution of curve shortening flow. As in Chapter 4 in [7], there exist for each k > 0, points z k ∈ γ t (S 1 ), t k ∈ [0, T − 1 k ], and scaling λ k > 0 such that β k s = λ k (γ T + s where s ∈ (a k , b k ). Moreover, lim k→∞ a k = −∞, lim k→∞ b k = ∞, and 0 < lim k→∞ sup (a k ,b k )×S 1 | − → k (β k s )| ≤ C. It is proved that β k s converge smoothly as k → ∞ to a non-compact flow (β s ) s∈R . We claim that lim k→∞ λ k z k = ∞. If not, then we could replace the points z k by z = 0 and obtain the same conclusion. This is impossible since central dilations converge to lines. Therefore, as k → ∞, d ds β s = − → k (β s ).
Acknowledgements. I would like to thank Jason Lotay for suggesting this problem and for his encouragement and support during this work.